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Derivatives and the Shape of a Graph

Explain how the sign of the first derivative affects the shape of a function’s graph.

The First Derivative Test

Corollary \(3\) of the Mean Value Theorem showed that if the derivative of a function is positive over an interval \(I\) then the function is increasing over \(I.\) On the other hand, if the derivative of the function is negative over an interval \(I,\) then the function is decreasing over \(I\) as shown in the following figure.

A continuous function \(f\) has a local maximum at point \(c\) if and only if \(f\) switches from increasing to decreasing at point \(c.\) Similarly, \(f\) has a local minimum at \(c\) if and only if \(f\) switches from decreasing to increasing at \(c.\) If \(f\) is a continuous function over an interval \(I\) containing \(c\) and differentiable over \(I,\) except possibly at \(c,\) the only way \(f\) can switch from increasing to decreasing (or vice versa) at point \(c\) is if \({f}^{'}\) changes sign as \(x\) increases through \(c.\) If \(f\) is differentiable at \(c,\) the only way that \({f}^{'}\) can change sign as \(x\) increases through \(c\) is if \({f}^{'}(c)=0.\) Therefore, for a function \(f\) that is continuous over an interval \(I\) containing \(c\) and differentiable over \(I,\) except possibly at \(c,\) the only way \(f\) can switch from increasing to decreasing (or vice versa) is if \(f'(c)=0\) or \({f}^{'}(c)\) is undefined. Consequently, to locate local extrema for a function \(f,\) we look for points \(c\) in the domain of \(f\) such that \(f'(c)=0\) or \({f}^{'}(c)\) is undefined. Recall that such points are called critical points of \(f.\)

Note that \(f\) need not have local extrema at a critical point. The critical points are candidates for local extrema only. In , we show that if a continuous function \(f\) has a local extremum, it must occur at a critical point, but a function may not have a local extremum at a critical point. We show that if \(f\) has a local extremum at a critical point, then the sign of \({f}^{'}\) switches as \(x\) increases through that point.

Using , we summarize the main results regarding local extrema.

If a continuous function \(f\) has a local extremum, it must occur at a critical point \(c.\)
The function has a local extremum at the critical point \(c\) if and only if the derivative \({f}^{'}\) switches sign as \(x\) increases through \(c.\)
Therefore, to test whether a function has a local extremum at a critical point \(c,\) we must determine the sign of \({f}^{'}(x)\) to the left and right of \(c.\)

This result is known as the first derivative test.

We can summarize the first derivative test as a strategy for locating local extrema.

Now let’s look at how to use this strategy to locate all local extrema for particular functions.

Condensed — the full section is in OpenStax Calculus Volume 1.

Concavity and Points of Inflection

We now know how to determine where a function is increasing or decreasing. However, there is another issue to consider regarding the shape of the graph of a function. If the graph curves, does it curve upward or curve downward? This notion is called the concavity of the function.

(a) shows a function \(f\) with a graph that curves upward. As \(x\) increases, the slope of the tangent line increases. Thus, since the derivative increases as \(x\) increases, \({f}^{'}\) is an increasing function. We say this function \(f\) is concave up. (b) shows a function \(f\) that curves downward. As \(x\) increases, the slope of the tangent line decreases. Since the derivative decreases as \(x\) increases, \({f}^{'}\) is a decreasing function. We say this function \(f\) is concave down.

In general, without having the graph of a function \(f,\) how can we determine its concavity? By definition, a function \(f\) is concave up if \({f}^{'}\) is increasing. From Corollary \(3,\) we know that if \({f}^{'}\) is a differentiable function, then \({f}^{'}\) is increasing if its derivative \({f}^{″}(x)>0.\) Therefore, a function \(f\) that is twice differentiable is concave up when \({f}^{″}(x)>0.\) Similarly, a function \(f\) is concave down if \({f}^{'}\) is decreasing. We know that a differentiable function \({f}^{'}\) is decreasing if its derivative \({f}^{″}(x)<0.\) Therefore, a twice-differentiable function \(f\) is concave down when \({f}^{″}(x)<0.\) Applying this logic is known as the concavity test.

We now summarize, in , the information that the first and second derivatives of a function \(f\) provide about the graph of \(f,\) and illustrate this information in .

Sign of \(f'\)	Sign of \({f}^{″}\)	Is \(f\) increasing or decreasing?	Concavity
Positive	Positive	Increasing	Concave up
Positive	Negative	Increasing	Concave down
Negative	Positive	Decreasing	Concave up
Negative	Negative	Decreasing	Concave down

Condensed — the full section is in OpenStax Calculus Volume 1.

