Average Rate of Change

Suppose you are planning to go on a road trip and want to estimate how much fuel your car will need so that it will not stop midway. Driving it everyday, you get a rough estimation over several days that the average distance your car covers in 1 litre of fuel is 15 kms. But how is this helpful? You can stretch this average rate to estimate if your car can go through the road trip in a whole fuel tank. What we have found out is the Average Rate of Change of distance relative to the fuel consumed. This is a fairly simple scenario, but it also has applications elsewhere, like calculating the average rate of change of electric current going through a circuit in unit time to work out the power and resistance in the circuit. 

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Contents
Table of contents

    We shall discuss the mathematical side of the above scenarios which can then be applied to a plethora of real-world problems.

    Average Rate of Change definition and formula

    The word ’average’ suggests that the rate of change we are looking for will be over a considerable interval. The reason why the average rate of change is useful is that it can be extended over a large interval to get fairly accurate results.

    The Average Rate of Change of a quantity relative to another is the measure of how much the quantity changes in a given interval per unit change of the other quantity.

    In a mathematical sense, the quantities are replaced by ‘functions’ but the narrative remains the same.

    Let y=fx{"x":[[173,173,173,173,174,176,177,178,181,184,187,193,195,203,206,212,215,221,227,232,235,242,244,249,250,253,254,254,254,254,254,254,254,254,254,255,255,255,254,250,249,242,240,231,223,219,207,202,191,184,179],[351,354,356,365,369,377,386,389,392,399,403,406,411,413,416],[353,350,348,345,344,344,345,347,353,356,365,368,372,379,384,388,397,401,408,414,418,420,424,426,427,429],[552,550,549,548,546,545,543,542,539,538,537,535,533,531,530,528,525,523,522,522,523,527,529,532,533,534,534,534,532,531,528,526,524,521,519,517,515],[500,500,500,499,500,501,507,509,512,520,524,529,533,541,546,553],[661,660,658,656,652,650,645,639,633,630,625,621,618,618,619,624,629,632,635,638,645],[683,683,682,680,680,681,682,688,693,701,705,717,721,728,735,736,738,739,739,738,734,732,726,724,718,717,713],[762,762,763,763,763,763,762,761,759,757,756,752,747,743,741,737,736,734,735,739,741,748,751,755,762],[817,818,819,821,823,831,835,838,844,849,854,856,857,858,858,855,852,851,844,841,834]],"y":[[260,261,266,281,287,303,309,312,319,324,326,330,330,329,328,324,322,316,309,301,297,285,281,269,266,259,257,260,265,269,279,289,302,323,330,351,357,374,380,394,398,410,413,420,423,423,424,423,418,413,407],[283,282,282,279,279,277,276,276,275,275,275,275,276,276,276],[319,320,321,321,321,320,320,320,319,318,316,316,315,313,313,312,310,310,308,308,308,308,308,308,308,309],[225,219,216,213,210,208,203,202,198,198,198,200,204,209,213,217,232,244,252,273,280,302,309,331,338,358,365,384,394,400,409,413,417,423,425,427,427],[328,326,325,325,324,324,323,323,323,322,321,320,319,318,317,316],[235,235,235,235,235,237,243,252,265,272,288,304,326,334,347,361,366,369,370,371,372],[294,293,292,289,288,284,282,278,276,273,272,271,271,272,275,277,281,284,289,291,300,303,311,314,320,322,325],[284,282,280,279,278,276,276,275,275,275,276,278,282,288,291,298,301,310,313,319,321,324,324,325,325],[232,231,229,228,228,230,232,235,241,249,258,268,273,292,297,315,327,332,347,352,361]],"t":[[0,4,14,29,33,43,49,49,57,66,78,83,98,99,111,116,116,126,134,142,149,159,167,177,184,198,200,225,232,242,249,259,266,277,284,298,301,311,316,327,334,342,349,360,366,374,386,397,405,413,416],[739,775,782,792,799,813,816,816,827,832,833,843,849,849,857],[1000,1007,1011,1019,1026,1034,1042,1049,1059,1066,1076,1083,1083,1093,1099,1100,1114,1116,1127,1133,1133,1144,1149,1150,1160,1166],[1522,1522,1526,1528,1533,1533,1544,1550,1560,1566,1577,1583,1594,1600,1600,1615,1621,1625,1634,1642,1650,1659,1667,1676,1684,1694,1699,1711,1717,1727,1733,1734,1744,1750,1750,1760,1764],[1930,1934,1944,1950,1960,1966,1976,1984,1984,1994,2000,2000,2010,2016,2017,2031],[2326,2328,2334,2345,2350,2350,2360,2368,2380,2383,2394,2400,2410,2422,2431,2433,2442,2450,2450,2460,2467],[2678,2684,2688,2693,2700,2708,2717,2731,2733,2742,2750,2760,2767,2775,2783,2793,2800,2801,2811,2817,2831,2836,2844,2850,2860,2867,2875],[3032,3036,3045,3050,3052,3058,3067,3076,3083,3084,3094,3100,3111,3117,3117,3130,3134,3145,3150,3161,3167,3177,3183,3184,3194],[3440,3445,3450,3461,3467,3478,3484,3484,3494,3501,3509,3517,3526,3534,3545,3551,3561,3567,3577,3584,3592]],"version":"2.0.0"} be a function that is defined over the closed interval a,b{"x":[[259,258,252,247,245,241,238,233,226,220,215,209,207,201,199,193,192,190,187,186,185,183,180,179,178,177,177,176,175,175,174,174,174,173,172,172,172,171,171,171,171,172,174,175,180,185,191,199,207,215,224,227,238,242,252,255,264,267,270,273,278,280,281],[336,333,331,331,329,328,326,324,323,319,317,311,309,301,299,296,292,290,288,287,284,283,283,283,285,287,291,295,299,302,310,313,322,324,332,337,339,342,346,349,350,351,355,358,361,363],[429,430,430,431,431,431,431,431,431,431],[509,509,509,509,508,507,506,504,503,503,502,502,502,501,501,502,504,506,508,511,515,521,524,528,537,542,547,548,551,552,552,550,547,542,537,535,532,529,524,521,519,515,513],[588,596,600,602,605,608,611,620,622,625,630,633,635,638,639,640,641,642,642,642,641,640,639,637,636,633,632,631,631,631,631,633,634,635,635,635,634,630,626,622,617,611,605,598]],"y":[[158,158,156,157,158,160,161,164,167,170,172,174,175,177,178,180,181,182,184,185,187,189,195,198,204,207,211,224,234,246,257,263,282,325,330,336,349,354,360,365,381,385,391,394,399,402,404,404,404,404,404,404,404,404,405,405,405,405,405,405,405,405,405],[310,302,298,296,290,285,281,277,275,270,270,269,270,277,281,285,293,298,303,308,317,325,331,333,335,336,336,335,333,331,324,322,315,313,311,312,314,319,325,331,334,336,341,344,346,347],[341,348,353,356,365,368,372,373,375,378],[222,223,225,235,245,256,266,276,285,293,300,304,307,311,313,317,318,316,314,311,306,300,299,297,296,297,299,301,306,308,314,318,322,326,329,331,332,333,335,335,335,333,329],[211,209,208,208,208,208,208,209,209,209,209,209,209,210,210,211,212,215,219,223,226,233,241,249,254,268,273,289,293,309,315,331,341,350,354,362,369,378,382,386,389,391,391,391]],"t":[[0,2,8,21,24,32,32,42,49,59,66,75,82,92,99,109,116,116,125,132,133,142,149,159,165,166,176,182,190,199,209,215,225,257,272,273,274,282,283,292,299,309,317,318,328,334,342,349,359,366,376,382,392,399,409,415,425,432,433,443,449,450,459],[752,760,767,774,783,792,799,809,816,825,833,843,850,858,866,866,875,882,883,891,899,909,916,918,928,933,943,951,961,966,975,984,995,1002,1009,1016,1027,1032,1042,1049,1050,1059,1066,1075,1082,1083],[1306,1315,1322,1324,1333,1343,1349,1350,1359,1366],[1630,1650,1658,1666,1676,1683,1694,1700,1710,1717,1727,1733,1736,1743,1750,1759,1776,1783,1794,1799,1811,1816,1825,1833,1845,1850,1859,1866,1877,1883,1895,1899,1912,1916,1929,1933,1933,1944,1950,1950,1959,1966,1975],[2303,2313,2318,2318,2325,2326,2333,2343,2349,2350,2360,2366,2367,2376,2383,2383,2393,2400,2408,2416,2417,2426,2434,2443,2450,2459,2467,2476,2484,2494,2501,2511,2516,2525,2533,2542,2550,2560,2567,2576,2584,2594,2600,2610]],"version":"2.0.0"} where a and b are real numbers. We want to know the rate of change of the function over that interval, the more rigorous definition of Average Rate of Change goes as follows:

    The Average Rate of Change of a function y=fx{"x":[[191,190,190,190,190,190,191,191,192,192,194,195,197,198,200,202,204,206,208,215,218,224,227,231,237,240,247,253,258,261,267,269,274,274,276,277,278,278,279,279,280,281,281,281,282,282,282,282,282,281,278,276,274,258,254,251,247,240,233,227],[372,373,374,377,379,386,399,404,418,423,437,441,445,452,455,462,468,469,471],[408,400,399,401,403,407,410,419,422,433,438,452,456,468,470,473,477,479,481],[593,597,597,598,598,598,597,595,594,593,592,588,584,580,576,573,570,569,569,569,568,568,568,567,565,564,562,558,556,551,549,547,543],[536,541,545,550,555,561,568,572,580,589,596,604,611,614],[670,671,670,668,663,661,657,652,645,642,641,641,642,650,656,659,668,675,683,686,689],[709,709,709,709,715,723,726,728,733,739,742,743,743,740,736,730,726,722,720,720],[770,771,772,772,771,767,763,756,754,749,747,749,751,753,760,765,775,783,797,801],[835,839,841,843,851,859,865,868,871,871,870,864,855,852,845,833,829]],"y":[[254,255,257,259,262,274,284,295,305,310,324,328,339,343,346,351,353,354,355,355,355,352,350,348,343,339,332,325,317,313,300,296,285,283,279,279,280,281,289,292,307,312,325,332,350,366,382,390,412,420,441,446,452,442,445,448,450,452,451,447],[292,292,292,292,291,290,288,288,288,288,288,288,289,290,290,292,295,296,299],[337,339,339,339,338,338,338,337,336,335,334,333,332,331,331,331,331,331,331],[246,231,228,225,220,217,216,215,215,216,218,222,230,240,253,266,281,298,323,340,357,365,382,398,412,419,425,443,447,460,463,465,468],[384,373,370,368,367,366,365,364,364,363,363,363,363,363],[271,262,260,261,266,268,275,284,303,318,341,347,359,376,381,384,389,390,391,391,390],[321,319,317,316,312,310,310,310,310,314,318,326,331,340,347,355,360,366,368,369],[322,319,316,315,315,317,321,330,333,344,355,362,364,366,369,370,370,369,364,362],[257,253,253,254,261,271,285,296,314,320,340,361,382,389,402,420,425]],"t":[[0,6,19,24,35,41,51,58,66,75,84,91,101,108,108,118,124,124,135,141,151,158,158,168,174,175,185,191,201,208,218,224,235,241,251,258,274,285,291,301,308,318,324,325,335,341,349,358,368,375,384,391,392,420,424,425,435,441,452,458],[773,780,784,791,799,813,818,826,834,841,850,858,858,867,874,884,891,901,908],[1040,1051,1058,1067,1075,1084,1092,1101,1109,1118,1126,1136,1141,1151,1158,1158,1168,1175,1179],[1407,1414,1418,1425,1433,1441,1442,1450,1458,1459,1468,1475,1485,1491,1501,1508,1518,1525,1535,1542,1551,1558,1568,1575,1584,1586,1592,1601,1608,1620,1625,1625,1637],[1773,1785,1792,1801,1808,1817,1825,1825,1835,1843,1851,1859,1872,1875],[2128,2139,2143,2161,2168,2175,2185,2192,2201,2209,2218,2231,2235,2251,2261,2270,2275,2285,2292,2302,2305],[2477,2480,2484,2486,2500,2517,2518,2527,2535,2542,2552,2569,2575,2585,2592,2609,2620,2625,2638,2646],[2772,2784,2788,2792,2802,2818,2825,2835,2842,2852,2870,2876,2884,2886,2892,2909,2918,2925,2942,2948],[3127,3135,3142,3150,3159,3169,3185,3192,3202,3212,3219,3239,3242,3253,3259,3268,3275]],"version":"2.0.0"} over an interval a,b is given as the ratio of change in the function over the interval to the change in the value of the endpoints.

    Translating the definition to an equation, the former part of the definition is ’the change in the function over the interval’ which becomes fb-fa{"x":[[184,179,176,174,171,169,165,161,158,152,149,147,145,145,145,145,145,146,147,148,149,149,149,149,148,147,146,145,143,138,135,131,129,125],[97,99,102,103,112,116,126,129,131,136,138,145,147,151,154,158,159,162,163,164],[253,244,240,237,235,234,232,231,229,228,227,227,228,231,235,237,245,249,252,258,261,269],[321,318,317,317,317,316,315,314,313,313,312,312,311,311,311,311,311,312,313,315,320,324,327,329,335,337,343,345,349,349,350,350,350,350,349,346,343,339,333,328,324,322],[417,422,423,426,430,432,433,434,436,437,437,436,434,429,427,425,420,418,412,410,405,404,403],[498,502,504,508,511,517,520,530,533,538,544,550,553,556,557],[679,681,681,681,681,678,676,674,670,668,661,659,654,652,648,647,645,644,644,644,644,645,646,647,647,647,647,647,644,643,639,637,631,629,627,622],[614,618,620,622,624,628,634,641,648,651,658,660,662],[759,759,758,757,756,754,752,746,743,738,732,726,721,717,714,713,714,716,720,726,728,730],[802,803,803,802,802,801,798,796,792,790,781,778,769,766,760,759,756,756,758,760,762,766,769,774,779,785,790,793,795,798,804,806,810,811,812,813,814,816,817,820,821,823,824],[865,868,869,871,875,876,881,882,884,884,884,883,882,879,877,875,873,869,864]],"y":[[130,123,120,119,118,118,118,120,123,131,137,146,157,170,177,200,207,224,233,242,265,273,289,303,317,330,335,341,350,360,364,366,366,365],[277,274,269,267,262,260,257,256,256,255,255,255,254,254,253,251,251,250,250,250],[149,149,153,159,163,168,179,185,191,204,217,232,253,267,279,283,295,298,301,304,305,306],[159,155,155,157,160,169,179,184,203,209,226,231,245,251,254,258,260,261,261,258,252,248,245,244,241,240,240,241,246,248,250,254,256,258,260,265,268,271,275,275,275,274],[155,157,158,163,171,175,179,184,193,202,212,222,233,250,255,260,271,275,287,290,298,300,302],[225,226,226,226,226,226,226,226,226,225,225,225,225,225,225],[163,153,151,146,145,140,139,138,137,137,137,138,143,145,154,158,168,178,185,199,212,227,243,256,263,270,284,290,307,313,328,332,343,345,346,349],[263,259,256,255,254,253,251,249,248,248,248,248,249],[175,173,171,170,169,169,170,178,182,192,205,220,233,247,260,277,282,290,297,304,305,305],[236,229,225,222,221,219,216,215,213,213,215,217,225,227,237,240,248,250,253,253,253,252,251,249,246,242,239,237,236,234,232,231,234,236,240,243,245,252,254,257,258,258,259],[182,176,177,180,188,192,206,211,227,232,247,253,258,268,273,279,284,293,302]],"t":[[0,7,12,13,18,28,34,44,51,61,69,77,85,94,101,111,118,126,128,134,144,151,161,168,178,184,185,194,201,211,219,228,234,244],[379,385,394,401,411,419,428,434,435,445,451,461,468,478,484,493,493,501,512,518],[875,886,893,901,901,911,918,918,928,934,944,951,961,969,980,985,998,1002,1005,1012,1018,1028],[1246,1256,1261,1269,1277,1285,1294,1302,1311,1319,1328,1335,1346,1353,1366,1368,1378,1385,1394,1401,1411,1419,1428,1436,1444,1452,1461,1469,1479,1485,1485,1496,1501,1502,1511,1518,1528,1535,1547,1552,1560,1569],[1771,1782,1785,1787,1794,1801,1802,1811,1818,1827,1835,1844,1852,1861,1870,1873,1881,1885,1897,1902,1913,1919,1923],[2217,2226,2230,2231,2235,2245,2252,2262,2268,2278,2285,2295,2303,2314,2319],[2596,2606,2610,2611,2619,2627,2635,2638,2645,2653,2662,2668,2678,2685,2695,2702,2713,2719,2719,2731,2736,2747,2752,2763,2769,2771,2781,2786,2797,2802,2815,2819,2831,2835,2836,2844],[3030,3042,3045,3047,3052,3054,3062,3070,3078,3086,3095,3102,3112],[3397,3403,3412,3419,3419,3429,3435,3446,3452,3462,3469,3479,3485,3495,3502,3512,3519,3527,3535,3545,3553,3557],[3763,3773,3777,3786,3786,3794,3802,3803,3812,3820,3829,3837,3846,3852,3862,3869,3878,3886,3895,3902,3914,3919,3919,3932,3936,3946,3954,3961,3963,3970,3979,3986,3996,4004,4011,4011,4019,4029,4036,4045,4046,4052,4053],[4226,4236,4244,4252,4263,4270,4279,4287,4299,4303,4313,4320,4321,4328,4336,4337,4346,4352,4368]],"version":"2.0.0"} and the latter part is ‘the change in the value of the endpoints’ which translates to b-a{"x":[[260,257,256,256,255,255,253,252,250,249,248,245,243,242,240,238,237,236,228,228,227,227,227,228,230,231,234,236,238,243,249,254,260,265,269,273,276,278,279,280,280,278,274,269,263,253,249,242,236,229,223,220],[380,379,380,381,384,390,404,409,423,428,441,445,456,460,469,472,475,478,480,481],[586,590,591,592,594,595,597,598,599,599,599,599,599,598,598,595,593,586,583,574,571,561,558,549,547,545,541,540,540,540,540,542,545,547,550,555,563,570,577,580,588,590,593,594,595,595,595,595,595,595,595,595,596,599,607,615,625,631,638]],"y":[[153,155,156,158,161,164,181,188,204,212,220,244,254,262,279,288,296,305,345,349,352,355,357,356,354,352,347,344,341,335,331,327,324,324,324,327,332,337,343,349,355,362,368,373,378,382,382,381,377,372,365,360],[309,309,309,309,309,309,308,307,307,307,307,306,306,306,305,305,305,305,305,305],[300,299,298,298,296,295,292,290,288,286,283,281,280,276,274,272,271,269,269,270,272,280,284,297,302,307,317,322,325,329,334,336,337,337,336,334,327,320,314,310,302,300,299,299,303,305,308,315,323,327,336,341,344,348,349,347,343,341,338]],"t":[[0,5,9,15,17,23,34,40,48,51,57,67,73,74,84,90,91,101,128,132,132,140,150,157,167,175,182,184,191,198,207,217,223,234,240,250,257,267,273,284,290,300,307,317,323,334,342,351,357,367,373,378],[718,729,740,742,750,757,766,775,783,792,800,808,817,826,833,840,850,857,867,874],[1114,1119,1125,1127,1134,1140,1150,1157,1157,1167,1174,1174,1184,1191,1191,1199,1208,1217,1225,1237,1241,1250,1257,1271,1274,1274,1282,1290,1291,1300,1307,1321,1324,1324,1335,1341,1353,1358,1369,1374,1387,1391,1400,1407,1419,1424,1424,1438,1441,1452,1457,1458,1467,1474,1484,1492,1506,1508,1512]],"version":"2.0.0"}. Now the average rate of change of a function is the ratio of these, which gives us:

