Hiking can be a way of connecting calculus with nature. Walking up a hill feels completely different from walking it down, and in most instances, it is easier to go up than go down. Not to mention that some hills can be steeper than others!
By just taking a look at a hill you can know beforehand if it will be easy to climb or not, its shape is actually giving you information!
Rather than thinking of a hill, you can picture a graph instead. The shape and the derivatives of a function can give you some crucial information about its behavior, so here we will look at how Derivatives and the Shape of a Graph are related.
Derivatives and the Shape of a Graph Meaning
Talking about the shape of a graph might sound a little vague at first. What kind of shapes are present in a graph?
In Calculus, the shape of a graph refers to important features of different sections of the graph:
- Is it increasing or decreasing?
- Is it above or below the x-axis?
- Does it have a steep slope?
- Its tangent lines lie below or above the graph?
The above features might apply at different sections, or intervals, of the graph. A graph can be increasing in one interval and then decrease in another!
Relationship between Derivatives and the Shape of a Graph
Now that you have seen the meaning of the shape of a graph in Calculus, you might be wondering how derivatives are involved.
Derivatives measure change, so having the derivative of a function is key to knowing how its graph is changing.
The first derivative tells you whether the graph is increasing or decreasing.
The second derivative tells you whether the graph curves up or curves down.
Knowing this information is enough to roughly sketch the graph of a function without using graphing software!
The First Derivative Effects on the Shape of a Graph
As stated before, the first derivative of a function tells you if a function is increasing or decreasing in a certain interval. Here is how it works:
- A function \(f (x) \) is increasing in an interval where its derivative is positive, that is \( f'(x) > 0\).
- A function is decreasing in an interval where its derivative is negative, that is \( f'(x) <0\).
You can also give a graphical interpretation to the above statements, that is
- A function is increasing in an interval where the slope of a line tangent to its graph is positive.
- A function is decreasing in an interval where the slope of a line tangent to its graph is negative.
Figure 1. Decreasing and increasing intervals of a function
The points where \( f'(x)=0 \) are known as critical points. In functions with continuous derivatives, critical points are likely to be points where a function switches from increasing to decreasing or vice-versa. For more information about this topic please take a look at our article about the First Derivative Test!
The Second Derivative Effects on the Shape of a Graph
The second derivative of a function \( f(x) \) is denoted as \( f''(x)\), and it can be found by differentiating the first derivative of the function, that is, by differentiating a function twice in a row. The second derivative of a function, assuming it exists, tells you which way the function bends. There are two special words in Calculus for this idea: Concave and convex.
A function is said to be concave down, or just concave, in an interval where its second derivative is negative. The lines tangent to the function's graph inside an interval where it is concave will lie above the graph.
Figure 2. A line tangent to a function in an interval where it is convex lies above the graph
And what if the tangent lines lie below the graph?
A function is said to be concave up, or convex, in an interval where its second derivative is positive. The lines tangent to the function's graph inside an interval where it is convex will lie below the graph.
Figure 3. A line tangent to a function in an interval where it is convex lies below the graph
The concavity of a graph is independent of whether it is increasing or decreasing! You can have, for instance, a concave decreasing interval or a convex decreasing interval. All four combinations are possible!
In the following graphs, you can look at the difference between concave and convex functions with graphs.
Both functions below are increasing. However, notice how they bend differently.
Figure 4. The tangent line is above the function, so it is concave
Figure 5. The tangent line is below the function, so it is convex
Now both functions below are decreasing. Pay close attention to the bending.
Figure 6. The tangent line is above the function, so it is concave
Figure 7. The tangent line is below the function, so it is convex
The points where a function switches from concave to convex, or vice-versa, are called inflection points. For more information about this topic, please reach out to our article about the Second Derivative Test.
Examples of first and second derivative effects on the shape of a graph
The first derivative of a function can be used to find intervals where a function is increasing or decreasing. Here is an example of how this is done.
Determine the intervals where the function
\[f(x)=\frac{1}{3}x^3-4x+1\]
is increasing and/or decreasing.
Solution:
Since you need to find the intervals where the given function is increasing and/or decreasing, you should begin by finding its derivative. You can achieve this with the Power Rule, that is
\[ f'(x) = x^2-4.\]
To find where the function is increasing, you need to solve the inequality
\[f'(x)>0,\]
that is
\[x^2-4>0,\]
which you can factor as
\[(x+2)(x-2)>0.\]
The above inequality states that the product of two expressions is greater than zero. This means that both expressions have the same sign, so either
\[ x+2>0 \quad \text{and} \quad x-2>0 \]
or
\[ x+2<0 \quad \text{and} \quad x-2<0.\]
Solving the above compound inequality tells you that \( x>2 \) or \(x<-2\), so the function is increasing in the interval \( (-\infty,-2) \) and in the interval \( (2,\infty)\).
To find where the function is decreasing, you can solve the inequality
\[ f'(x) <0,\]
but since you already solved an equality like this, the remaining interval is the decreasing interval, so the function is decreasing in the interval \( (-2,2) \).
To see if your result makes sense, you should finish by taking a look at the graph of the given function.
Figure 8. The function has one decreasing interval and two increasing intervals
You can also find the concavity of the given function by using its second derivative.
Determine the intervals where the function
\[f(x)=\frac{1}{3}x^3-4x+1\]
is concave and/or convex.
Solution:
Previously, you found the derivative of the given function using the Power Rule, that is
\[ f'(x)= x^2-4.\]
By using the Power Rule again, you can find the second derivative, so
\[ f''(x)=2x.\]
To find the intervals where the function is concave you need to solve the inequality
\[ f''(x) < 0,\]
that is
\[2x <0,\]
whose solution is
\[ x<0.\]
This means that the function is concave in the interval \( (-\infty,0) \).
To find the intervals where the function is convex, you need to solve the inequality
\[ f''(x) >0, \]
but since you just solved a similar inequality you can just flip the inequality sign, so
\[ x>0\]
gives you the interval where the function is convex, that is \( (0,\infty)\).
Figure 9. The function is concave in one interval, and convex in another
Derivatives and the Shape of a Graph - Key takeaways
- The derivative of a function can tell whether a function is increasing or decreasing in an interval.
- If \( f'(x) >0\), \( f(x) \) is increasing.
- If \( f'(x) <0\), \( f(x) \) is decreasing.
- Depending on how it bends, a function can be concave down (or just concave), or it can be concave up (or convex).
- The second derivative of a function can tell whether a function is concave or convex.
- If \( f''(x) <0\), \( f(x) \) is concave.
- If \( f''(x) >0\), \( f(x) \) is convex.
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