The Second Derivative Test

The first derivative test provides an analytical tool for finding local extrema, but the second derivative can also be used to locate extreme values. Using the second derivative can sometimes be a simpler method than using the first derivative.

We know that if a continuous function has local extrema, it must occur at a critical point. However, a function need not have local extrema at a critical point. Here we examine how the second derivative test can be used to determine whether a function has a local extremum at a critical point. Let \(f\) be a twice-differentiable function such that \({f}^{'}(a)=0\) and \({f}^{″}\) is continuous over an open interval \(I\) containing \(a.\) Suppose \({f}^{″}(a)<0.\) Since \({f}^{″}\) is continuous over \(I,\) \({f}^{″}(x)<0\) for all \(x\in I\) (). Then, by Corollary \(3,\) \({f}^{'}\) is a decreasing function over \(I.\) Since \({f}^{'}(a)=0,\) we conclude that for all \(x\in I,{f}^{'}(x)>0\) if \(xa.\) Therefore, by the first derivative test, \(f\) has a local maximum at \(x=a.\) On the other hand, suppose there exists a point \(b\) such that \({f}^{'}(b)=0\) but \({f}^{″}(b)>0.\) Since \({f}^{″}\) is continuous over an open interval \(I\) containing \(b,\) then \({f}^{″}(x)>0\) for all \(x\in I\) (). Then, by Corollary \(3,{f}^{'}\) is an increasing function over \(I.\) Since \({f}^{'}(b)=0,\) we conclude that for all \(x\in I,\) \({f}^{'}(x)<0\) if \(x0\) if \(x>b.\) Therefore, by the first derivative test, \(f\) has a local minimum at \(x=b.\)

Note that for case iii. when \({f}^{″}(c)=0,\) then \(f\) may have a local maximum, local minimum, or neither at \(c.\) For example, the functions \(f(x)={x}^{3},\) \(f(x)={x}^{4},\) and \(f(x)=\text{-}{x}^{4}\) all have critical points at \(x=0.\) In each case, the second derivative is zero at \(x=0.\) However, the function \(f(x)={x}^{4}\) has a local minimum at \(x=0\) whereas the function \(f(x)=\text{-}{x}^{4}\) has a local maximum at \(x,\) and the function \(f(x)={x}^{3}\) does not have a local extremum at \(x=0.\)

Let’s now look at how to use the second derivative test to determine whether \(f\) has a local maximum or local minimum at a critical point \(c\) where \({f}^{'}(c)=0.\)

Condensed — the full section is in OpenStax Calculus Volume 1.

Key Concepts

If \(c\) is a critical point of \(f\) and \({f}^{'}(x)>0\) for \(xc,\) then \(f\) has a local maximum at \(c.\)
If \(c\) is a critical point of \(f\) and \({f}^{'}(x)<0\) for \(x0\) for \(x>c,\) then \(f\) has a local minimum at \(c.\)
If \({f}^{″}(x)>0\) over an interval \(I,\) then \(f\) is concave up over \(I.\)
If \({f}^{″}(x)<0\) over an interval \(I,\) then \(f\) is concave down over \(I.\)
If \({f}^{'}(c)=0\) and \({f}^{″}(c)>0,\) then \(f\) has a local minimum at \(c.\)
If \({f}^{'}(c)=0\) and \({f}^{″}(c)<0,\) then \(f\) has a local maximum at \(c.\)
If \({f}^{'}(c)=0\) and \({f}^{″}(c)=0,\) then evaluate \({f}^{'}(x)\) at a test point \(x\) to the left of \(c\) and a test point \(x\) to the right of \(c,\) to determine whether \(f\) has a local extremum at \(c.\)

Derivatives and the Shape of a Graph

For the following exercises, analyze the graphs of \({f}^{'},\) then list all intervals where \(f\) is increasing or decreasing.

For the following exercises, analyze the graphs of \({f}^{'},\) then list all intervals where

\(f\) is increasing and decreasing and
the minima and maxima are located.

For the following exercises, analyze the graphs of \({f}^{'},\) then list all inflection points and intervals where \(f\) is concave up and concave down.

For the following exercises, draw a graph that satisfies the given specifications for the domain \(xϵ[-3,3].\) The function does not have to be continuous or differentiable.