    ΔyΔx=fb-fab-a{"x":[[186,185,183,180,176,174,167,165,159,156,154,149,144,141,136,134,133,129,128,126,124,124,122,121,120,119,119,119,119,119,120,121,122,124,126,131,132,134,137,141,146,152,160,167,174,178,190,194,207,211,224,228,236,242,248,252,255,257,258,259,259,259,259,257,255,251,250,244,242,236,233,229,223,218,214,209,204,200,196,192,188,185,182,179,177,175],[308,309,310,311,312,314,314,315,317,319,321,323,326,329,332,336,338,340,343,346,350,351,353,355,357,358,358,357,356,355,354,353,352,351,350,350,349,349,349,348,348,347,346,344,339,337,332,326,318,310,301,293,281],[135,150,154,164,170,184,199,216,235,263,273,293,313,333,351,369,387,395,402,416,422,428,437,441,445,450,454,454],[319,318,318,318,320,321,323,326,331,335,341,344,345,346,346,346,344,341,336,334,331,326,320,316,313,311,310,308],[362,363,364,364,364,365,366,366,365,362,359,352,350,343,341,339,337,334,333,332,334,337,344,349,356,360,372,376,389,393,402,405],[182,181,180,175,172,162,156,150,148,146,144,141,139,137,133,133,131,131,131,133,135,136,143,145,153,157,168,171,175,184,187,191,198,202,209,219,222,228,234,238,244,247,249,250,251,250,249,248,247,245,244,243,242,239,236,233,228,224,220,217,210,208,200,197,190,188,181,177,173,171,169],[547,549,550,551,553,559,562,572,576,585,588,595,600,602,605,609,612,615],[556,556,558,559,568,572,584,588,600,603,611,613,617],[726,726,726,726,726,725,725,723,722,720,719,714,713,708,707,704,703,701,701,700,700,700,701,702,702,703,704,705,706,706,706,705,704,701,699,694,691],[660,662,663,664,668,672,675,677,683,689,691,697,702,706,710,712,714,715],[775,775,775,773,771,767,765,760,754,749,744,741,739,738,738,738,739,743,744,751],[813,814,814,814,814,813,813,812,811,810,810,809,809,808,808,808,809,810,811,812,814,815,820,822,827,829,834,836,840,841,842,842,842,841,839,835,831,827,822,818,816,810,809,808],[865,864,863,862,862,862,863,865,869,872,874,875,876,876,875,874,873,871,868,866,865],[915,917,918,925,927,933,939,944,951,952],[1006,1008,1009,1009,1009,1007,1006,1004,1003,999,998,994,992,988,986,985,985,986,987,990,992,994,995,998,999,1000,1001,1001,1001],[978,978,978,979,982,984,991,994,1003,1006,1008,1013],[1056,1055,1054,1052,1051,1048,1045,1042,1040,1035,1034,1032,1032,1033,1033,1034,1037,1039,1040],[1089,1089,1089,1088,1086,1085,1083,1081,1076,1074,1069,1067,1064,1064,1063,1063,1064,1066,1068,1072,1075,1079,1083,1086,1089,1093,1095,1097,1099,1100,1100,1100],[1112,1117,1118,1119,1120,1123,1127,1128,1130,1131,1131,1131,1129,1126,1121,1116,1110],[742,736,735,734,736,742,750,756,776,794,813,831,851,862,874,895,907,919,931,965,987,1006,1023,1037,1048,1052],[812,812,812,812,812,812,811,811,811,810,809,808,807,806,805,809,814,816,822,825,827,831,833,835,836,838,838,839,838,837,834,830,826,822,819,812,810,805,804,804,806,807],[898,895,893,891,888,887,886,886,888,890,896,898,907,909,919,923,931,936,940],[997,999,999,1000,1001,1001,1002,1001,999,997,994,991,989,984,982,978,976,974,973,973,974,975,976,980,981,984,986,988,994,996,1000,1001,1005,1007,1009,1010,1013,1014,1018,1020,1022,1023,1028,1029,1031]],"y":[[112,112,115,120,127,131,143,148,160,165,170,180,190,195,205,209,213,220,224,227,232,235,239,242,245,246,249,250,252,253,253,253,253,253,253,251,250,249,248,247,245,245,244,244,244,244,242,242,239,238,235,234,232,230,229,228,227,226,225,223,221,220,219,216,213,207,205,198,195,187,184,179,173,166,159,152,145,139,133,128,124,119,115,112,109,106],[149,152,155,160,167,176,180,185,189,193,195,196,197,197,195,193,191,190,185,181,176,173,171,166,161,158,156,155,154,154,155,158,162,167,174,181,191,202,214,227,232,238,249,260,275,280,289,296,303,308,311,311,309],[317,316,315,315,315,314,312,310,308,306,305,304,303,302,301,300,300,300,299,299,299,299,299,299,299,300,302,303],[422,422,421,420,419,418,418,417,417,418,422,427,430,436,445,448,453,458,463,465,468,472,476,478,479,480,480,478],[430,430,429,428,427,426,426,425,425,426,428,433,435,442,445,448,451,458,464,469,473,476,479,479,479,478,475,473,468,466,461,459],[409,409,410,417,421,438,449,459,463,467,471,479,483,487,497,501,506,509,512,515,517,517,518,518,517,517,515,514,514,513,513,512,512,512,512,511,511,511,510,510,509,509,509,509,509,508,506,505,504,501,499,497,495,491,486,482,473,467,460,457,446,443,434,431,425,423,417,414,412,411,410],[274,274,274,274,274,273,273,272,272,271,271,271,271,271,271,271,272,272],[310,311,311,311,311,310,307,306,303,302,300,300,299],[155,150,147,146,144,142,141,139,138,137,136,135,135,135,136,138,140,147,150,163,168,173,179,191,197,203,213,227,235,241,243,251,253,258,259,261,261],[210,207,205,203,200,198,197,195,193,191,190,189,188,188,187,187,187,188],[154,151,150,148,148,148,149,152,158,165,174,183,193,202,206,219,222,230,232,235],[160,156,154,153,152,154,157,162,168,175,183,188,196,204,210,212,215,217,217,218,217,217,214,213,210,209,207,207,208,209,210,212,213,215,217,221,223,225,226,227,227,227,226,225],[156,156,156,157,158,164,167,174,184,191,199,202,211,214,224,227,229,235,239,242,243],[205,205,205,204,203,202,202,201,200,200],[164,158,155,151,150,148,146,145,145,145,146,151,154,165,173,183,193,203,208,218,225,228,231,237,239,241,245,247,249],[219,218,217,216,215,215,213,213,211,210,209,208],[171,167,165,165,164,165,166,170,172,181,184,195,199,206,208,210,212,213,214],[193,191,190,189,188,188,187,187,189,190,195,196,202,203,205,207,208,208,208,206,205,202,200,198,197,196,196,197,202,207,210,212],[169,169,168,169,170,173,180,183,189,192,196,207,214,221,228,234,240],[316,315,314,314,313,311,310,309,307,305,305,304,304,304,303,303,302,302,301,300,299,299,299,299,299,299],[338,337,336,337,340,345,353,367,375,381,384,388,391,393,393,388,383,382,378,377,377,377,377,378,379,382,384,386,391,392,397,400,402,404,406,407,408,407,404,401,396,394],[376,378,378,378,378,378,378,377,376,376,375,375,374,374,374,374,375,375,375],[378,378,377,376,374,373,370,366,364,362,361,360,360,363,365,371,374,379,384,386,390,393,393,394,393,391,390,389,384,383,380,379,377,377,378,379,384,386,391,393,395,395,395,394,393]],"t":[[0,36,46,53,64,73,79,86,96,103,103,113,120,130,136,137,146,153,153,163,170,170,180,186,196,203,213,220,230,246,253,263,270,279,288,300,303,303,315,320,330,336,346,354,363,371,379,388,396,404,413,420,429,436,446,453,463,470,479,487,496,498,505,513,520,531,539,546,556,563,570,583,588,598,603,614,623,630,636,646,653,663,670,680,687,697],[1120,1130,1137,1146,1153,1163,1170,1180,1187,1199,1203,1217,1220,1229,1237,1246,1249,1255,1263,1270,1279,1283,1290,1301,1303,1313,1320,1330,1337,1347,1353,1363,1370,1380,1387,1397,1403,1413,1420,1429,1433,1438,1447,1453,1463,1470,1480,1487,1496,1503,1512,1520,1530],[1965,1975,1979,1980,1987,1996,2004,2013,2020,2030,2039,2047,2054,2064,2070,2080,2087,2087,2098,2104,2104,2116,2120,2121,2131,2137,2147,2156],[3990,4001,4021,4030,4038,4038,4048,4054,4065,4071,4081,4091,4097,4105,4114,4123,4131,4138,4147,4150,4154,4165,4171,4180,4188,4188,4198,4204],[4344,4349,4356,4364,4367,4373,4382,4389,4398,4414,4421,4431,4439,4448,4454,4455,4465,4471,4481,4488,4498,4504,4515,4523,4532,4539,4548,4556,4569,4572,4582,4588],[6164,6197,6205,6215,6222,6237,6239,6251,6255,6256,6264,6272,6272,6280,6289,6303,6305,6306,6315,6323,6332,6339,6349,6356,6365,6372,6382,6389,6389,6399,6405,6406,6416,6422,6432,6439,6449,6455,6466,6472,6482,6490,6499,6506,6522,6539,6548,6555,6557,6566,6572,6574,6583,6589,6599,6605,6616,6623,6632,6640,6649,6657,6665,6673,6682,6690,6698,6705,6715,6722,6732],[7530,7540,7548,7548,7556,7566,7573,7583,7589,7599,7606,7617,7622,7623,7636,7639,7649,7656],[7814,7824,7831,7839,7849,7857,7866,7874,7883,7889,7899,7906,7916],[10764,10775,10780,10784,10790,10792,10801,10807,10808,10818,10824,10834,10841,10850,10859,10867,10874,10884,10890,10901,10907,10907,10918,10924,10924,10935,10940,10951,10958,10970,10974,10984,10990,11003,11007,11018,11025],[11157,11161,11167,11169,11174,11184,11190,11191,11201,11207,11218,11224,11234,11240,11251,11258,11267,11284],[11639,11650,11654,11658,11667,11674,11674,11685,11691,11701,11708,11718,11724,11734,11741,11751,11758,11768,11776,11786],[15603,15615,15618,15619,15626,15650,15659,15669,15675,15686,15692,15703,15709,15719,15725,15726,15737,15742,15753,15759,15769,15776,15786,15793,15805,15809,15819,15825,15837,15842,15854,15859,15859,15867,15876,15886,15893,15903,15910,15922,15926,15936,15942,15946],[16116,16120,16126,16142,16151,16159,16170,16176,16186,16193,16204,16209,16220,16226,16236,16242,16243,16256,16260,16270,16276],[16562,16592,16601,16609,16619,16626,16637,16642,16653,16659],[16903,16914,16918,16926,16926,16937,16943,16953,16960,16970,16976,16987,16993,17003,17009,17020,17026,17034,17043,17053,17059,17060,17071,17076,17076,17088,17093,17093,17104],[17237,17247,17252,17268,17285,17293,17303,17310,17320,17326,17326,17341],[17654,17664,17668,17676,17679,17687,17693,17703,17711,17720,17726,17737,17743,17754,17759,17760,17772,17776,17777],[17941,17952,17960,17970,17976,17977,17988,17993,18004,18010,18021,18026,18037,18043,18043,18055,18060,18070,18076,18087,18093,18104,18110,18122,18127,18137,18143,18155,18176,18185,18193,18201],[18348,18349,18353,18354,18360,18370,18376,18387,18393,18393,18405,18410,18421,18427,18438,18443,18451],[18876,18886,18890,18894,18903,18911,18921,18927,18938,18943,18955,18960,18971,18977,18977,18985,18993,18994,19004,19011,19021,19027,19038,19043,19055,19060],[19358,19360,19365,19385,19394,19404,19411,19427,19437,19444,19444,19455,19460,19471,19478,19510,19522,19527,19538,19544,19544,19556,19560,19561,19572,19577,19578,19589,19594,19602,19610,19621,19627,19638,19644,19655,19660,19672,19677,19688,19694,19702],[19876,19887,19890,19895,19904,19912,19919,19927,19937,19944,19955,19960,19972,19977,19988,19994,20005,20010,20022],[20301,20312,20315,20320,20328,20338,20346,20357,20362,20372,20377,20389,20394,20405,20411,20422,20428,20439,20444,20447,20456,20461,20472,20477,20489,20494,20494,20506,20511,20522,20527,20528,20540,20544,20555,20561,20572,20577,20586,20594,20605,20610,20622,20627,20628]],"version":"2.0.0"}

    Which is the equation for finding the average rate of change of y w.r.t (abbreviation of ‘with respect to’) x over the interval a,b{"x":[[293,294,294,294,294,292,291,289,283,278,274,269,266,265,261,260,259,258,258,257,257,257,258,258,259,260,261,262,264,265,266,267,268,269,270,273,274,278,280,286,289,292,295,300,303,305,311,313],[396,392,390,389,387,382,380,372,370,361,356,351,346,345,344,344,344,344,345,348,351,355,360,365,367,374,377,385,387,389,393,394,397,399,401,404,407,408],[474,475,476,476,476,475,475,474,473,473],[554,556,558,558,559,559,559,559,558,556,555,554,552,551,549,547,544,542,542,541,541,543,544,546,549,551,553,558,560,565,569,571,573,574,575,575,574,572,570,564,561,553,550,543,540],[629,630,632,634,636,640,642,649,651,658,661,666,668,669,672,673,674,674,675,676,676,675,673,669,666,664,662,660,658,656,655,655,655,655,655,655,653,651,650,645,642,640,633,629,621,617]],"y":[[284,281,279,277,276,275,275,276,279,281,284,287,290,291,294,295,296,298,300,307,310,317,326,330,335,345,350,360,370,381,390,399,404,415,419,429,432,439,441,446,447,448,449,450,450,450,450,450],[354,347,345,343,341,338,337,338,339,348,355,364,373,376,381,383,385,386,386,386,385,382,378,373,371,365,363,359,359,359,363,365,373,376,381,385,388,389],[383,384,386,390,394,398,403,405,411,413],[298,294,290,289,288,290,292,294,297,305,309,313,323,328,333,343,356,363,368,370,374,375,374,374,372,371,370,369,368,368,369,370,371,374,375,378,381,383,385,387,387,388,388,385,381],[269,268,267,266,266,266,266,265,265,265,265,265,265,265,265,266,266,267,269,272,277,279,284,294,301,310,314,324,334,349,353,365,369,381,389,392,400,403,406,412,415,417,421,422,425,425]],"t":[[0,13,16,20,21,36,46,53,63,70,79,86,96,105,115,120,120,130,137,146,153,163,170,170,180,186,187,196,203,215,220,231,237,247,253,265,270,280,286,296,303,304,315,320,320,333,337,337],[664,675,678,682,687,696,703,713,720,730,737,746,753,763,770,770,780,787,787,799,804,815,820,832,837,848,855,863,870,870,879,887,896,904,912,920,932,934],[1221,1225,1245,1254,1263,1271,1280,1287,1297,1305],[1601,1613,1617,1620,1630,1647,1654,1654,1664,1670,1671,1680,1687,1687,1696,1704,1713,1721,1733,1737,1749,1766,1770,1771,1780,1787,1787,1797,1804,1814,1820,1821,1830,1837,1837,1850,1854,1866,1870,1884,1887,1897,1905,1913,1920],[2140,2147,2154,2164,2171,2180,2187,2197,2205,2214,2220,2230,2237,2238,2247,2254,2254,2264,2271,2281,2287,2297,2304,2313,2322,2330,2337,2351,2354,2364,2371,2383,2388,2399,2404,2417,2421,2421,2433,2437,2438,2448,2454,2455,2466,2471]],"version":"2.0.0"}. We can also find a similar expression for the average rate of change of x w.r.t. y over the same interval, which will be just the reciprocal of what we got earlier:

    ΔxΔy=b-afb-fa{"x":[[302,311,313,316,319,320,321,322,322,321,320,314,302,299,291,289,285],[352,350,348,347,345,340,334,330,329,332,333,340,345,352,367,375,384],[188,195,202,213,225,239,254,261,277,302,309,325,334,349,364,378,385,399,403,409],[143,148,150,154,156,161,166,171,176,181,183,191,193,201,203,211,213,215,217,221,222,224,227,228,229,230,230,230,231,231,232,233,234,237,238,243,245,250,252,256,259,260,261,262,262,261,257,253,250,240,236,222,217,213,202,198,184,179,170,166,162,154,149,146],[286,285,285,284,284,284,284,284,284,286,287,290,293,297,299,303,307,310,317,322,324,330,331,335,336,337,337,337,335,332,331,327,324,323,320,318,316,315,312,311,308,302,300,299,294,292,290],[466,472,477,479,489,498,506,508,513,515,522,524,527,528],[473,476,482,488,491,500,503,512,514,519,523],[705,705,705,705,704,703,701,700,699,698,698,697,697,697,697,698,699,700,702,704,707,711,713,716,721,723,725,732,735,736,737,737,735,734,729,727,724,720,717,715,710,706,704,698,696,695],[802,806,809,811,816,821,824,827,834,837,839,847,852,854,856,857],[944,947,949,950,951,951,951,950,949,947,944,940,938,931,925,918,917,915,915,916,917,921,922,928,930,934,941,944,951,953,956,958,961,963,963,964,966,967,970,973],[647,653,657,660,665,675,680,691,704,718,735,752,762,793,804,814,838,850,863,887,899,912,937,947,969,988,1005,1019,1031,1036,1043,1046],[623,618,617,615,614,613,611,609,605,603,601,599,598,597,597,597,597,598,598,599,600,600,600,600,598,596,594,593,588,586,582],[570,571,576,578,581,588,595,602,606,610,614,620,622,625,630,632,636,639,640],[679,676,675,673,671,670,666,664,661,659,658,657,658,659,663,665,671,674,677,683],[721,721,722,722,720,719,718,717,714,713,712,711,711,711,712,713,715,717,721,724,728,731,734,737,739,739,740,740,738,737,732,728,723,719,718,716],[780,787,788,792,794,797,797,798,798,797,796,795,792,789,785,784,782,781,777,775,774],[864,864,865,866,867,869,873,876,879,883,888,892,894,896,898],[954,952,951,950,947,946,943,942,940,940,940,940,940,940,941,942,944,946,947,947,947,945,943,942,938,936],[931,935,937,940,942,945,949,953,955,958,960],[1019,1015,1013,1011,1007,1004,999,997,994,994,993,993,993,996,997,1000,1003,1005,1009],[1041,1039,1038,1037,1034,1031,1029,1027,1026,1023,1022,1020,1019,1019,1019,1022,1024,1027,1029,1032,1035,1036,1041,1042,1044,1046,1048,1050,1051,1052,1052,1053],[1072,1075,1078,1080,1083,1085,1086,1087,1087,1087,1087,1086,1086,1085,1083,1081,1080,1079],[132,132,132,133,133,134,135,136,138,139,141,144,147,152,155,160,166,169,173,176,181,184,191,194,201,203,206,210,212,215,218,219,221,223,224,225,226,226,226,226,226,226,226,227,227,227,227,228,229,230,230,231,234,235,237,237,238,240,241,243,244,247,247,249,249,251,251,251,251,252,252,252,252,252,252,252,252,252,251,250,249,247,246,244,241,239,236,231,224,221,213,209,205,193,185,182,171,166,161,158,153,151]],"y":[[177,180,181,183,187,189,192,196,199,204,207,216,227,230,236,238,240],[187,188,188,188,190,196,207,221,229,235,236,239,240,240,239,238,237],[295,295,294,293,293,292,291,291,290,288,288,287,286,285,284,284,284,285,285,287],[475,466,463,457,453,445,436,427,418,409,405,394,390,379,376,367,365,362,360,357,356,356,356,357,360,362,365,371,374,382,389,396,400,410,413,422,425,433,436,442,446,449,450,453,455,456,458,461,462,466,467,472,474,476,480,481,485,486,487,487,487,486,485,484],[396,396,395,395,399,402,411,414,425,430,432,434,435,435,435,433,431,429,423,419,417,409,407,402,401,399,398,401,405,415,418,431,440,445,455,460,468,472,482,484,491,497,497,497,497,495,494],[259,258,257,257,257,257,258,259,259,260,260,261,261,261],[298,297,296,295,294,293,292,292,291,291,290],[145,144,145,154,159,174,186,195,203,210,213,218,224,225,227,227,228,227,226,225,222,219,217,216,214,213,212,212,213,214,217,219,223,224,228,229,230,231,232,232,232,232,231,226,224,222],[205,204,204,204,204,204,204,204,204,204,204,204,203,203,203,203],[207,206,202,200,197,194,191,188,186,185,184,185,186,191,198,207,209,216,217,220,220,221,221,220,219,218,215,215,213,213,213,214,216,219,221,225,229,230,232,233],[308,305,305,304,304,303,303,303,304,304,305,305,305,305,305,305,305,305,305,304,304,304,303,303,303,303,304,305,307,307,309,310],[377,374,373,373,373,374,376,378,384,389,396,405,414,424,429,445,450,466,471,485,489,494,502,505,512,519,522,524,529,531,534],[463,462,460,459,459,458,458,457,457,457,456,455,455,454,453,452,452,451,450],[398,397,397,400,402,405,413,417,426,435,443,450,456,458,464,465,467,467,467,465],[389,391,394,399,407,411,416,419,429,431,437,438,439,441,441,441,441,440,438,436,433,433,433,433,435,436,440,441,445,447,451,453,454,455,455,455],[388,390,391,394,397,406,409,418,422,431,434,436,442,447,451,453,455,456,458,460,460],[421,422,422,422,422,422,421,421,421,420,419,418,417,416,415],[361,357,356,356,359,360,367,370,380,384,389,398,402,406,415,419,426,438,441,452,458,463,468,471,476,477],[439,436,434,433,432,431,431,430,430,430,430],[385,387,389,391,398,402,414,418,428,431,433,438,440,444,445,446,447,447,447],[417,417,415,415,415,415,417,418,419,421,422,424,427,428,430,430,430,429,428,427,427,426,426,426,426,426,427,430,432,435,436,437],[396,395,394,394,396,401,406,414,418,429,433,443,446,449,456,459,461,464],[256,255,254,254,255,255,255,255,253,252,250,245,239,231,228,220,211,207,199,195,187,183,173,169,159,157,154,149,146,142,138,137,135,132,132,131,131,132,134,136,137,141,143,148,149,151,155,160,166,168,172,175,185,188,194,197,200,205,208,213,216,224,226,232,234,240,242,244,247,249,250,251,255,256,257,258,259,258,257,255,254,253,253,252,252,252,252,252,252,252,253,254,255,257,259,260,260,260,259,259,258,258]],"t":[[0,10,14,18,25,26,35,42,42,51,60,68,85,91,101,108,120],[343,350,359,367,375,385,401,418,435,442,451,458,468,475,492,508,520],[843,856,860,861,867,875,883,892,902,908,918,925,925,935,942,951,958,971,975,984],[2992,3001,3005,3006,3010,3019,3026,3036,3042,3052,3059,3069,3076,3088,3092,3102,3109,3109,3118,3127,3128,3139,3142,3153,3159,3159,3169,3176,3176,3186,3192,3202,3209,3219,3226,3236,3242,3253,3259,3269,3276,3285,3292,3302,3309,3319,3326,3338,3342,3355,3359,3371,3376,3376,3385,3392,3401,3409,3419,3426,3427,3437,3442,3443],[3777,3791,3795,3801,3819,3826,3836,3842,3852,3859,3869,3876,3886,3893,3894,3905,3909,3919,3926,3943,3943,3954,3959,3972,3976,3976,3986,4018,4027,4037,4042,4055,4059,4070,4076,4076,4086,4093,4103,4109,4121,4140,4142,4143,4151,4159,4164],[4592,4602,4606,4610,4619,4636,4643,4653,4659,4669,4676,4686,4693,4709],[4908,4922,4926,4927,4934,4943,4953,4959,4970,4976,4987],[5949,5955,5970,5976,5987,5993,6003,6011,6020,6027,6037,6043,6053,6060,6072,6076,6088,6093,6104,6110,6118,6127,6127,6140,6143,6144,6154,6160,6171,6177,6187,6193,6203,6210,6218,6226,6227,6237,6243,6244,6254,6260,6270,6276,6287,6289],[6675,6685,6689,6693,6704,6710,6710,6721,6727,6727,6738,6743,6754,6760,6770,6777],[7085,7092,7109,7112,7120,7127,7141,7144,7154,7160,7171,7177,7187,7204,7210,7220,7227,7237,7244,7254,7260,7271,7277,7287,7293,7304,7310,7321,7327,7337,7344,7344,7354,7360,7371,7377,7387,7394,7404,7410],[7901,7910,7914,7915,7920,7927,7927,7938,7944,7954,7960,7971,7977,7987,7994,7997,8004,8010,8011,8022,8027,8028,8038,8044,8054,8060,8071,8077,8086,8088,8094,8104],[8781,8792,8795,8797,8799,8805,8811,8821,8827,8838,8844,8854,8861,8871,8877,8888,8894,8905,8911,8920,8927,8928,8938,8944,8955,8961,8961,8972,8977,8978,8990],[9171,9186,9202,9211,9211,9221,9227,9238,9244,9245,9255,9261,9261,9272,9277,9278,9288,9294,9305],[9545,9554,9558,9559,9563,9571,9578,9578,9588,9594,9605,9611,9621,9628,9638,9644,9653,9661,9661,9672],[9867,9886,9894,9905,9911,9920,9925,9929,9938,9944,9956,9961,9962,9973,9988,9994,10005,10011,10022,10028,10041,10044,10055,10061,10072,10078,10088,10094,10105,10111,10120,10128,10142,10144,10155,10158],[10340,10351,10354,10356,10361,10371,10378,10388,10394,10405,10411,10412,10422,10428,10438,10444,10445,10456,10461,10462,10473],[10727,10737,10745,10755,10761,10772,10778,10786,10795,10805,10812,10822,10828,10831,10836],[11093,11104,11107,11112,11121,11128,11142,11145,11155,11161,11162,11170,11178,11179,11189,11195,11206,11211,11223,11228,11239,11245,11256,11261,11272,11278],[11431,11442,11446,11451,11453,11461,11473,11478,11489,11495,11495],[11848,11858,11862,11867,11871,11878,11890,11895,11906,11912,11912,11920,11928,11937,11945,11955,11962,11962,11970],[12160,12165,12171,12179,12189,12195,12206,12212,12212,12220,12228,12239,12245,12256,12262,12279,12289,12295,12306,12312,12323,12328,12340,12345,12346,12357,12362,12373,12378,12387,12395,12399],[12568,12574,12579,12590,12595,12606,12612,12623,12628,12639,12645,12656,12662,12662,12674,12679,12679,12686],[18237,18247,18272,18297,18314,18324,18341,18358,18374,18380,18389,18397,18407,18414,18414,18425,18431,18441,18447,18448,18458,18464,18475,18480,18491,18497,18498,18509,18514,18525,18530,18531,18542,18547,18558,18564,18575,18591,18597,18608,18614,18625,18630,18642,18647,18648,18659,18664,18675,18680,18681,18693,18697,18708,18714,18715,18726,18730,18732,18745,18747,18758,18764,18775,18781,18792,18797,18798,18809,18814,18815,18826,18830,18842,18847,18848,18859,18905,18914,18924,18931,18941,18947,18948,18959,18964,18965,18976,18981,18992,18997,18998,19009,19019,19025,19031,19042,19047,19058,19064,19075,19081]],"version":"2.0.0"}

