In our discussions on derivatives, you learned about the Mean Value Theorem - an important theorem that claims that a function will take on its average value over an interval at least once. The Mean Value Theorem also has an application for integrals that is a consequence of the Mean Value Theorem and the Fundamental Theorem of Calculus.
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Jetzt kostenlos anmeldenIn our discussions on derivatives, you learned about the Mean Value Theorem - an important theorem that claims that a function will take on its average value over an interval at least once. The Mean Value Theorem also has an application for integrals that is a consequence of the Mean Value Theorem and the Fundamental Theorem of Calculus.
The Mean Value Theorem for integrals states that if a function f is continuous on the closed interval [a, b], then there is a number c such that
Clearly, the left-hand side of the equation is the area under the curve of f on the interval (a, b). The right-hand side can be thought of as the area of a rectangle. So, the theorem states that the area under the curve is equal to the area of a rectangle with a width of the interval (b - a) and a height equal to the average value of the function f. Rearranging this equation to solve for f(c), the average value, we find
Let us visualize the Mean Value Theorem for integrals geometrically.
and
Because F is continuous over the closed interval [a, b] and differentiable over the open interval (a, b), we can apply the Mean Value Theorem, which says there is a number c such that and
Using the outcomes of the Fundamental Theorem of Calculus
For the function over the interval [1, 4], find the value c (the x-value where f(x) takes on its average value).
Since f(x) is a polynomial, we know it is continuous over the interval [1, 4].
So, the average value that f(x) takes on is 14.5.
In Step 2, we found that the area under the curve is . To find the area of the rectangle, we multiply the width by the height.
Thus, the Mean Value Theorem for integrals holds.
Since and we want to find c, we can set f(x) equal to 14.5.
To solve for x, we apply the quadratic formula.
Since is outside of the interval, .
For the function , find the x-value where f(x) takes on the average value over the interval
The function sin(x) is continuous everywhere.
Use your knowledge of the unit circle to solve the trigonometric equations! Remember, is just a multiple of .
So, the average value that f(x) takes on is .
In Step 2, we found that the area under the curve is units2. To find the area of the rectangle, we multiply the width by the height.
units2
Thus, the Mean Value Theorem for integrals holds.
Since and we want to find c, we can set f(x) equal to .
Solving this equation graphically, we find that .
As a reminder
Geometrically speaking, the area under the curve is equal to the area of a rectangle with a width of b - a and a height of the average value of f(x), f(c)
The Mean Value Theorem for integrals is a consequence of the Mean Value Theorem for derivatives and the Fundamental Theorem of Calculus
The Mean Value Theorem for integrals states that if a function f is continuous on the closed interval [a, b], then the area under the curve is equal to the are of a rectangle with width b - a and height equal to the average value of the function f.
To use the Mean Value Theorem, integrate the function f over the given interval (a, b). Multiply the area under the curve by 1/(b-a) to find the average value over the given interval.
To find the value c, apply the Mean Value Theorem for integrals to find the function value at c. Then, set f(x) equal to f(c) and solve for x.
A simple example of the Mean Value Theorem for integrals is the function f(x)=x over the interval [0, 1] has an average value of 1/2 at x = 1/2. This means that the area under the of f(x) over the interval [0, 1] is equal to the area of a rectangle with a width of 1 and a height of 1/2.
The formula for the Mean Value Theorem for integrals says that the definite integral from a to b is equal to f(c)(b - a) for some c value.
The Mean Value Theorem for integrals is derived from which two theorems?
State the Mean Value Theorem for integrals in words.
The area under a function curve is equal to the area of a rectangle whose width is the interval of the function and whose height is the average value of the function over the interval.
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