For the following exercises, determine

intervals where \(f\) is increasing or decreasing and
local minima and maxima of \(f.\)

If needed, use a calculator to graph the functions, but show your work.

For the following exercises, determine a. intervals where \(f\) is concave up or concave down, and b. the inflection points of \(f.\)

intervals where \(f\) is increasing or decreasing,
local minima and maxima of \(f,\)
intervals where \(f\) is concave up and concave down, and
the inflection points of \(f.\)

intervals where \(f\) is increasing or decreasing,
local minima and maxima of \(f,\)
intervals where \(f\) is concave up and concave down, and
the inflection points of \(f.\) Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator.

Condensed — the full section is in OpenStax Calculus Volume 1.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Use the first derivative test to find the location of all local extrema for \(f(x)={x}^{3}-3{x}^{2}-9x-1.\) Use a graphing utility to confirm your results.

Gosi nzaghachi

Step 1. The derivative is \({f}^{'}(x)=3{x}^{2}-6x-9.\) To find the critical points, we need to find where \({f}^{'}(x)=0.\) Factoring the polynomial, we conclude that the critical points must satisfy

\[3({x}^{2}-2x-3)=3(x-3)(x+1)=0.\]

Therefore, the critical points are \(x=3,-1.\) Now divide the interval \((\text{-}\infty ,\infty )\) into the smaller intervals \((\text{-}\infty ,-1),(-1,3)\ \text{and}\ (3,\infty ).\)

Step 2. Since \(f'\) is a continuous function, to determine the sign of \({f}^{'}(x)\) over each subinterval, it suffices to choose a point over each of the intervals \((\text{-}\infty ,-1),(-1,3)\ \text{and}\ (3,\infty )\) and determine the sign of \({f}^{'}\) at each of these points. For example, let’s choose \(x=-2,x=0,\ \text{and}\ x=4\) as test points.

Interval	Test Point	Sign of \({f}^{'}(x)=3(x-3)(x+1)\) at Test Point	Conclusion
\((\text{-}\infty ,-1)\)	\(x=-2\)	\((\text{+})(\text{-})(\text{-})=+\)	\(f\) is increasing.
\((-1,3)\)	\(x=0\)	\((\text{+})(\text{-})(\text{+})=\text{-}\)	\(f\) is decreasing.
\((3,\infty )\)	\(x=4\)	\((\text{+})(\text{+})(\text{+})=+\)	\(f\) is increasing.

Step 3. Since \({f}^{'}\) switches sign from positive to negative as \(x\) increases through \(-1,f\) has a local maximum at \(x=-1.\) Since \({f}^{'}\) switches sign from negative to positive as \(x\) increases through \(3,f\) has a local minimum at \(x=3.\) These analytical results agree with the following graph.

Use the first derivative test to locate all local extrema for \(f(x)=\text{-}{x}^{3}+\frac{3}{2}{x}^{2}+18x.\)

Gosi nzaghachi

\(f\) has a local minimum at \(-2\) and a local maximum at \(3.\)

Use the first derivative test to find the location of all local extrema for \(f(x)=5{x}^{1\text{/}3}-{x}^{5\text{/}3}.\) Use a graphing utility to confirm your results.

Gosi nzaghachi

Step 1. The derivative is

\[{f}^{'}(x)=\frac{5}{3}{x}^{-2\text{/}3}-\frac{5}{3}{x}^{2\text{/}3}=\frac{5}{3{x}^{2\text{/}3}}-\frac{5{x}^{2\text{/}3}}{3}=\frac{5-5{x}^{4\text{/}3}}{3{x}^{2\text{/}3}}=\frac{5(1-{x}^{4\text{/}3})}{3{x}^{2\text{/}3}}.\]

The derivative \({f}^{'}(x)=0\) when \(1-{x}^{4\text{/}3}=0.\) Therefore, \({f}^{'}(x)=0\) at \(x=\text{\pm }1.\) The derivative \({f}^{'}(x)\) is undefined at \(x=0.\) Therefore, we have three critical points: \(x=0,\) \(x=1,\) and \(x=-1.\) Consequently, divide the interval \((\text{-}\infty ,\infty )\) into the smaller intervals \((\text{-}\infty ,-1),(-1,0),(0,1),\) and \((1,\infty ).\)

Step 2: Since \({f}^{'}\) is continuous over each subinterval, it suffices to choose a test point \(x\) in each of the intervals from step \(1\) and determine the sign of \({f}^{'}\) at each of these points. The points \(x=-2,x=-\frac{1}{2},x=\frac{1}{2},\ \text{and}\ x=2\) are test points for these intervals.