    But how can this average rate of change be interpreted in a more visual or graphical sense? Let’s take a look.

    Average Rate of Change on a Graph

    Let’s take the same familiar function, y=fx{"x":[[168,167,166,166,165,165,166,167,169,170,174,176,181,183,190,192,204,206,213,214,217,218,220,220,219,218,215,213,212,210,210,208,208,207,206,205,204,203,199,193,190,186,183],[316,317,324,334,338,348,354,360],[333,331,330,333,334,341,346,353,357,367,374],[520,520,520,520,518,517,512,510,504,502,497,496,490,488,486,484,483,481,481,480,480,480,481,481,482,484,484,485,485,485,485],[470,473,476,482,485,490,498,500,503,506,510,510],[600,599,597,594,591,587,584,577,575,569,568,569,569,571,574,575,582,587,592,594],[621,620,619,619,621,624,626,630,636,639,641,643,643,642,638,637,632,631,628,628,627],[656,657,658,658,658,657,655,654,651,650,648,647,648,649,654,656,664,667,677],[711,712,712,714,716,719,724,726,727,727,727,725,725,723,721,719,718,716,711,708]],"y":[[222,221,224,226,236,254,258,265,272,273,275,275,275,274,270,268,257,254,246,244,239,238,235,239,244,248,269,289,296,317,323,342,347,352,357,366,370,374,382,390,391,390,387],[248,248,247,247,247,249,250,250],[293,294,294,294,294,294,293,292,291,289,287],[200,196,194,190,187,186,183,182,181,181,181,181,186,191,197,205,210,227,233,240,254,268,281,289,294,316,322,334,338,347,349],[285,283,280,277,276,275,273,273,273,272,272,273],[205,205,205,205,208,213,217,233,240,269,275,289,293,296,302,304,307,308,308,308],[257,256,254,253,253,252,252,252,254,257,261,265,269,276,282,284,289,290,292,293,293],[259,258,255,254,255,256,262,264,272,279,285,294,300,301,303,303,302,300,295],[199,199,198,197,197,199,207,219,224,239,244,259,263,277,287,295,299,304,319,326]],"t":[[0,22,39,48,56,75,84,89,100,106,115,122,131,139,148,156,175,184,189,200,206,206,215,239,248,256,276,283,289,300,306,315,322,323,336,340,341,348,356,376,383,389,400],[729,785,786,789,798,806,815,823],[1025,1035,1044,1076,1085,1089,1101,1106,1115,1123,1132],[1479,1488,1493,1501,1506,1515,1523,1532,1540,1552,1556,1556,1575,1585,1590,1601,1606,1615,1623,1623,1632,1640,1652,1656,1656,1675,1685,1690,1701,1706,1714],[1925,1934,1940,1948,1956,1966,1978,1984,1990,2001,2006,2016],[2313,2317,2324,2333,2340,2348,2357,2366,2373,2395,2400,2407,2415,2419,2423,2432,2440,2449,2457,2466],[2656,2665,2694,2700,2707,2718,2723,2733,2740,2748,2757,2769,2774,2793,2801,2807,2818,2823,2833,2840,2844],[2967,2974,2993,2998,3023,3032,3040,3049,3057,3069,3074,3095,3100,3107,3118,3123,3133,3140,3150],[3334,3340,3341,3357,3369,3374,3394,3401,3407,3418,3424,3433,3440,3450,3457,3469,3473,3474,3495,3501]],"version":"2.0.0"} over the interval a,b{"x":[[309,308,306,305,301,299,290,262,259,256,253,244,241,235,233,229,227,226,225,224,222,222,222,222,222,221,221,221,221,221,221,221,220,220,222,223,228,230,236,242,249,256,259,269,272,280],[359,354,352,348,345,338,336,328,323,318,315,313,313,313,315,316,317,320,325,330,335,340,345,350,354,357,362,365,367,370,372,376,377],[460,460,460,460,460,460,460,460,460,460],[564,566,566,566,566,565,563,561,557,556,553,551,550,550,550,550,551,552,554,557,560,565,570,575,580,585,587,593,594,594,594,591,587,581,579,573,568,565,563,557,555],[650,650,650,650,653,655,661,664,673,675,683,687,689,693,695,696,700,700,701,702,702,702,701,700,698,697,695,694,691,690,689,686,684,682,677,674,666,661,655,652,650,644,639,636,633]],"y":[[164,164,164,164,165,166,167,174,175,176,177,181,182,185,186,189,190,191,194,195,208,215,225,236,242,262,270,290,297,317,324,342,353,363,388,391,400,402,408,411,413,415,415,416,416,417],[278,271,270,269,269,272,274,283,291,301,311,319,325,327,329,329,329,328,326,323,320,317,314,312,311,311,315,319,324,328,331,333,333],[339,341,343,345,347,353,359,362,371,373],[216,210,208,209,212,218,226,230,248,255,275,286,295,299,302,307,309,314,315,316,315,314,313,312,311,311,312,315,317,321,323,327,330,333,334,336,337,337,337,336,334],[210,206,205,203,200,200,199,199,199,199,199,199,199,200,201,201,204,206,207,212,218,225,234,239,256,263,283,290,311,318,325,338,351,357,375,381,393,400,406,408,409,411,412,412,412]],"t":[[0,27,60,70,77,86,93,130,135,135,149,152,164,170,177,187,193,194,204,216,234,236,249,255,260,270,277,287,293,303,310,319,327,337,363,368,377,387,394,404,410,420,427,437,443,453],[850,865,868,869,877,886,894,903,910,922,927,936,944,954,961,969,972,978,989,994,1003,1010,1019,1027,1039,1044,1055,1060,1070,1083,1091,1096,1103],[1354,1369,1372,1377,1378,1386,1394,1403,1411,1419],[1720,1735,1738,1744,1753,1761,1770,1777,1788,1794,1806,1811,1823,1827,1828,1837,1844,1856,1861,1873,1878,1889,1894,1906,1911,1924,1928,1940,1944,1953,1961,1970,1977,1988,1994,2003,2011,2015,2020,2027,2035],[2338,2353,2357,2361,2370,2378,2387,2395,2407,2411,2423,2428,2437,2444,2445,2457,2461,2470,2475,2478,2487,2494,2507,2511,2520,2528,2537,2544,2553,2561,2562,2576,2578,2587,2594,2603,2611,2620,2628,2637,2639,2645,2654,2661,2665]],"version":"2.0.0"}. Now let us graph the endpoints to get a better grasp of what’s happening.

    Average Rate of Change, average rate of change over two points, StudySmarterThe Average rate of change over two points, StudySmarter Originals

    Let the two endpoints be A and B which have coordinates a,fa{"x":[[180,178,174,172,170,167,159,152,148,144,136,118,114,106,97,93,92,91,93,98,108,116,127,143,155,167,173],[233,232,231,230,229,226,222,212,210,205,202,198,194,192,192,193,194,197,201,207,211,216,220,227,232,237,240,243,248,249,250,252,253,255,257,260,261,263,265],[320,321,322,323,324,324,323,322,320,319,319,318],[451,451,450,448,447,445,442,436,428,423,419,414,410,409,409,410,410,411,412,412,412,412,411,411,409,408,406,405],[387,387,387,389,393,398,401,404,408,416,419,426,427,429,431],[524,514,512,505,502,495,490,489,488,490,493,500,503,512],[610,615,618,620,621,624,625,625,622,618,615],[658,666,670,672,678,680,679,678,673,667,665,659,651,648,647],[576,576,576,576,575,575,575,575,574,574,573,571,570,566,565,560,559,554,553,551,550,550,551,552,555,556,559,561,564,568,572,574,579,581,585,586,588,588,588,588,588,588,588,588,589,589,591,592,594,598,600,602]],"y":[[226,226,225,224,224,224,227,232,237,242,255,290,301,320,352,370,378,387,408,419,433,439,444,448,448,447,445],[331,328,324,322,319,315,313,312,314,318,320,326,332,338,348,352,354,356,356,354,352,349,345,339,335,332,331,331,337,339,342,349,355,361,364,365,366,366,366],[352,352,352,352,358,368,371,378,389,393,395,397],[262,259,255,249,248,247,246,246,250,255,262,271,289,302,309,330,338,352,368,382,396,412,420,423,430,431,432,432],[356,353,352,348,346,345,344,343,342,340,340,339,339,340,340],[272,274,276,287,292,309,326,331,349,362,366,369,370,371],[284,287,291,294,297,312,320,334,347,357,362],[253,259,266,270,285,308,326,332,351,368,374,389,405,412,416],[320,319,318,317,316,314,313,310,309,307,306,305,305,305,307,312,315,323,326,335,338,340,342,343,344,344,343,342,340,337,334,333,327,325,321,321,320,322,324,330,332,334,336,339,341,342,345,345,346,347,347,346]],"t":[[0,9,11,19,20,30,35,45,55,57,61,80,87,96,102,112,118,128,135,145,152,162,169,186,196,202,209],[488,493,502,511,519,529,535,545,552,562,569,579,586,599,602,610,619,628,635,652,662,669,679,686,699,702,713,729,735,736,745,752,762,769,779,786,799,802,804],[931,938,952,962,978,999,1002,1014,1019,1029,1035,1038],[1322,1328,1335,1346,1352,1353,1362,1369,1386,1399,1402,1414,1419,1429,1436,1446,1452,1462,1469,1479,1486,1499,1502,1514,1519,1519,1536,1542],[1660,1664,1675,1686,1699,1702,1714,1719,1719,1729,1736,1746,1753,1762,1769],[2090,2099,2108,2111,2120,2130,2146,2152,2163,2179,2196,2202,2204,2219],[2913,2919,2925,2928,2936,2944,2961,2969,2986,3003,3019],[3271,3282,3288,3296,3303,3320,3336,3349,3353,3362,3371,3379,3398,3413,3418],[1654887116196,1654887116201,1654887116211,1654887116217,1654887116227,1654887116234,1654887116244,1654887116251,1654887116251,1654887116261,1654887116267,1654887116278,1654887116284,1654887116294,1654887116302,1654887116311,1654887116317,1654887116328,1654887116334,1654887116344,1654887116351,1654887116361,1654887116367,1654887116368,1654887116384,1654887116394,1654887116401,1654887116401,1654887116411,1654887116418,1654887116427,1654887116434,1654887116446,1654887116451,1654887116461,1654887116467,1654887116478,1654887116494,1654887116501,1654887116513,1654887116517,1654887116518,1654887116528,1654887116534,1654887116534,1654887116544,1654887116551,1654887116551,1654887116561,1654887116568,1654887116578,1654887116581]],"version":"2.0.0"} and b,fb{"x":[[249,247,246,243,241,240,237,234,232,230,226,223,221,214,204,197,190,187,178,175,170,169,168,169,170,173,178,185,189,196,205,209,219,229,238],[294,293,292,292,291,290,290,290,291,292,293,295,296,297,298,300,301,303,305,306,309,311,317,319,326,328,337,339,347,348,351,353,353,352,350,346,344,340,332,329,321,320,316,316],[438,438,438,438,439,439,439],[571,570,570,568,566,563,560,558,554,548,547,542,541,540,539,539,538,538,538,538,538,539,538,536,535,533,530,528,524,522],[520,526,530,533,537,548,554,557,564],[645,642,641,640,636,634,627,622,619,618,616,616,616,619,620,624,626,632,637,643,646],[702,700,700,699,698,695,693,691,688,686,685,683,683,681,681,681,683,685,690,692,694,699,702,710,712,716,718,719,720,720,720,718,715,711,707,703,701,695,693,688,687,686],[775,782,786,788,793,794,795,796,796,795,793,791,788,784,779,776,769,767,765,761,759,757],[840,841,843,844,847,850,854,858,860,865,866,869,869,870,870,870,868,864,860,857,852,846,844,841,836]],"y":[[206,204,202,201,199,198,198,198,198,198,200,202,204,209,221,233,247,255,282,291,321,330,357,365,373,388,400,412,418,426,433,436,439,440,439],[245,246,246,247,248,252,256,267,279,286,292,308,318,327,333,338,342,344,345,345,343,341,334,331,324,322,318,318,318,320,323,329,334,338,343,347,348,351,355,356,356,355,349,347],[330,337,342,348,354,358,366],[247,244,242,238,235,234,234,235,239,249,254,271,278,293,302,317,333,350,357,377,383,398,403,416,419,422,427,428,430,430],[350,347,346,345,344,341,340,339,338],[272,272,272,272,274,276,287,298,309,314,332,342,348,361,364,368,370,372,372,371,370],[274,269,268,268,269,275,282,290,304,317,320,327,329,334,337,339,339,339,336,335,334,332,331,331,331,332,335,336,338,341,342,345,347,349,350,350,350,349,348,343,340,335],[254,254,257,260,270,274,287,291,300,305,313,322,332,340,348,352,361,363,365,369,369,370],[227,225,222,221,221,222,226,233,238,256,263,282,289,308,315,321,335,348,360,366,377,386,391,395,402]],"t":[[0,7,19,26,33,39,50,56,57,67,73,73,83,90,100,106,117,123,133,140,150,156,167,173,173,183,190,200,206,217,223,223,233,240,250],[587,594,603,606,607,616,623,633,640,640,650,656,666,673,683,690,700,706,716,723,733,740,750,756,767,773,783,790,800,807,816,823,833,840,850,858,867,873,883,890,900,906,917,923],[1260,1277,1280,1281,1290,1298,1307],[1641,1647,1652,1658,1667,1674,1683,1690,1700,1707,1717,1723,1734,1740,1741,1750,1757,1767,1773,1783,1790,1800,1807,1819,1823,1824,1832,1840,1850,1854],[2007,2021,2024,2025,2032,2040,2050,2057,2067],[2361,2371,2374,2375,2382,2391,2401,2407,2418,2424,2432,2440,2450,2457,2467,2474,2474,2484,2491,2501,2507],[2673,2692,2695,2699,2707,2717,2724,2734,2749,2757,2767,2774,2774,2784,2791,2801,2817,2825,2836,2840,2841,2850,2857,2867,2874,2884,2891,2891,2903,2907,2908,2918,2924,2933,2941,2951,2957,2968,2974,2984,2991,3000],[3158,3174,3178,3183,3191,3201,3207,3218,3224,3224,3234,3241,3251,3258,3268,3274,3286,3291,3294,3301,3307,3308],[3652,3657,3662,3669,3674,3685,3691,3701,3708,3718,3724,3734,3741,3751,3758,3758,3768,3774,3784,3791,3801,3808,3808,3817,3824]],"version":"2.0.0"} respectively. The average rate of change between the two points will be the slope of the line passing through A and B. The reason is that the ratio of the change in y to the change in x is nothing but the gradient of that line.

    In other words, the average rate of change over two points is the slope of the secant line through them.

    Note that the average rate of change of a function may differ over different intervals since the slope will be altered respectively. For a linear function, the average rate of change will always be the same for any interval, which can be justified graphically since the slope will always be the same line as the function itself.

    Average Rate of Change of a Function as Gradient

    The average rate of change between two points can also be equivalently understood as the gradient between the two points.

    Average Rate of Change, gradient over two points as a part of right triangle, StudySmarterThe gradient over two points as a part of a right triangle, StudySmarter Originals

    If we draw the change in y coordinates and the change in x coordinates, we get a right-angled triangle as shown above. From the laws of trigonometry, the gradient can be found as:

    ΔyΔx=m=slope

    Which is just a slightly different way of looking at the average rate of change between two points.

    If the two points A and B are close enough, the average rate of change is more accurate over a bigger interval. The closer the points, the more precise the rate of change is. If the points get close enough, the average rate of change becomes an instantaneous rate of change (discussed in the accompanying article. As Δy0{"x":[[87,87,89,91,95,99,102,106,113,116,120,125,130,134,138,143,148,152,156,160,163,166,169,171,174,176,179,182,185,188,192,195,199,203,207,211,214,217,220,221,222,222,220,217,213,208,201,193,185,175,165,154,143,132,121,112,103,95,88,81,76],[258,258,259,260,260,262,262,263,265,266,268,271,273,275,280,282,287,292,297,303,306,312,320,321,322,321,320,318,316,314,310,309,306,303,302,300,300,300,300,301,301,301,300,299,295,291,286,280,273,265,256,252,244],[779,779,778,777,775,774,772,771,771,771,772,774,778,779,784,790,793,798,801,804,810,814,819,825,831,833,837,838,839,840,839,834,829,822,818,814,810,802,797,793,787],[428,429,430,434,441,458,469,481,486,506,512,532,539,550,561,566,576,580,583,586,592,594,596,599,600,601],[577,577,578,579,580,581,586,588,590,592,597,601,605,607,609,610,611,611,611,610,609,607,603,602,598,596,595,592,591,590,589,588,587,586]],"y":[[259,258,254,251,245,239,234,227,214,209,199,189,179,169,159,149,140,133,126,122,119,117,117,118,122,126,132,140,148,157,167,177,189,200,210,219,227,234,241,247,252,256,260,264,266,269,270,271,272,273,273,273,274,275,275,276,276,276,276,277,277],[170,169,169,172,177,183,186,194,200,206,210,214,215,216,216,215,213,210,207,200,197,189,177,174,171,169,169,169,171,173,180,183,189,201,206,223,230,242,254,266,276,286,295,303,311,315,318,318,318,315,311,308,303],[256,258,260,262,269,274,286,297,313,317,327,336,347,349,354,358,359,359,359,357,353,350,343,334,323,317,305,298,292,279,268,254,247,243,241,241,241,241,243,244,249],[320,320,319,318,316,313,312,310,310,308,308,308,307,307,307,307,307,307,307,307,308,308,308,308,308,308],[274,275,276,277,278,279,282,283,285,286,288,290,292,293,294,295,296,297,300,303,306,308,315,317,325,328,330,334,336,337,340,341,342,343]],"t":[[0,20,32,34,46,53,62,67,79,84,96,101,113,117,129,134,146,151,163,168,179,184,196,201,213,218,229,234,246,251,263,268,279,284,296,301,312,318,329,334,346,351,362,371,380,384,396,401,413,418,429,434,446,451,463,468,479,484,496,501,513],[976,988,1001,1011,1018,1033,1036,1045,1051,1062,1068,1079,1084,1096,1101,1112,1118,1129,1137,1146,1151,1162,1179,1196,1201,1212,1218,1229,1235,1246,1251,1263,1268,1279,1286,1296,1303,1312,1318,1329,1335,1346,1351,1363,1368,1379,1385,1396,1401,1412,1418,1429,1435],[5403,5413,5416,5422,5431,5437,5448,5453,5465,5470,5485,5488,5498,5503,5515,5520,5531,5536,5537,5548,5553,5553,5565,5570,5581,5587,5595,5599,5603,5615,5620,5631,5636,5648,5653,5653,5665,5670,5670,5682,5686],[12533,12541,12556,12567,12572,12583,12590,12600,12607,12617,12624,12633,12639,12650,12656,12656,12667,12672,12673,12684,12689,12690,12701,12706,12706,12717],[13041,13044,13081,13097,13098,13107,13117,13122,13123,13134,13139,13151,13156,13167,13172,13173,13184,13189,13200,13207,13219,13223,13234,13239,13250,13256,13259,13267,13273,13273,13284,13289,13301,13306]],"version":"2.0.0"} and Δx0{"x":[[291,291,291,294,297,302,305,315,319,328,330,337,339,343,346,347,348,348,348,345,342,335,330,324,322,319],[379,380,378,377,376,374,369,364,358,350,345,342,341,340,340,343,345,353,357,360,368,373,377,387,392,396],[497,493,492,491,492,494,498,502,508,512,527,533,553,560,580,586,605,611,621,630,633,636,642,644,646,648,649],[624,626,627,627,629,630,635,636,642,643,646,647,647,646,644,643,641,640,638,635,632,630,628,627,626],[777,778,778,777,776,775,774,771,768,766,764,764,764,765,767,768,769,771,775,777,780,783,788,794,800,804,808,814,818,825,832,837,842,844,846,846,845,840,837,828,824,820,812,807,803,795,789],[63,63,63,65,67,71,77,80,87,89,97,100,104,109,115,119,125,127,132,137,140,143,148,150,152,153,156,157,160,161,165,167,170,171,173,174,178,182,186,192,196,200,205,209,212,218,220,227,229,234,236,240,241,242,242,242,241,239,236,234,229,224,217,213,201,196,181,176,170,159,153,148,143,133,128,123,114,109,105,96,92,83,75,67,61,55,53,49,48]],"y":[[253,248,246,243,240,238,237,235,235,236,237,243,245,252,257,262,265,271,276,281,287,294,298,301,302,303],[251,250,249,249,250,250,253,256,260,268,273,279,281,289,291,296,298,302,302,303,303,303,303,301,300,299],[260,260,260,260,260,260,260,260,260,260,260,260,259,258,257,257,256,256,256,255,255,255,255,255,255,255,255],[238,235,235,234,233,233,233,233,235,237,242,244,252,254,260,263,266,269,271,276,280,283,286,287,288],[222,219,218,218,219,221,224,230,238,247,258,262,276,280,287,290,293,296,301,303,305,306,308,308,307,306,304,300,297,289,279,269,258,252,242,228,224,215,213,207,206,205,205,205,207,210,215],[365,363,360,354,350,340,325,320,302,296,276,269,261,248,234,228,216,210,200,191,187,181,175,173,172,172,174,176,182,185,195,202,209,213,217,222,230,239,250,264,274,284,292,301,306,316,319,328,331,337,339,344,347,350,352,353,355,357,359,360,362,364,366,367,370,371,374,374,375,375,375,375,375,375,375,374,374,373,373,373,373,374,375,376,377,378,378,376,374]],"t":[[0,12,15,20,28,38,43,54,61,71,78,87,93,104,110,121,126,137,143,154,160,171,178,187,193,204],[382,386,402,410,410,421,426,438,443,454,461,471,478,488,494,505,510,521,526,527,540,543,544,554,560,560],[875,881,885,887,910,920,927,927,938,944,954,962,971,977,988,993,1004,1010,1021,1027,1027,1038,1043,1044,1055,1060,1071],[1220,1225,1229,1235,1236,1244,1255,1261,1271,1278,1288,1293,1305,1310,1319,1324,1327,1327,1338,1343,1355,1360,1368,1372,1377],[1897,1903,1908,1922,1929,1935,1936,1945,1955,1962,1972,1978,1988,1995,2002,2002,2010,2011,2022,2027,2028,2038,2044,2055,2060,2061,2072,2077,2078,2089,2095,2105,2111,2122,2127,2138,2145,2155,2160,2172,2177,2178,2189,2194,2194,2205,2211],[14054,14073,14077,14083,14092,14098,14110,14116,14126,14132,14143,14148,14149,14160,14165,14176,14182,14182,14193,14199,14210,14215,14226,14232,14243,14249,14260,14265,14276,14283,14293,14298,14307,14315,14316,14326,14332,14343,14348,14360,14366,14376,14383,14393,14398,14410,14416,14429,14432,14443,14448,14460,14465,14476,14483,14493,14498,14510,14515,14516,14527,14532,14543,14549,14560,14566,14577,14582,14583,14594,14598,14599,14610,14615,14616,14627,14632,14633,14644,14649,14650,14660,14668,14677,14683,14693,14699,14710,14727]],"version":"2.0.0"}, we get: ΔyΔxdydx{"x":[[91,91,91,91,93,96,100,102,109,114,119,122,124,129,132,137,143,149,153,159,164,169,171,174,176,179,180,182,184,185,187,188,190,192,194,196,197,199,202,204,208,212,216,220,224,227,232,234,238,240,241,244,247,249,250,252,253,253,253,254,253,252,250,248,241,239,230,227,214,209,204,192,182,176,158,153,142,130,119,105,97,90,87,80,78,76,76],[305,305,305,305,305,305,307,307,310,311,312,315,317,319,321,325,328,331,337,339,342,347,350,356,361,363,365,368,370,372,374,375,375,374,373,373,372,372,371,371,371,370,369,368,365,364,358,356,348,345,337,334,324,320],[76,75,75,74,74,76,78,81,83,90,92,101,107,112,115,118,124,127,129,132,138,141,143,150,153,156,159,162,163,164,167,167,170,172,174,176,177,179,183,186,187,190,194,197,200,203,204,207,212,213,216,217,218,220,221,221,222,221,220,219,215,212,208,199,195,182,178,162,152,141,135,130,120,110,105,96,92,88,79,75,73,72],[298,299,300,304,306,314,317,326,331,334,336,337,338,339,339,339,339,338,335,331,327,322,318,316,311,310,308],[364,363,360,359,358,354,351,349,345,343,342,340,340,340,340,340,341,343,345,347,350,354,362,367,374,377,380,384],[130,131,131,132,135,140,148,152,169,175,181,194,201,215,237,253,261,269,277,293,306,320,327,333,343,347,351,357,359,361,361,361],[473,474,474,474,474,475,478,479,481,485,487,490,492,497,500,502,507,510,512,517,520,525,531,536,541,547,550,559,562,568,576,580,581],[486,486,487,488,493,498,503,507,514,518,521,524,526,528,531,533,536,539,541,543,547,549,551,557,560,563],[752,758,758,759,760,760,760,760,759,756,752,747,741,735,728,722,719,711,709,706,705,705,707,708,709,713,715,718,721,727,730,733,739,742,749,754,759,761,765,766,768,769,769,769,769,768,767,766,764,762,762,761,761,761,761,761,764,764,767,767,769],[807,807,807,807,808,809,811,813,816,818,821,823,825,829,831,833,837,839,843,846,850,852,855,857,858,858,858,858,856,855,854,853,852,851,850,850,850,850,850,850,850,849,849,847,846,842,840,835,833,831,825,822],[737,736,735,736,738,744,753,760,767,782,790,797,811,817,822,833,838,845,848,856,857,858,858,857,856],[779,780,781,782,784,785,787,788,788,788,787,786,783,778,776,773,770,759,756,748,745,741,736,734,732,731,731,732,735,739,747,751,761,765,768,775,778,781,783,789,790,793,796,798,799,800,800,800,800,799,799,798,797,797,797,796,796,795,794,794,794,794,794,794,794,795,796],[840,840,844,848,854,861,864,869,871,872,875,875,875,875,874,872,871,870,868,863,861,857,856,853,851],[901,902,901,900,896,895,894,890,887,883,882,880,879,877,877,877,878,882,888,891,899,909]],"y":[[300,299,298,296,291,284,277,270,257,248,238,233,228,218,212,202,191,179,169,159,148,137,134,129,126,122,121,121,123,124,130,130,135,145,148,155,159,163,172,176,184,193,201,209,218,220,231,235,245,249,252,258,263,268,270,274,278,279,281,282,284,285,286,287,288,288,289,289,289,290,290,290,289,290,291,292,293,295,296,297,298,298,298,298,296,296,295],[193,194,196,198,206,210,221,224,233,236,238,241,242,243,244,244,244,244,243,242,241,238,236,231,226,223,220,213,210,204,197,195,191,190,192,194,199,205,212,220,230,241,258,269,278,283,297,300,311,313,319,320,321,321],[546,546,547,547,546,543,540,535,532,521,517,505,496,488,484,479,471,467,463,459,452,448,445,436,431,427,425,423,422,422,422,423,427,432,437,443,447,454,465,474,478,487,501,509,517,525,529,536,547,549,557,559,562,565,566,567,568,568,568,568,567,566,564,562,561,560,560,560,560,560,561,561,561,562,562,563,563,563,563,563,562,561],[482,480,479,475,474,471,471,471,474,478,480,483,485,491,494,497,502,505,510,516,520,525,527,529,531,531,531],[482,481,480,479,479,480,481,483,486,489,491,497,501,504,510,513,518,523,526,527,529,530,530,527,524,522,520,518],[370,370,371,371,371,372,372,372,373,373,373,373,372,372,371,370,369,368,368,366,365,363,363,362,362,362,362,362,362,364,366,367],[344,341,339,338,337,336,331,329,328,324,323,321,320,319,319,319,320,321,323,325,326,327,329,331,332,333,333,332,331,330,327,325,324],[380,379,376,375,370,366,363,361,359,359,360,362,363,365,368,369,371,373,375,375,375,375,375,374,373,372],[268,260,258,256,251,249,248,244,242,240,238,238,238,240,244,249,252,260,263,269,271,272,273,273,273,273,272,271,269,265,263,261,256,252,244,234,226,221,210,206,199,194,189,188,187,187,187,190,195,203,207,212,221,226,237,242,258,262,273,276,282],[247,248,249,251,252,258,262,267,271,272,273,274,274,274,274,273,271,269,267,263,259,257,250,247,243,241,238,236,236,237,239,243,246,250,256,259,263,272,276,284,292,299,302,310,313,317,319,320,320,320,318,316],[374,374,374,374,374,374,372,371,370,368,367,366,365,365,365,364,364,364,365,366,366,368,369,370,370],[457,457,457,457,457,457,456,455,452,451,449,448,446,445,445,445,445,447,449,454,456,460,466,470,476,482,485,487,489,489,486,485,478,475,472,464,460,456,451,441,437,429,421,415,411,410,407,405,404,405,408,414,422,427,432,441,451,456,467,471,475,485,487,491,492,494,495],[457,455,453,452,451,451,451,453,454,456,462,464,466,470,475,479,480,482,484,488,489,491,492,492,492],[455,455,455,455,456,456,457,459,463,468,470,473,476,483,485,489,491,493,494,494,494,492]],"t":[[0,8,24,32,43,50,60,66,81,83,94,98,99,109,115,126,132,143,148,159,166,180,182,194,198,210,215,226,243,248,260,267,280,282,294,298,299,309,315,316,326,333,343,350,360,367,379,382,395,398,399,410,415,426,432,443,448,449,460,467,480,485,494,499,510,516,526,533,543,548,550,560,568,580,582,594,598,610,615,626,633,643,649,660,669,680,686],[1042,1065,1075,1082,1094,1099,1110,1116,1126,1132,1133,1143,1149,1149,1160,1165,1166,1180,1182,1182,1195,1199,1199,1210,1217,1224,1224,1233,1243,1249,1260,1266,1280,1295,1310,1315,1327,1332,1343,1349,1357,1365,1374,1382,1394,1399,1410,1417,1427,1433,1443,1449,1460,1465],[2823,2833,2841,2842,2866,2876,2883,2893,2900,2912,2916,2927,2933,2944,2949,2950,2961,2966,2966,2978,2983,2983,2995,3002,3011,3016,3028,3033,3044,3049,3061,3067,3078,3083,3095,3102,3112,3116,3127,3133,3144,3150,3161,3166,3178,3183,3196,3203,3211,3216,3228,3233,3233,3245,3249,3250,3262,3283,3284,3291,3303,3312,3316,3328,3333,3345,3350,3361,3366,3378,3383,3384,3395,3404,3414,3416,3418,3429,3437,3444,3450,3461],[6924,6929,6938,6946,6952,6962,6968,6979,6984,6996,7001,7002,7013,7018,7018,7030,7034,7035,7046,7051,7063,7068,7079,7084,7096,7101,7104],[7374,7409,7426,7434,7435,7446,7451,7463,7468,7468,7480,7484,7485,7497,7501,7502,7513,7518,7530,7534,7546,7551,7563,7568,7580,7584,7585,7593],[9120,9127,9130,9135,9146,9153,9164,9169,9180,9185,9187,9197,9202,9214,9218,9227,9235,9236,9247,9252,9264,9268,9269,9281,9285,9286,9297,9302,9302,9314,9319,9326],[12163,12168,12171,12178,12179,12186,12198,12203,12203,12215,12220,12220,12232,12236,12237,12249,12253,12253,12265,12270,12270,12282,12286,12298,12304,12315,12320,12332,12340,12348,12353,12365,12370],[12761,12764,12778,12787,12795,12803,12812,12820,12831,12836,12849,12853,12853,12861,12870,12870,12886,12888,12895,12903,12912,12912,12920,12931,12937,12942],[14163,14170,14174,14180,14187,14188,14199,14204,14204,14216,14221,14229,14238,14248,14255,14266,14271,14282,14287,14299,14304,14304,14316,14320,14321,14333,14337,14338,14350,14354,14354,14367,14370,14371,14383,14388,14395,14404,14415,14421,14432,14437,14449,14454,14455,14470,14482,14487,14499,14504,14505,14516,14520,14521,14533,14538,14545,14555,14565,14571,14583],[14859,14864,14872,14884,14889,14899,14904,14916,14921,14929,14937,14938,14950,14954,14954,14967,14971,14971,14983,14988,14999,15004,15016,15021,15029,15030,15037,15046,15064,15071,15083,15087,15088,15100,15104,15105,15117,15121,15121,15133,15138,15150,15154,15166,15171,15183,15188,15200,15204,15205,15217,15221],[15491,15496,15500,15513,15521,15532,15538,15538,15550,15554,15555,15567,15571,15571,15584,15588,15588,15600,15604,15616,15621,15633,15638,15638,15651],[15879,15883,15886,15888,15896,15904,15919,15922,15929,15938,15946,15947,15954,15966,15971,15971,15984,15988,16000,16004,16005,16017,16021,16022,16034,16038,16050,16055,16067,16071,16084,16088,16100,16104,16105,16113,16121,16122,16133,16138,16138,16151,16155,16167,16171,16183,16188,16200,16205,16221,16233,16238,16250,16254,16255,16268,16272,16284,16288,16293,16301,16305,16317,16321,16322,16334,16339],[16576,16605,16621,16633,16638,16650,16655,16663,16668,16672,16684,16688,16688,16701,16706,16717,16721,16722,16735,16739,16751,16755,16755,16768,16772],[16927,16940,16965,16971,16984,16988,16989,17001,17005,17017,17021,17022,17035,17039,17051,17055,17055,17068,17072,17084,17088,17101]],"version":"2.0.0"}).

    Examples of Average Rate of Change

    Find the average rate of change of the function fx=2x2-1{"x":[[168,168,169,169,170,171,171,171,171,171,170,167,165,162,158,154,151,147,143,140,137,135,132,131,130,129,128,127,125,124,122,120,117,113,110,106,102,97,93],[89,90,91,93,94,102,105,108,115,118,125,130,135,139,143,146,147],[253,244,241,233,227,224,216,213,207,206,205,204,204,205,208,212,216,222,229,235,241,249,256,259],[265,269,273,278,284,290,296,300,304,306,306,306,304,301,298,295,292,289,288,288],[343,340,339,338,335,332,329,325,321,319,317,316,316,316,319,322,326,331,337,343],[386,386,386,387,390,394,398,403,406,409,409,409,407,405,402,398,393,389,385,382,381],[477,476,471,470,469,469,471,473,477,482,488,493,500,505,510,515,517,518],[476,475,476,479,484,491,499,506,511,515,517,518,519,520,520],[600,600,600,599,599,599,600,602,605,609,614,618,622,625,627,627,627,627,624,621,617,612,607,602,598,596,596,596,598,602,607,613,620,626,633,638,643,647],[676,676,676,676,677,678,681,684,689,694,700,704,707,709,710,709,708,706,703,699,695,690,686,682,680,679],[734,734,734,733,732,729,725,720,715,709,705,703,702,703,705,708,714,720,728,736,747,757],[770,770,770,770,770,771,772,773,777,781,786,792,797,802,806,808,809,809,808,807,804,803,801,796,794,792,788,784,782,781,783,787,795,799,810,814,823],[837,838,839,840,843,847,855,857,866,869,874,878,882,885,886],[949,949,949,949,949,949,948,948,947,947,947,947,947,946,945,945,944]],"y":[[219,214,212,208,206,202,197,193,189,185,181,178,175,173,172,172,174,178,184,191,202,208,230,249,270,290,311,332,350,366,380,391,401,409,414,419,421,421,419],[326,325,325,323,323,321,320,320,320,320,320,319,319,319,319,319,320],[230,231,232,238,244,248,261,266,284,289,295,306,318,328,337,345,351,356,359,360,360,358,355,354],[284,281,279,279,279,279,281,283,286,290,294,299,304,310,315,320,324,327,328,329],[287,287,287,287,288,291,294,298,304,308,313,318,322,326,329,331,332,332,332,329],[244,243,242,242,245,249,256,263,272,282,292,302,312,322,332,342,350,359,366,372,374],[281,281,280,280,280,279,279,280,280,280,280,280,280,280,280,280,280,281],[346,346,346,345,343,341,340,339,339,339,340,340,341,341,342],[286,283,282,279,278,275,273,270,268,266,265,265,265,267,270,274,279,284,289,295,300,306,311,316,319,322,324,325,326,326,327,327,327,327,328,329,329,330],[281,280,278,277,275,274,272,270,269,269,270,272,275,280,285,291,297,303,308,313,317,321,324,326,326,326],[276,275,274,273,273,273,275,278,283,289,295,301,306,311,315,318,321,322,323,323,321,318],[221,216,214,212,210,209,207,206,204,202,201,200,200,201,204,207,212,216,222,224,228,230,232,235,237,239,242,245,247,248,248,248,247,246,243,242,240],[293,294,294,294,294,294,293,293,292,292,292,292,291,291,291],[260,259,262,264,266,272,279,288,301,309,315,317,322,327,330,332,333]],"t":[[0,6,11,13,19,30,36,47,52,64,69,80,86,97,102,111,119,134,136,144,152,164,169,180,186,197,203,214,219,231,236,247,253,264,269,280,286,297,302],[496,501,508,512,521,532,536,536,547,552,566,569,580,586,598,603,615],[909,922,928,936,947,953,961,969,981,986,986,997,1003,1014,1019,1032,1036,1048,1054,1064,1070,1080,1086,1094],[1308,1320,1328,1336,1347,1353,1364,1370,1381,1386,1397,1403,1414,1420,1431,1436,1448,1453,1462,1470],[1636,1649,1652,1654,1661,1670,1681,1686,1698,1703,1714,1720,1731,1736,1748,1753,1765,1770,1781,1786],[2008,2021,2028,2048,2053,2065,2070,2081,2087,2098,2103,2115,2120,2135,2136,2147,2153,2165,2170,2181,2184],[4020,4039,4043,4048,4055,4065,4071,4082,4087,4099,4104,4116,4121,4132,4137,4146,4154,4166],[7194,7230,7239,7250,7255,7267,7272,7284,7289,7297,7305,7314,7322,7333,7339],[7908,7913,7916,7923,7932,7939,7950,7955,7964,7972,7983,7989,8000,8006,8017,8022,8030,8039,8050,8055,8067,8072,8084,8089,8100,8105,8117,8122,8134,8139,8154,8156,8167,8172,8184,8189,8201,8205],[8453,8458,8466,8474,8484,8491,8501,8506,8514,8522,8534,8539,8551,8556,8567,8572,8584,8589,8601,8606,8618,8622,8634,8639,8651,8655],[8813,8818,8825,8834,8839,8851,8856,8868,8872,8884,8889,8901,8906,8918,8922,8934,8939,8951,8956,8968,8972,8984],[11626,11634,11640,11641,11649,11653,11658,11669,11674,11685,11690,11702,11707,11719,11723,11735,11740,11752,11757,11769,11773,11774,11786,11790,11791,11803,11807,11819,11824,11832,11849,11857,11868,11875,11885,11890,11902],[12429,12433,12438,12440,12449,12457,12468,12475,12486,12490,12502,12507,12519,12524,12536],[14449,14462,14491,14492,14501,14508,14519,14525,14537,14542,14553,14558,14570,14575,14586,14591,14604]],"version":"2.0.0"} as x varies from 1 to 3.