Interval	Test Point	Sign of \({f}^{'}(x)=\frac{5(1-{x}^{4\text{/}3})}{3{x}^{2\text{/}3}}\) at Test Point	Conclusion
\((\text{-}\infty ,-1)\)	\(x=-2\)	\(\frac{(\text{+})(\text{-})}{+}=\text{-}\)	\(f\) is decreasing.
\((-1,0)\)	\(x=-\frac{1}{2}\)	\(\frac{(\text{+})(\text{+})}{+}=+\)	\(f\) is increasing.
\((0,1)\)	\(x=\frac{1}{2}\)	\(\frac{(\text{+})(\text{+})}{+}=+\)	\(f\) is increasing.
\((1,\infty )\)	\(x=2\)	\(\frac{(\text{+})(\text{-})}{+}=\text{-}\)	\(f\) is decreasing.

Step 3: Since \(f\) is decreasing over the interval \((\text{-}\infty ,-1)\) and increasing over the interval \((-1,0),\) \(f\) has a local minimum at \(x=-1.\) Since \(f\) is increasing over the interval \((-1,0)\) and the interval \((0,1),\) \(f\) does not have a local extremum at \(x=0.\) Since \(f\) is increasing over the interval \((0,1)\) and decreasing over the interval \((1,\infty ),f\) has a local maximum at \(x=1.\) The analytical results agree with the following graph.

Use the first derivative test to find all local extrema for \(f(x)=\sqrt[3]{x-1}.\)

Gosi nzaghachi

\(f\) has no local extrema because \({f}^{'}\) does not change sign at \(x=1.\)
For the function \(f(x)={x}^{3}-6{x}^{2}+9x+30,\) determine all intervals where \(f\) is concave up and all intervals where \(f\) is concave down. List all inflection points for \(f.\) Use a graphing utility to confirm your results.

Gosi nzaghachi

To determine concavity, we need to find the second derivative \({f}^{″}(x).\) The first derivative is \(f'(x)=3{x}^{2}-12x+9,\) so the second derivative is \({f}^{″}(x)=6x-12.\) If the function changes concavity, it occurs either when \({f}^{″}(x)=0\) or \({f}^{″}(x)\) is undefined. Since \({f}^{″}\) is defined for all real numbers \(x,\) we need only find where \({f}^{″}(x)=0.\) Solving the equation \(6x-12=0,\) we see that \(x=2\) is the only place where \(f\) could change concavity. We now test points over the intervals \((\text{-}\infty ,2)\) and \((2,\infty )\) to determine the concavity of \(f.\) The points \(x=0\) and \(x=3\) are test points for these intervals.
Interval Test Point Sign of \({f}^{″}(x)=6x-12\) at Test Point Conclusion
\((\text{-}\infty ,2)\) \(x=0\) \(-\) \(f\) is concave down
\((2,\infty )\) \(x=3\) \(+\) \(f\) is concave up.
We conclude that \(f\) is concave down over the interval \((\text{-}\infty ,2)\) and concave up over the interval \((2,\infty ).\) Since \(f\) changes concavity at \(x=2,\) the point \((2,f(2))=(2,32)\) is an inflection point. confirms the analytical results.
For \(f(x)=\text{-}{x}^{3}+\frac{3}{2}{x}^{2}+18x,\) find all intervals where \(f\) is concave up and all intervals where \(f\) is concave down.

Gosi nzaghachi

\(f\) is concave up over the interval \((\text{-}\infty ,\frac{1}{2})\) and concave down over the interval \((\frac{1}{2},\infty )\)
Use the second derivative to find the location of all local extrema for \(f(x)={x}^{5}-5{x}^{3}.\)