    Solution:

    Step 1: Calculate the value of the function at the endpoints, 1 and 3:

    f1=212-3=-1{"x":[[129,125,123,120,119,116,114,111,108,107,105,103,103,103,103,105,106,108,110,112,113,114,114,113,113,111,109,106,102,101,99,96],[73,75,78,80,82,85,96,100,108,115,121,127,131],[220,214,212,208,200,198,189,186,178,176,174,172,171,171,172,175,179,187,194,201,209,218],[253,253,253,253,253,253,254,254,254],[288,291,295,298,300,306,312,314,319,321,323,323,323,322,321,319,316,314,311,308],[385,387,391,396,400,410,414,420,426,430,436],[388,388,389,390,392,395,398,403,406,409,416,422,431,436,440,444],[519,519,518,518,519,521,525,530,534,536,541,542,544,545,544,542,539,537,535,533,528,526,523,519,514,513,513,514,515,519,528,531,541,545,556,562],[641,638,637,634,632,628,623,617,612,607,605,602,600,599,598,598,598,600,604,609,614,617,626,629],[679,679,679,678,678,678,678,678,678,678,678,678,678,678,678,678,679,681],[710,711,711,714,715,719,721,722,725,727,728,729,730,730,730,731,730,730,728,725,723,718,716,710,709,707],[741,743,744,745,746,748,750,754,756,761,762,764,764,764,764,764,763,761,760,758,754,753,751,751,752,755,760,766,769,779,783,786,793,796,801],[812,814,815,818,821,831,835,843,854,860,865],[932,932,932,932,932,932,932,933,936,938,942,944,945,948,949,950,950,949,948,945,944,941,939,937,933,931,927,925,924,924,926,927,933,935,941,943,945,947,950,951,952,953,953,953,952,951,950,946,944,939,934,929,926,919,918,917],[998,1004,1005,1009,1011,1018,1020,1026,1027,1029,1031],[998,998,998,999,1000,1003,1005,1013,1017,1020,1025,1028],[1066,1065,1066,1069,1072,1074,1080,1083,1085,1090,1092,1094,1095,1097],[1119,1119,1119,1119,1119,1119,1118,1118,1118,1118,1118,1118,1118,1119,1119,1119,1120,1120,1121,1121,1121,1121]],"y":[[166,163,162,160,160,161,163,167,175,179,190,204,221,239,260,282,293,314,334,348,355,373,377,390,394,397,402,406,407,407,407,406],[308,309,309,309,308,307,304,303,300,297,296,293,292],[207,205,205,205,210,213,228,233,254,261,269,276,290,302,312,320,327,333,336,338,338,336],[238,239,241,244,259,265,282,286,296],[186,187,192,195,200,210,226,231,247,252,268,274,290,296,301,311,319,323,331,337],[244,244,244,243,243,242,241,240,240,240,240],[282,283,283,283,283,283,283,282,282,281,280,279,278,277,276,275],[252,250,249,247,246,245,244,244,245,246,251,253,259,264,268,273,278,280,282,285,290,292,294,298,302,305,306,308,309,310,311,311,311,311,310,310],[228,227,226,226,226,228,232,240,250,260,266,276,282,287,296,300,304,314,320,324,328,329,331,331],[256,257,261,264,276,280,291,294,296,301,305,309,310,311,313,314,315,315],[233,233,234,236,237,244,247,250,257,261,265,274,278,283,287,295,299,303,310,319,322,329,331,334,335,336],[171,166,161,160,158,156,156,155,155,156,157,162,164,165,170,172,175,179,182,186,192,194,197,200,201,201,201,201,200,198,198,197,196,196,196],[267,267,267,267,266,266,266,265,265,265,265],[226,222,221,220,219,217,216,215,213,213,213,213,213,215,216,219,221,222,226,230,232,236,238,240,244,246,248,251,252,253,255,256,259,260,263,264,265,267,270,272,274,278,280,281,285,287,289,291,293,295,297,298,298,299,298,298],[245,245,245,245,246,247,248,249,249,250,250],[278,279,280,280,281,281,281,280,280,279,279,278],[276,277,276,276,275,275,274,274,274,274,273,273,273,273],[235,232,231,232,237,243,251,255,260,270,276,281,285,294,298,301,308,312,317,323,327,329]],"t":[[0,6,11,12,17,24,35,41,52,57,68,74,85,91,102,107,108,118,124,135,142,152,158,170,174,174,185,191,202,207,208,219],[480,496,499,508,508,518,524,535,541,552,558,569,574],[1018,1033,1036,1037,1049,1059,1069,1074,1086,1091,1092,1102,1108,1119,1124,1136,1141,1152,1158,1169,1175,1186],[1627,1646,1650,1658,1669,1675,1686,1691,1702],[2024,2038,2041,2042,2050,2058,2069,2076,2086,2093,2103,2110,2119,2125,2125,2133,2141,2153,2158,2170],[2551,2575,2586,2592,2608,2610,2620,2625,2636,2642,2654],[2840,2849,2853,2858,2870,2875,2886,2892,2892,2903,2908,2920,2931,2936,2942,2953],[3481,3493,3502,3509,3520,3525,3537,3543,3553,3560,3570,3577,3587,3592,3603,3609,3620,3625,3626,3637,3642,3642,3653,3659,3670,3677,3687,3692,3704,3709,3720,3726,3740,3742,3753,3759],[4029,4035,4042,4051,4054,4059,4070,4075,4091,4094,4103,4109,4110,4120,4125,4126,4138,4142,4154,4160,4170,4176,4187,4192],[4426,4442,4453,4459,4470,4476,4487,4492,4493,4504,4509,4517,4526,4526,4537,4543,4554,4559],[4765,4770,4779,4787,4793,4804,4809,4809,4821,4826,4826,4834,4842,4843,4854,4859,4860,4871,4876,4887,4893,4904,4910,4921,4926,4929],[5175,5179,5191,5193,5194,5201,5209,5221,5226,5237,5243,5254,5259,5260,5271,5276,5276,5284,5293,5301,5310,5321,5326,5338,5343,5354,5361,5371,5377,5388,5393,5396,5401,5409,5421],[5726,5733,5738,5744,5752,5760,5771,5776,5788,5793,5804],[6069,6074,6079,6085,6087,6093,6105,6110,6124,6127,6138,6143,6145,6155,6160,6172,6176,6177,6188,6193,6205,6210,6210,6222,6226,6227,6238,6243,6255,6260,6268,6278,6288,6294,6305,6310,6310,6322,6326,6327,6339,6343,6344,6355,6360,6360,6372,6376,6377,6388,6393,6405,6410,6422,6427,6430],[6853,6859,6863,6868,6878,6888,6894,6905,6910,6910,6922],[7149,7155,7165,7169,7177,7189,7194,7205,7210,7211,7222,7227],[8900,8905,8961,8969,8977,8978,8989,8994,8994,9003,9011,9011,9023,9027],[9246,9259,9267,9278,9289,9294,9306,9311,9311,9323,9328,9328,9340,9344,9345,9356,9361,9362,9373,9378,9389,9394]],"version":"2.0.0"}

    f3=232-3=15{"x":[[61,62,62,62,63,63,63,63,63,62,61,59,58,56,55,51,48,46,45,44,44,45,47,49,51,52,54,54,54,54,53,50,47,44,40,39,36,34,32,30],[24,23,22,21,20,22,24,30,32,40,44,47,50,54,56,58,59],[111,109,107,106,105,102,99,98,94,93,92,92,93,95,98,101,103,107,111,114],[131,131,131,131,134,137,139,145,147,152,154,156,158,157,155,154,153,151,149,147,146,143,142,142,141,141,143,145,147,153,154,160,162,166,167,167,167,166,163,160,157,151,149,147,146],[188,189,190,192,198,203,208,210,214,215,216,218,219,218,216,213,211,209,205],[263,264,265,268,272,277,282,285,292,294,300,301,302],[272,272,275,277,282,292,297,302,304,306],[367,367,367,367,368,369,374,377,381,383,386,387,387,387,386,384,381,377,375,372,366,364,361,361,360,361,362,365,366,373,375,382,384,392,394,401],[471,470,470,469,469,468,466,461,457,453,448,444,442,437,436,436,436,440,442,444,446,450,452,454,459],[476,476,475,475,475,476,478,479,481,484,486,489,493,495,497,497,498,497,496,493,492,489,487,486,484,483,483,485,487,490,493,496,498,500,501,502,502,502,502,499,498,497,494,491,490,487],[531,532,533,534,535,537,539,542,545,547,548,548,548,547,546,543,541,539,536,531,529,525,523,521,519],[544,544,544,544,545,547,548,552,553,557,558,560,561,561,562,562,562,561,560,559,558,556,555,554,551,550,548,546,546,547,549,551,557,560],[592,591,592,596,601,605,611,616,619],[662,662,662,663,663,665,666,670,671,675,677,681,682,683,683,683,683,682,681,679,678,675,673,671,670,668,668,669,670,674,675,677,680,682,685,686,687,687,688,688,688,687,684,682,679,677,671,669,663,662,659],[763,769,773,778,783,787,791,795,797,800,802],[758,757,758,762,767,775,779,789,792,799,801,802,805,806],[884,884,884,884,884,883,882,881,880,880,880,880,881,881,882,883,884,885],[960,960,958,957,952,951,945,943,942,940,938,937,936,935,934,933,932,932,931,930,930,929,928,928,927,927,926,926,927,929,931,934,937,939,944,945,947,949,951,951,951,951,950,949,947,946,944,943,939,936,934,927,922]],"y":[[229,227,226,223,222,220,218,215,213,210,209,206,206,206,206,211,218,225,230,241,253,267,280,300,313,325,335,344,349,360,363,372,375,377,378,378,376,375,374,372],[308,308,308,308,308,308,307,306,306,304,303,303,302,302,301,301,300],[240,240,241,242,243,247,254,257,271,276,291,296,312,321,330,338,341,345,348,349],[279,276,275,273,270,268,267,264,263,261,261,261,264,266,270,272,273,275,279,280,282,286,287,288,290,292,293,294,294,294,294,295,295,296,297,300,302,304,307,310,314,319,321,322,322],[238,235,235,235,239,244,251,255,264,269,273,281,295,304,313,322,329,333,341],[273,274,274,273,272,270,269,268,268,267,267,267,268],[296,297,297,297,296,295,294,294,294,293],[279,277,276,275,273,272,270,270,270,271,272,275,276,282,284,291,296,301,303,307,313,314,317,318,319,320,320,320,320,320,320,319,319,318,318,318],[257,255,254,253,252,252,252,254,258,264,273,282,287,302,307,319,324,333,335,338,340,344,345,346,347],[291,290,289,288,287,287,286,285,285,284,284,283,283,283,284,286,287,291,292,296,298,301,303,304,306,307,308,309,309,309,310,310,311,313,313,315,317,320,321,324,325,326,327,328,328,328],[269,269,269,269,269,271,274,278,284,290,296,300,303,310,314,324,327,330,335,339,341,344,345,346,347],[219,212,211,209,207,205,205,203,202,202,202,203,204,205,208,210,212,216,218,220,222,226,228,230,235,236,239,242,243,243,243,243,242,241],[299,299,299,300,300,299,298,298,298],[278,275,273,272,271,269,269,267,267,266,266,266,267,268,269,272,275,277,278,281,283,287,289,292,293,295,296,297,298,299,300,300,302,305,307,309,311,313,316,318,320,325,328,331,333,335,337,337,337,337,335],[295,294,294,295,295,296,297,298,299,300,300],[324,324,324,324,324,323,323,322,322,322,322,322,322,322],[267,266,267,269,277,285,294,303,306,316,319,322,326,329,332,334,335,336],[267,268,268,268,268,269,269,269,269,270,270,271,271,272,273,274,277,279,281,286,288,294,295,297,300,301,304,305,305,305,305,305,305,305,306,307,308,311,314,318,322,325,327,329,332,333,334,334,336,336,336,336,334]],"t":[[0,6,11,13,20,24,32,43,49,60,66,77,83,94,100,110,116,127,132,144,149,160,166,177,183,194,199,210,216,227,234,244,249,262,266,277,282,283,294,296],[488,493,500,510,527,544,550,560,566,577,583,594,599,611,616,627,633],[921,932,936,941,943,950,961,966,977,984,994,1000,1011,1016,1027,1033,1044,1049,1060,1064],[1345,1360,1363,1368,1383,1394,1401,1411,1416,1427,1433,1448,1461,1466,1475,1483,1483,1494,1500,1500,1511,1516,1517,1528,1533,1544,1551,1561,1568,1578,1584,1595,1600,1611,1616,1628,1633,1644,1650,1661,1666,1678,1683,1694,1700],[1876,1887,1900,1910,1916,1928,1933,1945,1950,1950,1961,1966,1978,1984,1995,2000,2011,2018,2028],[2387,2391,2408,2417,2428,2437,2446,2450,2461,2467,2478,2483,2487],[2667,2683,2700,2711,2717,2730,2734,2742,2750,2754],[3123,3135,3138,3143,3150,3162,3167,3178,3184,3195,3201,3212,3217,3228,3235,3245,3250,3262,3267,3278,3284,3296,3300,3311,3317,3328,3334,3345,3350,3362,3367,3379,3384,3396,3400,3409],[3722,3726,3731,3732,3734,3743,3750,3762,3768,3779,3785,3792,3801,3812,3818,3829,3834,3845,3850,3851,3863,3867,3868,3879,3884],[4120,4134,4138,4143,4151,4162,4167,4168,4179,4184,4184,4196,4201,4212,4218,4229,4234,4246,4251,4262,4267,4279,4284,4284,4292,4309,4317,4329,4336,4346,4351,4362,4368,4379,4384,4392,4401,4412,4417,4429,4434,4435,4449,4451,4462,4467],[4630,4635,4646,4652,4659,4668,4679,4685,4696,4701,4712,4717,4720,4730,4735,4746,4751,4751,4763,4767,4780,4784,4789,4797,4799],[5088,5102,5106,5111,5118,5126,5135,5146,5152,5163,5168,5180,5184,5185,5196,5201,5201,5213,5218,5218,5230,5234,5235,5247,5251,5251,5263,5268,5280,5285,5296,5301,5309,5319],[5649,5659,5685,5701,5713,5719,5730,5735,5747],[6006,6018,6022,6027,6035,6047,6052,6063,6068,6080,6085,6093,6101,6113,6118,6130,6135,6135,6147,6151,6152,6164,6168,6180,6185,6193,6203,6213,6218,6230,6235,6235,6247,6252,6260,6268,6269,6280,6285,6285,6297,6302,6314,6318,6330,6335,6347,6352,6364,6369,6380],[6791,6806,6812,6820,6830,6835,6843,6852,6863,6868,6881],[7037,7052,7068,7080,7086,7097,7102,7114,7119,7130,7135,7135,7147,7152],[9405,9421,9436,9449,9453,9466,9470,9481,9486,9498,9503,9503,9515,9519,9532,9536,9548,9553],[9782,9788,9811,9820,9831,9836,9845,9853,9853,9865,9870,9870,9882,9886,9887,9899,9903,9915,9920,9932,9936,9948,9953,9957,9966,9970,9982,9986,10031,10039,10046,10053,10065,10070,10081,10087,10090,10099,10103,10115,10119,10132,10136,10137,10149,10153,10154,10166,10170,10170,10182,10186,10198]],"version":"2.0.0"}

    Step 2: Find the change in x: 3-1=2{"x":[[186,187,188,190,192,195,206,210,222,229,236,240,243,244,244,244,244,242,239,237,233,229,225,223,218,217,216,217,221,223,231,238,241,247,252,256,258,260,260,260,259,256,251,248,237,232,228,217,213,199,190,187],[334,338,339,342,344,347,351,364,368,380,385,395,401,406,407,410,411,411],[482,482,483,483,484,486,487,489,490,491,491,491,491,490,489,489,488,487,486,486,486,486,488,488,490,491],[629,630,631,632,637,640,643,652,665,673,681,687,691,694,696],[625,634,637,644,648,653,656,664,667,671,678,681,683,688,690,693],[796,793,792,792,792,792,793,794,796,797,801,803,810,812,819,821,826,827,828,829,829,828,825,824,820,818,815,811,806,801,799,796,794,793,793,794,796,802,806,818,823,828,839,850,860,865,870,875,882]],"y":[[252,242,239,236,234,231,223,220,215,212,211,211,212,213,214,216,220,224,229,232,237,243,249,251,258,260,266,268,274,276,283,288,290,294,299,304,307,312,317,320,323,328,333,336,344,346,348,352,353,354,352,350],[300,298,298,297,297,296,296,295,295,294,294,293,293,293,293,293,293,294],[234,232,232,231,231,232,235,241,245,255,260,273,286,291,302,307,318,327,335,338,347,350,355,356,358,358],[266,266,266,266,265,264,264,263,261,260,259,259,259,259,260],[298,297,296,295,295,294,294,293,293,293,292,292,292,291,291,291],[264,259,257,254,253,250,248,246,243,242,239,238,236,236,236,238,243,246,249,255,262,269,276,280,287,290,293,299,304,310,311,315,318,320,321,322,323,323,323,322,321,321,320,319,318,317,317,317,316]],"t":[[0,12,16,22,23,30,41,47,58,63,75,80,91,97,97,108,114,125,130,130,141,148,160,164,175,180,191,198,208,213,223,230,241,247,258,263,264,275,281,288,289,297,308,315,325,332,335,341,347,358,364,375],[2384,2395,2398,2400,2405,2407,2414,2423,2431,2442,2449,2459,2464,2475,2481,2492,2498,2509],[2733,2738,2741,2742,2762,2765,2776,2781,2793,2798,2798,2809,2815,2826,2831,2832,2843,2848,2859,2864,2876,2882,2892,2899,2909,2914],[3191,3223,3231,3241,3248,3250,3260,3265,3276,3283,3293,3299,3309,3315,3323],[3505,3516,3519,3525,3531,3532,3543,3548,3548,3560,3565,3565,3577,3581,3582,3593],[3967,3980,3987,3998,3999,4010,4015,4027,4032,4032,4043,4049,4060,4065,4077,4082,4095,4099,4101,4110,4115,4126,4132,4143,4148,4149,4160,4165,4177,4182,4182,4194,4199,4210,4218,4227,4232,4243,4248,4260,4265,4266,4274,4282,4293,4298,4299,4310,4315]],"version":"2.0.0"}.

    Step 3: Take the ratio of the change in function to the change in x:

    ΔyΔx=f3-f13-1{"x":[[51,51,52,53,56,57,62,63,68,70,76,78,80,84,86,89,93,96,98,101,103,104,105,106,107,107,108,110,110,112,114,115,117,118,119,122,124,125,129,133,135,140,141,144,145,146,146,146,145,144,140,138,132,126,119,109,100,84,74,64,55,46,39,36,33],[199,199,199,200,200,201,201,202,203,204,206,209,212,213,215,219,225,230,235,237,244,246,252,254,258,259,260,261,261,260,259,257,256,255,254,254,252,252,251,250,250,250,248,247,244,242,237,235,233,228,222,219,213,210,207,200],[63,67,72,76,91,98,116,123,130,145,160,167,183,190,196,210,216,223,234,249,256,261,262,264],[46,45,44,45,45,49,50,53,57,61,66,68,73,76,82,84,89,91,96,99,102,104,106,107,108,109,109,111,113,115,116,118,120,122,124,127,129,132,134,136,138,139,140,139,138,136,131,117,105,96,92,79,74,64,61,58,53,51,49],[174,172,171,170,170,169,169,169,169,172,174,181,186,192,197,202,206,207,209,210,211,211,211,209,206,202,198,194,192,187,185,183,182],[226,229,231,232,233,230,229,225,223,220,217,216,214,212,212,212,212,214,217,221,227,229,235],[336,341,345,347,353,357,360,367,371,378,386,392,399,402,407,413,415,416],[335,335,337,339,344,350,357,365,373,377,388,391,397,400,405,408,409],[530,530,531,531,531,532,532,532,532,530,529,528,526,525,524,521,519,517,515,514,513,510,509,507,504,502,501,498,498,497,497,497,497,497,497,497,496,495,495,494,492,491,489,488],[472,473,476,478,485,487,490,495,500,502,506,508,510],[582,578,577,574,573,571,567,563,561,558,556,555,554,553,553,553,554,556,560,562,568,570,572],[601,601,600,600,600,600,602,604,605,608,610,613,614,618,619,620,623,623,624,625,625,625,625,625,624,622,620,619,617,616,616,616,617,618,619,620,622,624,625,628,629,631,633,633,634,634,634,633,630,629,624,623,617],[665,666,666,667,668,670,675,676,681,682,683,685,685,686,686,686,686,685,682,680,678,673,670,665,660,656,655,653],[742,741,740,743,747,752,755,759,766,770,774,777,782,784,786,788,789],[881,884,885,885,885,884,881,880,878,875,873,868,866,864,860,857,855,853,852,851,849,849,849,848,848,848,848,848,848,848,847,846,846,844,843,842],[832,831,831,831,831,831,833,836,842,845,849,859,866,871,874,877,879,880],[934,928,924,922,918,916,910,908,907,906,906,906,908,909,912,914,916,923,925],[961,962,962,963,963,963,963,963,963,963,963,963,963,964],[990,992,994,995,1000,1002,1007,1008,1010,1012,1013,1013,1013,1012,1012,1010,1007,1003,1001,997,995,991,989],[565,564,565,566,571,573,586,598,604,618,624,632,657,667,686,697,719,742,765,777,811,822,854,864,889,897,904,915,924,927,932,933,934],[625,626,627,629,632,635,637,643,646,649,653,657,658,658,658,655,654,649,647,643,640,638,637,637,637,637,638,640,641,644,647,650,652,654,655,655,655,654,651,649,647,645,641,639,635,632,629],[732,734,735,738,741,749,756,762,765,771,776,781,782,784],[847,849,849,850,850,850,850,850,849,848,847,847,846,846,846,846,846,846,847]],"y":[[264,262,259,257,248,244,231,227,214,209,194,189,185,175,172,164,157,151,146,139,136,133,132,131,131,132,133,136,139,147,154,158,166,171,175,186,190,195,204,217,221,232,236,245,248,255,258,261,262,263,264,265,266,267,268,269,269,271,273,275,277,278,280,281,281],[190,191,192,193,198,200,203,210,212,217,221,224,225,226,226,226,225,222,219,217,210,207,198,195,187,185,184,181,180,181,183,186,189,193,196,199,206,209,218,227,237,241,255,261,273,277,288,291,294,299,302,303,305,305,305,304],[340,341,340,340,338,338,337,336,336,335,334,334,333,332,332,331,331,330,330,329,329,329,329,330],[457,459,460,460,459,454,453,448,443,437,429,426,418,414,404,401,393,391,385,382,380,379,379,379,380,381,383,386,391,396,399,405,409,416,422,428,434,439,444,448,452,455,458,461,463,464,467,471,473,474,474,474,474,474,473,473,471,471,470],[426,426,425,425,424,424,422,421,420,417,416,414,413,413,415,418,421,423,427,430,432,436,438,443,447,452,456,459,460,463,463,464,463],[427,426,425,423,422,421,421,423,425,428,431,432,436,442,447,450,455,460,462,464,465,464,463],[307,308,308,308,307,307,307,307,307,307,307,307,307,307,307,307,307,307],[343,345,345,345,345,345,344,343,342,341,340,339,339,338,338,338,338],[168,166,165,164,163,160,158,154,152,147,146,145,142,141,140,138,138,138,139,140,142,147,151,161,174,191,200,228,238,246,261,268,274,285,290,294,302,306,308,311,315,316,317,317],[244,244,244,242,240,239,238,236,235,234,233,233,233],[180,179,179,179,180,181,185,191,195,205,210,215,225,230,239,245,248,253,259,260,263,263,263],[210,209,209,208,207,206,204,203,202,200,199,199,198,198,198,198,199,200,201,203,204,205,207,208,211,213,216,217,220,222,223,224,225,225,226,226,227,228,228,229,230,231,234,235,236,239,242,243,245,246,247,247,248],[183,182,181,181,181,181,184,186,195,199,203,210,214,218,221,227,231,234,241,244,248,254,258,263,267,270,271,272],[218,218,218,217,216,215,215,214,213,212,212,212,211,211,211,211,211],[177,170,167,165,162,161,157,156,155,153,153,153,153,154,157,161,164,171,176,180,189,195,201,220,233,246,258,263,278,282,291,294,296,300,302,302],[247,246,245,244,243,242,241,239,237,236,235,233,232,231,231,231,231,231],[183,185,186,188,193,196,207,214,221,224,228,230,237,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    ΔyΔx=15+12=8{"x":[[84,85,85,86,87,87,88,89,89,90,90,91,91,92,93,96,98,101,105,110,116,121,125,129,132,134,139,140,144,145,147,148,149,150,150,151,153,154,155,156,158,161,163,165,167,170,171,175,176,179,180,181,183,185,186,187,188,188,189,189,188,186,183,181,175,173,164,160,147,139,129,125,119,111,107,103,99,92,84,81,80],[243,244,244,244,245,247,248,250,251,252,254,255,257,260,264,268,272,274,280,282,288,289,291,293,294,295,295,294,293,292,289,287,286,284,284,283,283,283,283,283,283,282,280,278,277,273,270,268,265,258,255,252,243],[147,168,174,189,195,210,217,231,244,251,257,269,279,284,293,296,299,304,304],[118,117,117,117,119,120,121,122,125,128,131,137,141,145,146,151,152,156,157,160,161,163,164,164,165,166,167,168,169,170,172,174,176,178,180,181,182,184,185,186,186,186,186,185,183,182,180,176,173,171,167,161,157,154,146,142,134,127,120,114,109,107,105],[227,224,224,224,226,229,234,236,240,242,245,249,251,254,256,257,257,257,256,256,252,251,246,243,240,239],[281,282,282,280,279,277,276,273,270,267,266,263,263,263,264,267,271,276,278,281,284,291,295,298],[396,395,394,394,396,400,405,408,415,418,422,429,432,437,443,445,447,450],[395,394,393,391,390,392,393,395,399,401,406,412,418,425,430,433,439,441,444,445],[575,575,575,575,574,574,573,572,572,571,570,570,569,568,566,566,565,565,565,565,565,565],[655,655,654,651,649,642,637,633,629,626,624,623,622,621,620,619,619,619,619,620,620,621,622,623,623,624,624,624,624,623,623,623,624,625,627,629,632,634,639,640,642,645,646,650,650,651,651,651,651,649,646,643,641,640,635,632,628,626],[715,712,711,712,713,717,719,726,731,736,739,742,746,749,751,753,753,752,751],[738,737,737,737,737,737,736,734,732,732,731,731,731,731,731,732,733,734,734],[824,825,826,826,828,828,829,829,829,829,829,829,829,829,829,829,829,829],[644,659,664,678,692,700,708,734,743,752,771,781,790,807,824,839,852,858,868,873,880,885,888],[733,732,731,731,730,730,730,731,732,733,736,739,743,748,753,755,760,760,761,761,759,756,752,750,748,743,739,737,732,731,729,728,727,729,732,734,742,745,749,757,761,775,779,783,791,794],[945,945,946,948,949,954,957,964,971,977,983,985,989,992,993,994],[952,951,953,954,960,962,968,971,977,983,988,990,996,998,1002],[1071,1072,1073,1074,1076,1077,1079,1082,1084,1085,1088,1089,1089,1088,1085,1082,1077,1072,1070,1065,1062,1060,1054,1052,1051,1051,1051,1052,1057,1059,1067,1069,1072,1077,1082,1085,1088,1089,1089,1089,1089,1086,1081,1076,1071,1067,1064,1060,1059,1057,1056,1056,1058,1060,1063,1068,1076,1083,1087,1090,1093]],"y":[[267,268,269,272,272,273,273,273,272,270,269,267,265,261,258,249,244,236,228,219,207,198,190,182,173,169,158,154,145,143,138,136,135,134,133,133,134,136,138,140,144,153,159,166,170,181,185,198,202,214,217,221,227,234,236,241,245,249,252,254,257,258,260,260,261,262,263,263,265,266,267,268,269,271,272,273,273,275,276,277,277],[194,194,195,198,203,210,213,220,222,225,228,230,231,232,233,232,231,230,225,223,216,214,211,207,203,200,199,199,199,199,203,208,213,219,223,235,239,254,258,262,270,278,284,287,290,295,297,300,302,305,306,307,308],[366,364,363,362,362,360,359,357,356,355,354,353,352,352,351,351,351,352,353],[483,483,484,482,479,476,474,471,464,456,447,435,427,419,416,409,407,403,402,401,401,403,404,406,408,413,416,419,425,428,434,441,447,453,459,461,464,468,470,475,476,479,480,481,482,482,483,484,484,485,486,486,487,487,488,489,490,491,492,493,493,493,491],[441,439,437,436,434,433,431,431,431,431,432,435,437,442,447,452,454,457,463,465,472,474,480,482,484,484],[442,440,439,439,440,441,442,446,450,455,458,467,470,478,480,484,486,487,487,487,486,483,481,480],[315,315,315,314,314,313,313,313,312,312,312,311,311,311,311,311,311,312],[346,346,347,347,348,348,348,348,348,348,347,347,346,345,344,344,343,343,344,344],[192,193,196,198,204,207,211,220,225,229,237,241,249,252,261,264,271,273,277,278,279,280],[194,195,195,197,198,201,204,206,207,209,209,210,210,211,212,213,214,215,218,220,222,226,228,232,234,238,241,244,245,245,246,248,247,246,245,243,242,241,240,240,240,240,241,244,246,248,251,253,255,259,263,267,269,270,273,275,277,277],[228,228,227,227,227,227,227,226,226,225,225,225,225,225,225,225,226,226,226],[207,205,204,202,201,202,204,211,217,220,223,229,235,239,243,245,247,250,251],[196,196,196,197,200,203,209,212,220,227,233,238,242,244,249,251,255,256],[337,336,336,336,335,335,335,333,332,331,329,328,327,325,323,321,320,320,320,319,319,320,321],[405,405,404,403,403,402,401,400,399,398,396,395,394,393,393,394,398,400,406,408,416,421,426,428,430,435,438,440,443,444,445,447,447,447,448,448,447,447,446,445,444,442,442,441,440,440],[322,321,321,321,320,320,320,319,319,319,319,320,320,322,322,323],[350,350,350,350,349,349,348,347,346,345,344,344,344,344,343],[284,284,284,283,282,281,279,277,274,272,266,264,259,256,253,252,251,251,251,252,253,255,260,262,265,267,271,273,279,281,288,290,292,296,301,306,310,312,316,317,319,323,326,328,329,329,329,327,326,323,316,313,303,300,297,289,282,277,275,273,272]],"t":[[0,6,16,46,52,64,70,97,102,114,119,127,131,136,148,152,164,169,181,186,198,202,214,219,230,236,247,252,264,270,280,286,286,298,302,314,333,336,336,349,352,364,369,380,386,397,403,414,420,431,436,436,447,458,463,469,481,486,497,502,514,520,530,536,547,553,564,570,581,586,597,602,603,614,619,620,631,636,647,654,664],[1005,1021,1032,1039,1047,1053,1064,1069,1070,1081,1086,1086,1098,1103,1114,1121,1131,1136,1147,1154,1164,1169,1170,1178,1186,1197,1203,1211,1219,1231,1236,1248,1253,1264,1270,1281,1286,1298,1303,1306,1314,1321,1333,1336,1337,1348,1353,1353,1365,1369,1370,1385,1386],[1725,1738,1741,1747,1754,1761,1765,1770,1778,1786,1787,1798,1803,1815,1820,1820,1832,1836,1848],[2211,2215,2221,2236,2245,2253,2253,2262,2270,2281,2286,2298,2303,2315,2322,2334,2337,2348,2355,2365,2370,2381,2386,2387,2398,2403,2404,2415,2420,2420,2432,2437,2445,2454,2465,2470,2471,2482,2487,2498,2503,2515,2520,2520,2532,2536,2537,2545,2553,2554,2565,2570,2571,2582,2587,2587,2599,2604,2615,2620,2632,2637,2649],[2947,2964,2971,2981,2987,2995,3003,3015,3020,3021,3029,3037,3037,3048,3054,3065,3070,3073,3082,3087,3099,3104,3115,3120,3129,3139],[3282,3296,3312,3322,3334,3337,3337,3349,3355,3365,3370,3382,3387,3399,3404,3416,3420,3432,3437,3438,3449,3454,3454,3466],[5498,5507,5514,5521,5538,5549,5555,5566,5571,5572,5583,5588,5588,5600,5605,5616,5621,5633],[5811,5815,5821,5822,5830,5855,5855,5866,5871,5872,5884,5888,5900,5905,5917,5921,5933,5938,5950,5955],[6410,6421,6438,6450,6455,6455,6467,6471,6472,6484,6488,6489,6500,6505,6517,6522,6533,6538,6550,6555,6555,6567],[6791,6809,6822,6833,6838,6850,6855,6867,6872,6883,6888,6889,6901,6905,6905,6922,6930,6934,6939,6950,6955,6967,6972,6984,6989,6997,7005,7017,7022,7022,7034,7038,7063,7073,7083,7089,7100,7105,7117,7122,7125,7134,7139,7151,7155,7156,7168,7172,7172,7184,7188,7197,7205,7206,7217,7222,7234,7239],[10492,10507,10523,10548,10556,10568,10573,10585,10590,10602,10606,10607,10615,10623,10635,10640,10665,10673,10685],[10819,10824,10829,10834,10843,10873,10883,10890,10902,10906,10907,10919,10923,10935,10940,10940,10953,10956,10957],[11213,11221,11225,11232,11240,11252,11257,11257,11269,11274,11285,11291,11302,11307,11319,11324,11336,11342],[11861,11865,11870,11872,11882,11886,11890,11903,11907,11910,11920,11923,11924,11932,11940,11952,11957,11969,11974,11974,11986,11991,12003],[12387,12400,12407,12419,12424,12424,12432,12440,12441,12452,12457,12457,12470,12474,12486,12491,12503,12508,12519,12524,12536,12540,12553,12557,12558,12570,12574,12586,12591,12591,12603,12607,12608,12625,12635,12640,12653,12657,12658,12670,12675,12686,12690,12691,12704,12707],[13262,13299,13303,13311,13319,13324,13324,13337,13343,13354,13358,13370,13374,13387,13391,13403],[13578,13592,13624,13635,13641,13653,13658,13658,13671,13675,13687,13691,13703,13708,13720],[14067,14072,14086,14091,14108,14108,14120,14125,14137,14142,14154,14158,14170,14174,14187,14191,14204,14208,14220,14224,14225,14237,14242,14254,14258,14258,14271,14275,14287,14291,14304,14308,14309,14321,14325,14337,14343,14355,14358,14359,14371,14375,14387,14392,14404,14408,14420,14425,14437,14441,14454,14458,14470,14475,14475,14488,14492,14504,14508,14509,14521]],"version":"2.0.0"}

    Hence the average rate of change of the function w.r.t. x between the points 3 and 1 is 8.

    Find the average rate of change of the functiony=e2t-t32{"x":[[409,409,411,413,415,424,427,437,441,449,454,458,459,460,461],[403,402,403,408,414,421,425,433,438,441,444,448,450,453,455,456],[586,586,587,588,589,590,593,594,598,602,605,610,612,613,613,612,610,609,607,605,602,598,595,591,587,584,580,578,575,575,575,577,586,598,603,608,613,624,629,635,645],[621,621,621,621,622,623,625,628,630,632,634,635,635,635,634,634,631,630,626,624,623,621,619,618,618,617,617,618,619,622,625,629,631,637,639,645,647,648,651,653,655,656,657],[679,678,677,676,675,674,674,673,672,671,671,670,670,670,670,671,673,674,677,678,681,683,687,689,693],[658,660,661,663,673,681,688,694,700,705,707,708,709],[754,751,750,749,749,750,751,754,756,762,769,777,781,794,797,806,809,812,813],[915,914,913,911,909,908,907,907,907,909,910,913,914,915,918,920,921,924,928],[895,893,894,896,898,905,908,918,921,931,933,939,941,942],[958,958,958,958,958,960,962,965,966,968,972,974,976,979,980,983,983,984,984,983,982,979,977,975,973,971,970,970,970,971,972,974,976,979,984,987,989,994,994,995,995,993,992,990,986,981,977,974,971,969,963,962,959,958],[889,890,892,894,904,908,919,929,945,954,963,968,973,980,986,988,992,994,995,996],[912,911,911,912,913,915,916,917,919,922,924,926,929,930,933,934,935,935,933,932,927,925,919,915,912,910,909,908,907,906,908,911,913,918,924,930,934,944,947,955,957,961,963,964],[211,212,214,217,221,225,228,234,236,241,245,248,250,254,260,262,265,271,274,281,285,288,289,290,290,290,288,287,285,284,282,282,281,281,280,280,278,277,271,265,258,255,250,248,245,244]],"y":[[246,245,244,243,242,240,240,238,238,238,238,238,238,239,239],[271,271,271,271,270,270,269,269,268,268,268,268,268,268,268,268],[254,255,255,255,255,255,254,254,252,251,249,246,243,240,238,234,232,231,230,230,230,231,233,237,242,249,257,265,274,282,285,289,296,297,297,296,294,290,287,284,279],[180,179,178,177,176,175,174,173,173,173,174,175,177,179,183,184,188,190,195,196,198,199,202,203,204,206,207,207,208,208,208,208,207,206,206,205,204,204,204,204,204,204,204],[158,156,155,155,156,160,162,165,168,174,178,181,188,190,196,201,204,206,208,209,209,209,207,206,201],[179,179,179,179,177,176,173,171,170,168,168,167,167],[255,255,256,256,257,257,257,257,257,257,257,256,255,253,253,252,252,252,252],[205,207,212,227,237,247,255,262,265,273,275,279,280,281,282,283,283,282,280],[242,242,242,242,241,239,238,234,233,230,229,227,227,226],[174,167,165,164,163,162,160,159,158,157,156,156,155,155,155,156,157,159,160,164,165,169,172,175,178,180,182,183,184,185,185,185,185,186,186,187,187,189,190,193,194,197,198,200,202,204,205,206,207,207,208,208,207,206],[313,313,313,313,311,311,310,309,307,307,307,306,306,306,306,306,306,306,307,307],[354,354,352,351,351,349,349,348,347,347,347,347,348,349,351,353,356,357,363,365,371,373,379,382,385,387,388,388,390,390,391,391,391,389,388,386,386,384,384,384,384,384,385,386],[237,236,238,242,249,257,260,267,268,269,268,266,265,261,255,252,250,244,241,232,228,225,224,224,225,228,238,243,260,274,290,305,314,339,353,360,365,368,371,372,373,372,369,366,363,360]],"t":[[0,16,32,42,49,60,67,77,82,94,99,107,116,116,127],[317,341,357,376,382,394,399,411,416,416,427,432,433,444,449,460],[9680,9729,9736,9738,9744,9746,9753,9764,9769,9781,9786,9797,9802,9814,9819,9831,9836,9848,9853,9853,9864,9869,9881,9887,9897,9904,9914,9920,9929,9936,9947,9961,9969,9981,9986,9986,9998,10003,10003,10014,10019],[11024,11029,11038,11045,11053,11054,11065,11070,11081,11087,11098,11103,11115,11120,11128,11137,11148,11153,11165,11170,11170,11182,11186,11187,11198,11203,11203,11220,11220,11231,11237,11248,11253,11265,11270,11282,11286,11287,11298,11303,11315,11320,11332],[11569,11574,11587,11597,11603,11615,11620,11620,11632,11636,11637,11649,11653,11654,11665,11670,11682,11687,11698,11703,11715,11720,11732,11737,11749],[11922,11938,11941,11947,11954,11965,11971,11982,11987,11999,12003,12006,12011],[14286,14301,14313,14321,14338,14338,14349,14354,14355,14369,14372,14382,14388,14399,14404,14416,14421,14433,14438],[17003,17030,17039,17050,17055,17067,17072,17084,17089,17101,17105,17117,17122,17122,17134,17139,17139,17151,17155],[17333,17347,17357,17371,17372,17380,17389,17400,17407,17417,17422,17434,17439,17439],[17712,17725,17732,17740,17751,17756,17768,17772,17773,17785,17789,17789,17801,17806,17806,17818,17823,17834,17839,17851,17856,17868,17872,17884,17889,17897,17906,17917,17922,17923,17935,17939,17940,17951,17956,17968,17973,17984,17989,18001,18006,18018,18022,18023,18035,18039,18051,18056,18056,18068,18072,18085,18089,18090],[18828,18841,18851,18856,18868,18873,18885,18889,18902,18906,18918,18923,18923,18931,18939,18951,18956,18957,18969,18973],[22031,22049,22066,22074,22074,22086,22091,22091,22103,22107,22108,22120,22124,22124,22137,22141,22153,22158,22169,22174,22186,22191,22203,22207,22216,22224,22225,22236,22241,22241,22266,22274,22275,22286,22291,22299,22308,22316,22325,22336,22341,22353,22357,22358],[1655646619458,1655646619479,1655646619495,1655646619504,1655646619512,1655646619520,1655646619529,1655646619537,1655646619553,1655646619555,1655646619563,1655646619571,1655646619579,1655646619587,1655646619596,1655646619604,1655646619606,1655646619612,1655646619621,1655646619629,1655646619637,1655646619653,1655646619660,1655646619662,1655646619671,1655646619679,1655646619687,1655646619696,1655646619704,1655646619713,1655646619721,1655646619729,1655646619737,1655646619761,1655646619762,1655646619775,1655646619780,1655646619791,1655646619797,1655646619806,1655646619813,1655646619822,1655646619829,1655646619830,1655646619837,1655646619853]],"version":"2.0.0"} between the values of t:0 and 32.

    Solution:

    Step 1: Calculate the value of the function at the points given:

    f0=e20-032=1-0=1

    f32=e2.32-12323=e3-2716

    Step 2: Calculate the difference in the t coordinates: 32-0=32{"x":[[192,192,193,193,193,194,195,197,198,202,204,209,213,216,217,218,219,219,218,218,217,215,210,206,201,199,193,191,189,188,188,189,191,193,196,199,200,202,203,205,205,205,204,202,200,195,193,186,175,171,169],[155,153,152,153,157,159,164,170,183,192,201,211,214,221,225,231,236,239,240,241],[192,193,193,194,196,198,200,205,206,211,213,216,217,217,217,217,216,215,212,211,208,203,201,198,193,191,189,185,184,183,183,184,185,190,195,200,207,214,221,225,231,237,243,246,248,252],[339,337,336,338,340,344,350,357,365,370,379,388,396,400,409,412,419,421],[495,496,497,497,495,492,491,486,484,481,481,482,483,484,487,489,492,494,499,506,512,519,526,529,535,541,545,547,551,552,552,551,548,546,543,538,536,534,531,527,524,523,519],[643,651,657,660,666,677,683,686,688,693,695,697,698,700],[637,641,642,649,652,655,659,662,669,672,675,682,684,687,691,692,694,696,698],[831,832,833,833,837,840,845,849,854,858,860,864,865,865,866,865,863,860,857,854,852,848,846,844,844,844,845,848,850,855,857,862,863,865,865,865,864,861,857,854,852,849,844,841,838,832,830,826,823],[802,802,802,803,810,814,819,831,842,848,866,871,884,888,897,902,905,906,906,905],[844,842,841,840,840,841,842,846,848,850,854,856,857,859,861,862,863,863,863,861,858,855,853,849,846,842,840,839,839,841,842,846,852,855,858,866,870,875,886,894,899]],"y":[[185,184,182,180,179,177,175,173,172,169,168,166,165,165,166,168,171,172,176,178,180,185,193,198,203,205,212,213,217,219,221,222,224,226,229,232,234,237,239,242,246,249,253,257,258,263,265,268,271,271,271],[312,313,314,314,314,314,314,314,311,310,309,307,307,307,307,307,307,309,310,315],[386,385,384,382,380,379,378,376,376,377,377,380,382,383,387,389,391,394,399,402,406,413,416,419,424,427,430,434,435,438,439,441,441,441,441,440,439,438,436,435,434,433,431,431,431,430],[293,293,293,293,293,293,292,292,291,290,289,288,287,286,285,285,284,284],[250,250,250,251,254,259,263,276,282,298,303,317,320,324,330,333,335,337,339,341,340,338,334,331,325,316,307,302,285,279,264,255,249,247,244,241,240,239,239,239,239,240,243],[267,268,269,269,269,269,269,269,269,269,269,269,269,269],[304,305,305,305,304,304,304,303,302,302,302,301,300,300,300,300,300,300,300],[205,198,197,195,192,190,189,189,188,188,188,190,191,193,195,197,202,205,208,212,214,219,221,224,226,229,231,234,236,240,242,247,249,254,255,257,260,263,266,267,268,269,271,271,271,271,271,270,268],[330,331,332,332,332,332,331,330,329,328,327,326,325,325,325,325,325,326,327,328],[396,395,395,393,392,389,388,384,384,383,382,382,382,383,386,388,390,394,397,402,407,412,415,420,424,430,432,435,436,437,438,439,439,439,439,439,439,439,437,436,435]],"t":[[0,5,18,21,33,38,38,50,55,67,72,80,88,96,105,117,121,133,138,138,150,155,167,173,183,188,200,205,217,221,234,238,250,255,267,271,284,288,289,300,306,317,323,333,338,350,355,366,382,388,396],[640,647,655,672,683,688,700,705,713,722,733,740,750,755,756,767,772,783,789,797],[1040,1047,1055,1064,1073,1083,1089,1100,1105,1117,1122,1134,1139,1139,1150,1155,1156,1167,1172,1172,1184,1189,1189,1200,1205,1206,1217,1222,1222,1234,1239,1250,1256,1267,1272,1284,1289,1300,1305,1317,1322,1334,1339,1339,1351,1355],[1699,1708,1722,1747,1756,1767,1772,1784,1789,1789,1797,1806,1817,1823,1834,1839,1851,1856],[2083,2096,2099,2114,2124,2134,2139,2151,2156,2168,2173,2184,2189,2189,2197,2206,2206,2218,2223,2231,2239,2251,2256,2256,2268,2273,2284,2290,2301,2306,2318,2323,2334,2339,2340,2351,2356,2356,2368,2373,2373,2381,2389],[2748,2748,2752,2758,2768,2773,2785,2789,2790,2802,2806,2806,2818,2820],[3008,3013,3022,3023,3031,3032,3040,3042,3052,3056,3057,3068,3073,3073,3085,3090,3090,3098,3106],[3413,3426,3430,3435,3441,3452,3458,3468,3473,3485,3490,3502,3507,3510,3519,3523,3535,3540,3556,3557,3565,3573,3585,3590,3590,3602,3607,3618,3623,3635,3640,3652,3657,3669,3673,3674,3685,3690,3698,3707,3707,3719,3723,3724,3736,3740,3741,3752,3757],[4013,4023,4026,4032,4040,4049,4049,4058,4065,4075,4085,4092,4102,4107,4119,4123,4136,4140,4152,4157],[4372,4386,4393,4402,4408,4419,4424,4436,4440,4441,4453,4457,4457,4469,4474,4474,4486,4490,4491,4503,4508,4519,4524,4536,4540,4557,4558,4565,4574,4586,4590,4603,4607,4608,4620,4624,4624,4636,4641,4652,4657]],"version":"2.0.0"}.

    Step3: Take the ratio the the change in the function and the x coordiantes:

    ΔyΔt=f32-f032-0

    ΔyΔx=e3-2716-132{"x":[[44,46,47,53,58,63,67,69,76,78,85,87,92,97,100,105,109,111,117,119,123,125,127,128,130,131,134,134,137,137,139,140,141,142,143,144,145,146,147,150,151,152,154,156,157,161,164,166,168,169,171,172,173,173,173,172,172,169,166,164,155,152,141,136,127,123,112,100,89,83,72,63,52,49,46],[195,195,196,196,196,198,199,203,204,207,210,213,214,217,219,222,226,231,233,237,239,241,242,242,242,242,241,240,239,238,238,237,237,236,236,235,234,233,232,230,228,225,224,223,221,218,217],[85,89,92,102,106,124,129,142,147,159,172,177,183,193,198,203,207,215,218,224,228,231],[188,187,186,182,180,180,181,182,185,187,189,191,194,195,200,201,203,205,205,205,205,204,203,201,200,198,197,196,195,194,194,196,198],[225,226,226,227,227,227,226,224,224,221,219,218,216,215,213,212,212,213,214,217,219,222,227,230,236,244,247],[315,314,313,313,316,317,323,325,331,334,340,345,350,352,357,359,360,361,361],[315,311,310,310,313,316,318,320,322,328,331,334,340,347,353,355,360,365,367],[440,439,440,442,446,450,454,458,462,463,466,466,466,466,465,463,462,460,456,452,448,446,444,442,438,436,434,431,430,429,429,432,436,445,448,461,466,470,479],[512,511,510,509,509,509,510,511,512,514,516,518,521,523,527,530,532,532,533,533,533,532,531,526,525,523,520,517,514,514,513,513,514,515,517,519,520,523,524,527,529,530,530,530,529,525,523,520,515,512,511,510,509],[564,563,564,566,569,574,576,585,589,596,599,604,606,607],[653,652,652,653,654,657,658,659,661,666,668,672,673,674,675,675,674,672,670,665,661,658,653,650,649,648,647,647,648,650,652,654,660,665,671,676,679,686,688,689],[713,715,715,716,718,721,725,727,731,735,742,744,748,749,751,752,753,754,754,754,754,754,753,751,750,748,747,746,744,744,742,742,741,741,741,741,741,742,743],[729,731,734,738,744,750,755,758,763,764,766,767],[661,660,661,662,670,680,691,697,702,712,721,725,732,735,739,747,750,753,755,756],[692,693,693,693,693,693,693,693,693,693,692,691,691,691,691,692,694],[736,737,738,738,736,733,732,728,726,722,721,720,718,718,718,719,720,721,723,725,727,732,736,739,741,742,743,743,740,739,733,729,726,724,723,720,720],[808,807,806,805,804,804,806,809,812,817,821,826,832,834,838,839,844],[896,898,898,899,899,899,897,896,895,895,895,895,894,894,894,894,894,894,894],[472,476,478,480,484,493,505,512,526,543,553,572,583,594,629,641,679,705,729,754,767,799,809,836,843,862,870,876,878],[620,620,620,619,619,619,621,623,625,632,634,639,643,646,648,649,649,649,649,648,647,645,644,642,641,640,640,640,641,642,643,646,648,650,651,651,652,651,649,647,642,640,634,629,625,620,619],[589,590,591,598,602,616,626,637,645,650,663,667,677,683,687,689,691,692],[633,632,631,631,631,632,634,637,641,645,647,649,653,656,657,658,658,657,654,651,648,646,641,639,636,634,633,634,635,636,637,643,646,655,659,668,675,678,686],[56,54,53,52,53,55,56,60,63,65,69,71,73,78,81,85,89,93,96,99,102,103,103,105,105,105,105,105,104,104,104,105,105,106,108,110,111,112,115,118,121,122,125,126,127,129,130,132,133,134,135,136,137,136,135,135,133,132,130,129,126,123,116,110,103,99,92,89,82,78,72,65,55,49,43,38,36,36]],"y":[[263,262,261,254,248,240,233,230,217,213,199,194,184,174,170,161,155,152,144,142,137,136,134,134,133,133,134,135,139,140,147,153,159,162,165,172,176,179,183,190,194,197,204,207,210,219,225,230,235,237,243,245,249,250,252,252,253,254,255,255,256,256,256,255,255,255,255,256,258,259,262,265,267,267,267],[192,193,194,195,196,203,206,213,214,217,218,218,218,217,216,213,210,204,202,194,190,186,183,182,183,184,187,192,198,204,211,215,228,232,239,246,255,257,259,262,264,265,265,265,264,261,259],[333,333,332,331,330,329,328,327,326,325,323,323,322,320,319,319,318,317,316,316,316,316],[421,421,421,420,418,416,415,413,412,410,410,410,410,410,413,414,421,426,430,433,435,438,442,444,446,450,451,453,454,455,454,452,450],[421,419,418,418,417,416,416,416,417,420,421,423,427,429,437,442,447,451,452,454,455,455,455,454,452,448,447],[280,280,280,279,279,279,279,279,278,278,278,277,277,277,278,279,280,281,282],[315,317,318,319,319,319,319,318,318,317,316,315,313,311,309,308,306,305,304],[263,263,263,262,260,258,256,253,251,249,244,243,237,235,233,231,230,230,230,231,233,236,238,241,248,252,257,265,269,276,279,283,286,289,289,287,286,284,280],[142,141,140,138,135,134,134,133,133,131,131,131,130,130,130,132,134,136,140,142,144,147,149,155,156,158,161,163,165,166,166,167,168,168,169,170,171,173,174,176,179,182,183,187,188,191,192,194,196,196,196,195,194],[251,252,252,252,252,252,252,252,251,250,250,249,249,249],[207,206,205,203,202,202,201,201,201,201,201,203,204,207,208,211,215,219,221,228,231,235,239,242,244,245,246,248,249,250,250,250,250,249,248,247,247,246,246,246],[188,188,187,187,187,187,187,188,188,188,188,188,187,187,187,187,188,189,190,191,194,196,198,205,211,216,222,225,232,235,242,244,245,248,251,252,253,253,254],[224,224,223,223,223,223,223,223,223,223,224,224],[288,288,288,288,288,287,285,284,284,283,282,281,280,280,280,279,279,279,279,279],[311,312,313,315,317,319,324,332,334,342,346,349,351,352,353,352,350],[311,311,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    ΔyΔt=16e3-4324

    Hence the average rate of change of the function is 16e3-4324{"x":[[240,239,238,237,237,236,236,236,236,236,236,235,235,234,234,234,233,233,233,233,234,236],[338,338,337,336,335,332,329,326,324,320,315,307,304,302,298,295,294,291,290,289,289,291,295,297,299,304,307,313,318,322,325,329,330,331,332,332,331,328,326,320,317,314,310],[405,402,401,403,405,408,411,415,417,423,424,429,430,430,430,430,429,429,426,425,423,420,418,416,411,409,405,401,398,397,397,398,401,405,414,418,429,434,437,446,450],[486,482,481,481,481,481,482,483,487,489,491,492,496,497,499,501,502,502,503,503,502,501,499,498,497,494,494,491,491,491,491,492,493,497,500,502,505,511,513,516,520,522,524,524,525,524,522,520,518,510,508,499,496,493],[552,550,551,554,560,573,579,596,602,614,618,621,628,630,633,635,639,640,641],[709,711,712,712,713,712,711,710,708,707,703,702,700,697,695,694,693,692,692,693,696,698,702,706,714,717,727,730,739,742,745,750,751,752,754,756],[729,730,730,730,730,730,730,728,727,727,726,726,725,725,725,725],[835,835,835,835,836,839,841,843,848,851,856,861,866,868,871,871,871,870,865,863,861,856,852,850,846,844,843,842,842,843,844,847,850,855,857,862,864,868,868,869,869,868,864,862,860,857,852,848,844,832,822,817],[355,361,367,372,383,403,412,419,436,445,475,486,521,546,571,584,597,611,635,660,683,693,703,730,746,758,764,773,784,786,790],[501,502,503,503,504,505,505,506,508,510,513,515,521,523,525,529,531,533,536,536,537,538,539,539,539,539,539,538,537,536,535,532,531,527,524,521,519,515,514,513,513,514,515,516,519,524,526,532,536,539,547,550,554,562,566,569,576,579,582],[612,613,613,613,614,614,614,614,613,611,610,608,607,606,606,606,607,610,613,618,627,630,638,644,651,658,661,663,668,671,673,677,679],[643,645,645,645,645,646,646,646,646,647,647,647,647,646,646,646,646,646,646,647,648,648]],"y":[[176,175,174,175,176,185,189,208,222,234,240,246,259,269,273,282,290,296,304,307,309,309],[162,161,160,159,157,157,158,162,165,172,181,199,206,212,224,231,237,249,255,266,274,280,285,286,288,289,289,289,286,282,279,269,266,263,257,254,250,247,247,247,247,248,249],[243,245,246,246,246,244,243,240,239,235,233,227,225,223,220,219,218,217,217,216,216,217,219,220,224,226,233,241,250,254,267,270,276,281,285,286,285,284,283,278,276],[156,151,149,147,146,145,142,141,137,136,135,135,134,133,133,133,134,135,137,139,140,141,145,147,148,151,152,156,157,159,161,162,163,164,165,166,167,169,170,171,174,175,178,180,184,186,188,190,191,194,195,197,197,197],[242,242,242,242,241,239,238,236,235,234,234,234,234,234,234,234,234,234,234],[178,178,178,179,182,188,194,198,201,205,213,216,219,225,229,231,235,238,242,244,247,248,249,250,250,250,248,247,246,245,245,244,244,244,244,244],[222,223,224,230,234,247,251,263,267,271,277,280,282,287,289,290],[209,206,204,203,201,200,199,199,198,197,197,197,198,199,202,204,210,212,219,221,224,228,233,234,238,239,240,243,244,246,247,248,249,251,252,255,256,261,263,268,270,271,274,276,277,278,280,280,281,282,282,282],[380,379,377,376,376,375,375,375,374,374,373,373,371,369,367,366,365,363,362,360,358,358,357,357,356,356,356,357,358,359,359],[459,456,455,451,448,447,446,443,440,438,435,435,433,433,432,433,433,434,437,439,441,446,449,451,454,459,462,465,470,472,475,479,481,485,488,492,493,496,497,499,500,501,502,502,502,503,503,502,502,501,500,500,499,498,497,497,496,495,494],[420,419,417,416,415,418,422,428,434,440,448,454,459,464,468,471,473,474,475,476,476,476,475,474,473,472,472,471,470,470,470,469,469],[439,440,441,442,443,445,453,457,461,465,473,477,481,488,491,494,501,503,505,510,516,517]],"t":[[0,6,11,17,25,40,42,54,58,71,75,75,87,92,104,108,119,125,137,143,154,171],[397,402,411,423,426,437,442,454,459,470,475,487,492,495,504,508,509,520,525,537,542,554,558,559,567,575,576,587,593,604,609,620,625,628,637,649,650,658,659,670,675,676,687],[981,993,1009,1033,1043,1054,1060,1071,1076,1087,1092,1104,1108,1109,1121,1125,1126,1137,1142,1142,1154,1159,1159,1171,1175,1176,1188,1192,1204,1209,1221,1226,1237,1242,1254,1259,1271,1275,1276,1288,1292],[1523,1539,1542,1550,1551,1559,1571,1575,1588,1592,1592,1604,1609,1609,1621,1625,1626,1638,1643,1650,1655,1659,1671,1676,1679,1688,1692,1704,1709,1709,1717,1725,1726,1738,1742,1743,1755,1759,1760,1771,1775,1777,1788,1794,1807,1810,1817,1819,1826,1838,1843,1854,1859,1860],[2215,2230,2243,2253,2259,2271,2276,2288,2293,2305,2309,2309,2322,2326,2326,2338,2342,2343,2355],[2575,2579,2588,2595,2605,2609,2622,2626,2626,2639,2643,2643,2651,2659,2660,2671,2676,2676,2688,2693,2705,2710,2722,2726,2738,2743,2755,2759,2772,2776,2777,2789,2794,2795,2808,2810],[2970,3001,3010,3021,3026,3038,3043,3055,3059,3060,3072,3076,3076,3088,3093,3093],[3362,3370,3384,3394,3405,3409,3410,3422,3426,3427,3439,3444,3455,3461,3472,3477,3489,3493,3505,3509,3510,3518,3533,3538,3543,3543,3555,3559,3572,3576,3577,3589,3593,3605,3610,3622,3627,3639,3643,3655,3660,3661,3672,3676,3677,3689,3693,3694,3706,3710,3722,3727],[6230,6242,6248,6253,6261,6273,6277,6281,6290,6294,6307,6311,6323,6327,6340,6344,6344,6357,6361,6373,6377,6378,6390,6394,6406,6411,6423,6427,6440,6444,64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between the points 0 and 32.

    A bus covers a distance of 60 km from its first stop to its last stop. It takes the bus about 2 hours for the whole journey. The bus driver says that he had several stops in between and drove at different speeds at different times. The bus driver has to report the average speed he was driving at so that the authorities can be assured that he was driving under the speed limit on average. The speed limit is 35 km/h. What is his average speed under it?

    Solution:

    Step 1: Convert the given data into mathematical information. Let d be the distance covered by the bus and t be the time it takes to cover the distance.

    Step 2: Find the change in net distance covered: Δd=60-0=60{"x":[[32,4,5,24,31,28,18,7],[54,42,40,41,48,59,68,69,68,66,54],[83,103],[131,119,117,118,125,136,145,146,145,143,131],[161,187],[161,188],[231,205,203,210,227,229,219,206],[254,242,240,241,248,259,268,269,268,266,254],[-223.6922828456312,-224.513701962725,-225.3351210798188,-226.15654019691263,-227.79937843110022,-231.9064740165692,-233.54931225075683,-235.19215048494442,-239.2992460704134,-240.94208430460102,-242.58492253878862,-244.2277607729762,-247.51343724135143,-248.33485635844522,-249.97769459263282,-252.4419519439142,-254.0847901781018,-257.370466646477,-259.83472399775843,-262.2989813490398,-264.7632387003212,-266.4060769345088,-267.2274960516026,-270.5131725199778,-271.3345916370716,-273.79884898835303,-275.4416872225406,-277.0845254567282,-277.0845254567282,-277.905944573822,-278.7273636909158,-278.7273636909158,-278.7273636909158,-279.5487828080096,-279.5487828080096,-279.5487828080096,-279.5487828080096,-278.7273636909158,-277.905944573822,-277.0845254567282,-276.2631063396344,-275.4416872225406,-273.79884898835303,-272.1560107541654,-270.5131725199778,-269.691753402884,-268.8703342857902,-266.4060769345088,-263.9418195832274,-259.83472399775843,-258.1918857635708,-256.5490475293832,-251.6205328268204,-249.15627547553902,-246.6920181242576,-241.76350342169482,-238.4778269533196,-232.727893133663,-229.44221666528782,-221.22802549434982,-217.9423490259746,-212.19241520631803,-207.26390050375522,-199.04970933281723,-196.58545198153584,-194.94261374734822,-190.83551816187924,-188.37126081059782,-185.90700345931643,-182.62132699094124,-180.97848875675362,-180.15706963965982,-178.51423140547223,-176.87139317128464,-174.40713582000325,-173.58571670290942,-173.58571670290942,-172.76429758581563,-172.76429758581563,-173.58571670290942,-174.40713582000325,-175.22855493709704,-176.87139317128464,-176.87139317128464,-179.33565052256603,-180.15706963965982,-182.62132699094124,-185.08558434222263,-186.72842257641022,-188.37126081059782,-190.83551816187924,-193.29977551316063,-194.12119463025442,-196.58545198153584,-199.04970933281723,-201.51396668409862,-203.1568049182862,-205.62106226956763,-206.44248138666143,-209.7281578550366,-210.5495769721304,-213.01383432341183,-213.83525344050562,-216.29951079178701,-216.29951079178701,-217.9423490259746,-217.9423490259746,-217.9423490259746,-217.9423490259746,-217.9423490259746,-217.1209299088808,-215.47809167469322],[-114.44354027215586,-116.08637850634345,-116.08637850634345,-116.90779762343725,-116.90779762343725,-120.19347409181245,-121.01489320890624,-125.12198879437526,-125.94340791146905,-130.87192261403186,-132.51476084821945,-134.15759908240705,-137.44327555078223,-139.08611378496985,-139.90753290206365,-141.55037113625124,-143.19320937043884,-144.01462848753263,-144.01462848753263,-144.01462848753263,-143.19320937043884,-141.55037113625124,-139.08611378496985,-135.80043731659464,-132.51476084821945,-130.87192261403186,-124.30056967728144,-122.65773144309385,-116.08637850634345,-111.97928292087445,-108.69360645249925,-107.05076821831166,-102.94367263284265,-101.30083439865506,-100.47941528156126,-99.65799616446746,-98.01515793027986,-96.37231969609226,-95.55090057899845,-94.72948146190465,-93.90806234481086,-93.90806234481086,-93.08664322771706,-93.08664322771706,-93.08664322771706,-93.08664322771706,-93.90806234481086,-93.90806234481086,-94.72948146190465,-94.72948146190465,-95.55090057899845,-95.55090057899845,-96.37231969609226,-96.37231969609226,-97.19373881318606,-97.19373881318606,-98.01515793027986,-98.83657704737365,-98.83657704737365,-99.65799616446746,-100.47941528156126,-100.47941528156126,-100.47941528156126,-99.65799616446746,-98.83657704737365,-98.01515793027986,-95.55090057899845,-94.72948146190465,-93.08664322771706],[-57.76562119268367,-52.837106490120874,-47.90859178755807,-42.158657967901476,-40.515819733713876,-37.230143265338675,-35.587305031151075,-33.944466796963475,-30.658790328588278,-29.837371211494478,-29.015952094400678,-26.551694743119278,-25.730275626025477],[-64.33697412943407,-61.05129766105887,-59.40845942687127,-58.58704030977747,-55.30136384140227,-53.65852560721467,-49.551430021745674,-47.90859178755807,-42.98007708499527,-39.69440061662007,-36.40872414824488,-34.76588591405728,-31.48020944568208,-30.658790328588278,-29.015952094400678]],"y":[[8,45,110,118,92,59,50,59],[115,112,62,18,8,8,20,54,93,112,116],[82,82],[115,112,62,18,8,8,20,54,93,112,116],[73,73],[92,92],[8,46,92,118,111,65,48,59],[115,112,62,18,8,8,20,54,93,112,116],[-0.2492720162225955,0.5721471008712042,2.214985335058804,4.679242686340203,7.143500037621602,15.357691208559599,17.821948559840997,21.107625028216198,27.678977964966595,30.143235316247996,32.60749266752939,35.071750018810796,40.82168383846739,43.28594118974879,45.750198541030194,51.50013236068679,53.96438971196819,59.714323531624785,65.46425735128139,70.39277205384418,76.14270587350079,81.07122057606358,83.53547792734499,90.92824998118918,93.39250733247057,100.78527938631477,104.8923749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km. And the time taken to cover that distance: Δt=2 hrs.

    Step 3: The average speed of an object is given by the ratio of the net distance covered by the bus to the time taken to cover the distance, hence:

    Vav=ΔdΔt{"x":[[111,112,112,113,115,118,123,127,129,136,138,145,147,153,155,157,161,164,166,169,170,172,175,177,179,184,187,192,198,205,208,211,217,220,227,231,233,234,234,236,236,236,237,236],[273,269,268,265,264,259,258,251,249,246,241,236,233,232,231,230,229,229,229,229,230,231,234,235,238,243,247,249,256,258,264,266,271,274,277,278,279,281,282,283,285,287,289,290,292,294,295],[322,322,322,323,323,324,326,327,329,329,330,332,334,335,339,340,342,346,349,353,357,360,363,366,368,370,371],[487,488,490,495,501,509,516,523,529,531,540,542],[494,493,495,497,506,513,517,528,531,540,542,545],[778,778,777,774,771,768,762,755,753,744,740,735,726,722,717,714,711,709,707,706,706,707,709,710,713,715,717,721,724,726,731,735,742,749,757,761,773,776,787,790,793,799,805,811,813,821,825,827,830,832,833,833,832,830,826,823,821,816,814,810,808,807,804,802,801,799,795,793,792,788,786,785,782,781,778,777,775,774,773,773],[912,908,907,906,904,900,896,894,885,882,875,872,867,867,866,866,868,870,872,874,879,884,891,900,906,912,914,920,922,924,925,925,926,926,926,925,924,923,921,919,918,916,915,913,911,909,908,907,907,906],[744,740,742,749,753,771,779,805,814,823,840,858,877,885,894,912,921,930,937,960,966,979,984,993,997,1004],[717,716,715,714,712,711,710,709,708,708,708,709,711,712,714,716,720,722,726,731,736,741,746,749,753,758,762,764,765,769,770,774,775,778,779,780,782,784,786,788,789,792,793,793,796,797,799,802,805,806,811,813,815,818,820,825,826,830,832,832,833,832,831,827,820,814,808,800,791,786,771,766,751,743,734,729,721,713,709,704,698,695,688,688],[912,911,909,908,908,907,907,907,908,909,911,912,916,920,923,927,932,934,942],[883,879,884,888,897,914,927,937,942,952,955,962]],"y":[[200,200,201,203,208,218,234,252,262,291,301,327,334,352,356,360,365,367,367,366,363,360,351,345,338,322,312,293,272,249,241,232,216,209,191,181,174,172,170,167,165,164,164,165],[363,356,355,352,351,350,350,352,354,357,363,371,381,386,391,395,404,408,411,414,415,415,415,415,412,408,403,400,390,387,378,374,367,365,364,366,368,374,377,384,391,397,402,405,408,409,409],[369,372,373,379,382,385,395,397,403,404,405,405,405,404,399,397,393,386,381,373,365,356,350,344,339,337,337],[279,279,278,277,275,274,273,273,273,273,274,275],[309,309,309,308,306,304,304,301,301,300,299,299],[194,195,196,202,206,211,223,234,239,256,263,274,291,300,308,315,320,322,327,328,329,329,329,328,327,327,326,325,324,323,322,321,320,319,318,317,316,316,316,316,316,317,317,317,317,316,315,314,313,312,311,310,309,308,306,304,303,298,296,289,286,282,274,270,266,262,253,249,246,239,235,232,226,223,218,215,213,212,211,210],[267,262,261,260,259,258,258,259,265,268,278,282,291,294,296,298,299,299,298,297,294,290,285,277,268,259,254,240,236,226,223,221,217,216,215,214,215,216,222,230,235,245,250,261,274,285,295,302,305,307],[395,394,394,393,393,391,390,387,386,385,383,381,378,378,377,376,375,374,374,372,372,371,371,370,370,370],[538,538,538,538,538,538,538,538,538,537,536,535,533,531,528,526,521,519,514,508,502,495,488,485,479,474,469,467,466,463,461,459,458,457,457,457,457,457,458,460,461,464,466,468,472,475,481,487,495,498,509,513,517,523,525,532,533,536,537,538,539,539,539,539,539,538,538,538,538,538,537,537,536,536,536,536,538,540,542,543,545,547,548,546],[435,438,445,457,464,472,489,503,516,522,533,537,544,548,550,551,549,546,537],[476,477,476,475,472,467,463,460,458,456,456,456]],"t":[[0,14,19,29,36,45,52,62,69,79,86,96,103,112,119,119,129,135,147,152,152,164,169,169,179,186,186,199,203,214,219,222,231,236,247,252,264,269,269,281,285,286,298,316],[1078,1095,1103,1113,1120,1129,1136,1146,1153,1153,1163,1169,1181,1186,1186,1197,1202,1203,1215,1219,1219,1231,1236,1236,1246,1253,1265,1270,1279,1287,1296,1303,1316,1319,1330,1336,1345,1353,1353,1363,1369,1382,1386,1396,1403,1414,1420],[1675,1711,1719,1728,1728,1736,1746,1753,1763,1769,1770,1780,1786,1796,1803,1803,1813,1819,1820,1830,1837,1847,1854,1863,1870,1880,1896],[2373,2395,2403,2413,2421,2429,2437,2446,2454,2462,2478,2478],[2669,2676,2703,2715,2720,2728,2738,2747,2754,2763,2771,2774],[3223,3240,3253,3264,3270,3270,3281,3287,3297,3303,3314,3320,3330,3337,3346,3354,3364,3370,3380,3387,3397,3417,3420,3432,3437,3437,3447,3453,3454,3466,3470,3471,3478,3487,3497,3503,3514,3521,3531,3537,3538,3549,3553,3564,3570,3581,3587,3597,3603,3613,3620,3631,3638,3647,3654,3664,3670,3681,3688,3697,3703,3704,3712,3722,3723,3734,3737,3738,3748,3753,3754,3765,3770,3771,3781,3787,3798,3804,3814,3821],[4344,4353,4356,4362,4363,4370,4381,4387,4398,4405,4414,4421,4431,4437,4437,4448,4464,4470,4471,4479,4487,4498,4504,4514,4521,4531,4538,4548,4554,4563,4571,4574,4581,4587,4588,4598,4620,4621,4631,4637,4646,4654,4654,4665,4671,4681,4688,4698,4705,4708],[5112,5129,5138,5148,5155,5165,5172,5182,5187,5188,5199,5204,5215,5221,5221,5232,5237,5238,5249,5254,5265,5271,5271,5282,5287,5295],[5776,5781,5788,5799,5804,5815,5830,5838,5854,5913,5923,5925,5930,5938,5946,5949,5954,5965,5971,5982,5988,5996,6004,6005,6015,6021,6032,6038,6040,6049,6054,6065,6072,6082,6088,6089,6099,6104,6115,6121,6132,6138,6139,6149,6154,6155,6166,6171,6182,6188,6199,6204,6207,6216,6221,6232,6239,6249,6254,6266,6271,6288,6298,6304,6315,6322,6334,6338,6349,6354,6366,6371,6382,6388,6399,6404,6413,6421,6422,6432,6438,6439,6456,6471],[6798,6838,6846,6855,6865,6868,6872,6883,6888,6899,6905,6905,6916,6921,6932,6939,6949,6956,6966],[7155,7172,7188,7199,7205,7216,7221,7233,7238,7249,7255,7266]],"version":"2.0.0"}=602kmhr

    Vav=30 km/hr

    Hence the average speed at which the driver was driving was 30 km/hr, which is below the average speed limit of 35 km/hr.

    Average Rate of Change - Key takeaways

    • The Average Rate of Change of a quantity relative to another is the measure of how much the quantity changes in a given interval per unit change of the other quantity.
    • In a mathematical sense, the Average Rate of Change of a functiony=fx{"x":[[191,190,190,190,190,190,191,191,192,192,194,195,197,198,200,202,204,206,208,215,218,224,227,231,237,240,247,253,258,261,267,269,274,274,276,277,278,278,279,279,280,281,281,281,282,282,282,282,282,281,278,276,274,258,254,251,247,240,233,227],[372,373,374,377,379,386,399,404,418,423,437,441,445,452,455,462,468,469,471],[408,400,399,401,403,407,410,419,422,433,438,452,456,468,470,473,477,479,481],[593,597,597,598,598,598,597,595,594,593,592,588,584,580,576,573,570,569,569,569,568,568,568,567,565,564,562,558,556,551,549,547,543],[536,541,545,550,555,561,568,572,580,589,596,604,611,614],[670,671,670,668,663,661,657,652,645,642,641,641,642,650,656,659,668,675,683,686,689],[709,709,709,709,715,723,726,728,733,739,742,743,743,740,736,730,726,722,720,720],[770,771,772,772,771,767,763,756,754,749,747,749,751,753,760,765,775,783,797,801],[835,839,841,843,851,859,865,868,871,871,870,864,855,852,845,833,829]],"y":[[254,255,257,259,262,274,284,295,305,310,324,328,339,343,346,351,353,354,355,355,355,352,350,348,343,339,332,325,317,313,300,296,285,283,279,279,280,281,289,292,307,312,325,332,350,366,382,390,412,420,441,446,452,442,445,448,450,452,451,447],[292,292,292,292,291,290,288,288,288,288,288,288,289,290,290,292,295,296,299],[337,339,339,339,338,338,338,337,336,335,334,333,332,331,331,331,331,331,331],[246,231,228,225,220,217,216,215,215,216,218,222,230,240,253,266,281,298,323,340,357,365,382,398,412,419,425,443,447,460,463,465,468],[384,373,370,368,367,366,365,364,364,363,363,363,363,363],[271,262,260,261,266,268,275,284,303,318,341,347,359,376,381,384,389,390,391,391,390],[321,319,317,316,312,310,310,310,310,314,318,326,331,340,347,355,360,366,368,369],[322,319,316,315,315,317,321,330,333,344,355,362,364,366,369,370,370,369,364,362],[257,253,253,254,261,271,285,296,314,320,340,361,382,389,402,420,425]],"t":[[0,6,19,24,35,41,51,58,66,75,84,91,101,108,108,118,124,124,135,141,151,158,158,168,174,175,185,191,201,208,218,224,235,241,251,258,274,285,291,301,308,318,324,325,335,341,349,358,368,375,384,391,392,420,424,425,435,441,452,458],[773,780,784,791,799,813,818,826,834,841,850,858,858,867,874,884,891,901,908],[1040,1051,1058,1067,1075,1084,1092,1101,1109,1118,1126,1136,1141,1151,1158,1158,1168,1175,1179],[1407,1414,1418,1425,1433,1441,1442,1450,1458,1459,1468,1475,1485,1491,1501,1508,1518,1525,1535,1542,1551,1558,1568,1575,1584,1586,1592,1601,1608,1620,1625,1625,1637],[1773,1785,1792,1801,1808,1817,1825,1825,1835,1843,1851,1859,1872,1875],[2128,2139,2143,2161,2168,2175,2185,2192,2201,2209,2218,2231,2235,2251,2261,2270,2275,2285,2292,2302,2305],[2477,2480,2484,2486,2500,2517,2518,2527,2535,2542,2552,2569,2575,2585,2592,2609,2620,2625,2638,2646],[2772,2784,2788,2792,2802,2818,2825,2835,2842,2852,2870,2876,2884,2886,2892,2909,2918,2925,2942,2948],[3127,3135,3142,3150,3159,3169,3185,3192,3202,3212,3219,3239,3242,3253,3259,3268,3275]],"version":"2.0.0"} over a interval a,b is given as the ratio of change in the function over the interval to the change in the value of the endpoints.
    • The rate of change of a function between two points is equal to the gradient of the line formed by joining the two points.
    • The formula for the Average rate of change of a function between two points is given by ΔyΔx=fb-fab-a{"x":[[39,40,42,43,45,47,50,52,57,61,65,70,76,79,85,89,93,96,98,101,103,104,105,105,105,106,106,106,106,107,109,111,112,117,118,121,121,123,124,126,127,128,128,129,129,127,124,123,117,114,105,102,91,84,77,69,66,56,53,45,43,41,38,37],[172,173,173,173,175,175,178,179,183,185,188,190,192,196,199,201,206,208,212,213,215,217,217,218,218,218,218,218,217,216,215,214,213,211,211,208,207,203,200,195,191,188,181,178,170,169],[77,84,87,94,98,103,113,124,129,141,147,154,166,172,178,195,205,214,221,224,231,233,234],[47,48,50,53,57,60,62,66,68,73,74,77,79,82,85,87,88,92,93,96,97,98,99,100,101,101,103,103,105,105,107,109,109,112,113,116,117,119,119,121,121,121,121,121,120,116,114,109,106,100,93,86,82,71,68,59,56,51],[181,181,182,182,185,188,193,198,202,204,208,208,207,204,201,196,194,188,186,181,180,179,178,178],[234,234,234,233,232,231,229,228,224,222,218,216,213,212,212,213,214,219,223,229,233,236,243,246,250],[298,297,296,295,295,296,298,300,305,307,315,318,327,331,336,337,339,341,343,344,346,347],[298,298,299,301,305,308,311,315,326,329,335,339,344,349,353],[429,428,427,426,423,421,417,413,408,404,400,397,395,392,391,391,391,391,391,392,393,393,393,394,393,393,391,389,387,386,385,384],[375,376,378,380,384,389,391,394,396,402,403,405],[449,445,444,440,439,435,433,431,430,430,431,432,437,439,446,449,454,456,458,461],[478,478,478,477,476,475,474,473,473,473,473,473,473,473,473,474,476,478,479,484,485,491,493,498,501,503,503,503,502,501,496,492,488,484,481,479,479],[531,532,534,535,536,537,538,539,539,539,538,537,536,533,530,528,522,521,517],[558,561,566,568,574,578,582,585,588,589,590,593,594],[661,662,662,662,661,659,657,656,652,651,647,645,642,640,639,637,636,636,635,635,635,635,635,635,636,636,635,633,633,631],[628,630,631,632,634,638,643,645,651,653,657,660],[694,692,690,689,684,682,680,677,675,673,672,671,671,672,673,677,679,683,686,690],[716,716,716,716,716,716,716,715,714,714,713,709,708,703,701,698,696,694,694,694,695,698,700,703,707,710,712,714,716,719,721,722,726,727,730,732,733,733,734,734,735],[742,743,744,746,748,751,753,754,756,757,757,756,755,751,748,745],[393,393,394,395,400,407,415,426,436,448,454,474,481,506,523,542,562,582,602,613,641,650,675,682,688,699,707],[493,492,491,491,490,489,487,486,486,486,486,486,486,487,489,492,495,497,503,506,512,517,520,522,523,523,524,524,524,522,521,520,514,510,506,503,498,495,494],[556,557,558,562,566,571,576,580,585,587,593,595],[657,659,659,659,658,657,654,649,647,637,630,628,624,623,622,622,624,626,628,631,634,636,638,640,645,647,653,654,659,662,663,667,668,672,673,678,679,681]],"y":[[323,318,315,312,307,302,295,290,277,267,257,248,240,235,226,221,217,215,214,211,211,211,212,213,216,218,226,233,240,249,258,267,272,282,285,291,294,298,302,306,309,311,312,314,315,315,316,316,317,317,319,320,323,324,326,328,329,330,330,330,330,330,330,330],[265,265,268,273,281,285,295,297,301,301,302,302,301,300,297,295,289,286,279,277,274,270,268,267,266,267,270,272,282,290,298,306,310,323,326,338,341,348,353,356,358,358,358,358,353,352],[397,397,397,397,396,396,396,395,394,393,393,392,391,390,390,388,388,388,387,387,389,389,391],[536,530,525,519,512,506,504,495,493,484,481,474,470,464,459,455,453,448,447,445,444,444,444,444,446,447,453,455,462,465,475,481,485,496,499,508,511,519,521,527,531,532,535,536,537,540,542,544,545,546,547,548,548,548,548,547,547,547],[488,487,486,485,483,482,481,481,482,484,491,497,504,511,518,525,528,536,538,542,542,542,542,540],[494,493,492,491,490,490,488,488,489,490,498,501,513,521,527,531,533,536,537,537,537,537,536,535,533],[356,356,355,354,353,353,352,351,350,350,350,350,350,350,350,350,350,351,352,352,353,354],[392,391,391,389,388,387,386,384,381,380,379,378,377,376,376],[243,237,233,231,229,229,229,231,233,236,240,245,249,262,267,279,300,316,331,338,356,361,374,378,389,392,400,403,406,406,406,406],[326,326,325,323,322,321,321,320,320,320,320,319],[276,278,278,284,287,301,311,321,331,335,345,347,350,351,352,352,351,351,350,349],[285,283,282,287,290,301,310,318,321,325,330,331,333,334,337,337,335,333,332,329,328,326,325,325,326,328,330,331,335,336,340,342,344,346,346,346,345],[279,279,279,280,282,284,288,300,305,317,321,330,333,340,344,347,351,353,355],[317,316,315,315,314,314,313,313,312,312,312,312,312],[276,266,265,261,258,256,255,254,253,253,255,258,263,270,274,289,295,314,321,335,341,352,361,369,372,381,384,389,391,393],[341,340,339,338,336,335,334,333,331,330,329,328],[289,288,288,288,293,296,300,308,312,316,323,329,335,339,340,344,345,346,346,346],[318,317,315,313,311,310,309,307,306,305,305,305,305,310,313,318,323,326,329,330,331,331,330,328,326,324,322,321,320,318,317,317,317,317,321,324,326,329,330,332,332],[292,292,295,300,307,314,318,322,326,335,338,345,348,354,360,364],[447,446,445,444,440,438,436,435,434,434,434,435,436,436,436,434,432,429,427,425,423,422,420,420,420,420,420],[483,490,493,497,501,512,524,532,536,544,546,548,549,549,547,544,541,539,536,535,534,534,536,537,539,540,543,545,546,549,550,551,555,557,558,558,557,555,554],[523,524,525,525,524,523,522,522,521,521,521,521],[521,517,514,510,507,504,503,502,502,506,511,514,521,524,530,532,535,536,536,536,535,534,533,532,528,527,524,523,522,523,524,529,530,533,534,535,534,534]],"t":[[0,16,20,24,34,41,50,58,69,74,86,91,105,108,119,124,136,142,150,158,167,174,200,208,218,225,234,242,250,258,270,275,287,291,301,308,308,320,324,333,341,352,359,369,375,417,428,433,441,451,458,467,475,485,491,503,508,521,524,535,541,542,552,558],[986,1004,1010,1018,1026,1035,1041,1051,1058,1069,1075,1075,1084,1092,1103,1108,1117,1125,1136,1141,1142,1150,1158,1159,1168,1184,1191,1200,1208,1217,1225,1234,1243,1255,1258,1268,1275,1285,1292,1301,1309,1325,1330,1335,1341,1350],[1695,1708,1711,1717,1725,1725,1735,1743,1751,1758,1759,1772,1775,1775,1784,1792,1803,1809,1819,1825,1835,1842,1854],[2323,2329,2335,2342,2351,2358,2368,2375,2385,2392,2401,2409,2409,2421,2425,2438,2442,2451,2459,2468,2475,2488,2492,2504,2509,2517,2525,2537,2542,2552,2559,2572,2575,2587,2592,2602,2609,2618,2625,2635,2642,2652,2659,2659,2669,2675,2685,2692,2693,2702,2709,2722,2725,2735,2742,2753,2759,2771],[5368,5386,5393,5393,5403,5410,5419,5426,5438,5443,5454,5460,5469,5476,5486,5493,5507,5510,5520,5526,5535,5543,5543,5552],[5757,5774,5778,5785,5793,5803,5810,5820,5826,5840,5843,5856,5862,5872,5877,5887,5894,5903,5910,5919,5927,5927,5936,5943,5944],[8798,8810,8813,8828,8844,8861,8875,8881,8887,8894,8904,8911,8920,8928,8937,8944,8945,8956,8963,8971,8978,8986],[9321,9344,9353,9361,9373,9378,9378,9387,9394,9404,9411,9412,9423,9428,9438],[10041,10048,10054,10061,10072,10078,10088,10095,10107,10111,10123,10128,10141,10145,10154,10161,10171,10178,10190,10195,10208,10211,10220,10228,10239,10245,10255,10262,10271,10279,10287,10291],[10476,10485,10489,10495,10505,10511,10520,10522,10528,10538,10545,10550],[10895,10897,10903,10912,10923,10928,10939,10945,10955,10962,10973,10978,10988,10995,11007,11012,11021,11028,11029,11037],[11228,11244,11254,11272,11278,11292,11295,11307,11312,11312,11321,11329,11329,11340,11345,11371,11378,11390,11395,11407,11412,11424,11429,11439,11445,11455,11462,11472,11479,11488,11495,11507,11512,11522,11529,11539,11546],[11734,11750,11754,11755,11762,11763,11771,11779,11789,11795,11806,11812,11822,11829,11840,11845,11859,11862,11870],[12142,12160,12170,12179,12189,12195,12206,12212,12223,12229,12229,12238,12246],[12545,12562,12571,12579,12589,12596,12609,12612,12623,12629,12638,12646,12656,12663,12671,12679,12688,12696,12705,12712,12713,12724,12729,12742,12746,12756,12762,12774,12779,12787],[12941,12946,12951,12955,12963,12973,12979,12990,12996,13006,13012,13023],[13328,13347,13354,13363,13373,13379,13380,13390,13396,13396,13405,13413,13422,13430,13438,13446,13456,13463,13476,13479],[13670,13674,13677,13680,13689,13696,13696,13707,13713,13713,13724,13729,13740,13746,13757,13763,13772,13779,13789,13796,13809,13822,13829,13839,13846,13855,13863,13863,13872,13880,13880,13892,13896,13906,13913,13923,13929,13940,13946,13957,13963],[14108,14122,14130,14139,14146,14156,14163,14163,14172,14180,14189,14196,14197,14205,14213,14221],[14704,14720,14723,14730,14739,14747,14757,14763,14774,14780,14792,14796,14805,14813,14825,14830,14839,14846,14857,14863,14877,14880,14890,14898,14899,14910,14913],[15313,15330,15333,15337,15338,15347,15358,15363,15374,15380,15391,15397,15407,15413,15424,15430,15441,15447,15458,15463,15473,15480,15490,15497,15497,15509,15513,15514,15523,15530,15531,15541,15547,15558,15563,15573,15580,15590,15597],[15825,15843,15856,15864,15874,15880,15891,15899,15906,15914,15924,15930],[16234,16243,16248,16257,16264,16276,16281,16290,16297,16314,16324,16331,16340,16347,16357,16364,16373,16380,16391,16397,16406,16414,16415,16425,16430,16441,16448,16460,16464,16475,16480,16490,16497,16507,16514,16526,16531,16535]],"version":"2.0.0"}.
    Frequently Asked Questions about Average Rate of Change

    What is the meaning of average rate of change?

    The Average Rate of Change of a quantity relative to another is the measure of how much the quantity changes in a given interval per unit change of the other quantity.

    How do you find average rate of change?

    The Average Rate of Change of a function y =  f(x) over an interval [a, b] is given as the ratio of change in the function over the interval to the change in the value of the endpoints.

    What is an example of average rate of change?

    An example of the average rate of change is the average distance covered in a certain time interval by a car. 

    What is the formula for calculating the average rate of change?

    The formula to find the average rate of change is to take the ratio of the change of the two variables over the interval. 

    How to graph the average rate of change?

    To graph the graph of average rate of change between two points, plot the two endpoints and draw a line joining them. The slope of the line will be the average rate of change.

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