Gosi nzaghachi

To apply the second derivative test, we first need to find critical points \(c\) where \({f}^{'}(c)=0.\) The derivative is \({f}^{'}(x)=5{x}^{4}-15{x}^{2}.\) Therefore, \({f}^{'}(x)=5{x}^{4}-15{x}^{2}=5{x}^{2}({x}^{2}-3)=0\) when \(x=0,\text{\pm }\sqrt{3}.\)
To determine whether \(f\) has local extrema at any of these points, we need to evaluate the sign of \({f}^{″}\) at these points. The second derivative is
\[{f}^{″}(x)=20{x}^{3}-30x=10x(2{x}^{2}-3).\]
In the following table, we evaluate the second derivative at each of the critical points and use the second derivative test to determine whether \(f\) has a local maximum or local minimum at any of these points.
\(x\) \({f}^{″}(x)\) Conclusion
\(\text{-}\sqrt{3}\) \(-30\sqrt{3}\) Local maximum
\(0\) \(0\) Second derivative test is inconclusive
\(\sqrt{3}\) \(30\sqrt{3}\) Local minimum
By the second derivative test, we conclude that \(f\) has a local maximum at \(x=\text{-}\sqrt{3}\) and \(f\) has a local minimum at \(x=\sqrt{3}.\) The second derivative test is inconclusive at \(x=0.\) To determine whether \(f\) has local extrema at \(x=0,\) we apply the first derivative test. To evaluate the sign of \({f}^{'}(x)=5{x}^{2}({x}^{2}-3)\) for \(x\in (\text{-}\sqrt{3},0)\) and \(x\in (0,\sqrt{3}),\) let \(x=-1\) and \(x=1\) be the two test points. Since \({f}^{'}(-1)<0\) and \({f}^{'}(1)<0,\) we conclude that \(f\) is decreasing on both intervals and, therefore, \(f\) does not have local extrema at \(x=0\) as shown in the following graph.
Consider the function \(f(x)={x}^{3}-(\frac{3}{2}){x}^{2}-18x.\) The points \(c=3,-2\) satisfy \({f}^{'}(c)=0.\) Use the second derivative test to determine whether \(f\) has a local maximum or local minimum at those points.

Gosi nzaghachi

\(f\) has a local maximum at \(-2\) and a local minimum at \(3.\)
If \(c\) is a critical point of \(f(x),\) when is there no local maximum or minimum at \(c?\) Explain.
For the function \(y={x}^{3},\) is \(x=0\) both an inflection point and a local maximum/minimum?

Gosi nzaghachi

It is not a local maximum/minimum because \({f}^{'}\) does not change sign
For the function \(y={x}^{3},\) is \(x=0\) an inflection point?
Is it possible for a point \(c\) to be both an inflection point and a local extremum of a twice differentiable function?

Gosi nzaghachi

No
Why do you need continuity for the first derivative test? Come up with an example.
Explain whether a concave-down function has to cross \(y=0\) for some value of \(x.\)

Gosi nzaghachi

False; for example, \(y=\sqrt{x}.\)
Explain whether a polynomial of degree \(2\) can have an inflection point.
\(f(x)>0,{f}^{'}(x)>0\) over \(x>1,-3
\({f}^{'}(x)>0\) over \(x>2,-3

Gosi nzaghachi

Answers will vary
\({f}^{″}(x)<0\) over \(-10,-3
There is a local maximum at \(x=2,\) local minimum at \(x=1,\) and the graph is neither concave up nor concave down.

Gosi nzaghachi

Answers will vary
There are local maxima at \(x=\text{\pm }1,\) the function is concave up for all \(x,\) and the function remains positive for all \(x.\)
\(f(x)=\text{sin}\ x+{\text{sin}}^{3}x\) over \(\text{-}\pi

Gosi nzaghachi

a. Increasing over \(-\frac{\pi }{2}\frac{\pi }{2}\) b. Local maximum at \(x=\frac{\pi }{2};\) local minimum at \(x=-\frac{\pi }{2}\)
\(f(x)={x}^{2}+\text{cos}\ x\)
\(f(x)={x}^{3}-4{x}^{2}+x+2\)

Gosi nzaghachi

a. Concave up for \(x>\frac{4}{3},\) concave down for \(x<\frac{4}{3}\) b. Inflection point at \(x=\frac{4}{3}\)
\(f(x)={x}^{2}-6x\)
\(f(x)={x}^{3}-6{x}^{2}\)

Gosi nzaghachi

a. Increasing over \(x<0\) and \(x>4,\) decreasing over \(02,\) concave down for \(x<2\) d. Infection point at \(x=2\)
\(f(x)={x}^{4}-6{x}^{3}\)
\(f(x)={x}^{11}-6{x}^{10}\)

Gosi nzaghachi

a. Increasing over \(x<0\) and \(x>\frac{60}{11},\) decreasing over \(0\frac{54}{11}\) d. Inflection point at \(x=\frac{54}{11}\)
\(f(x)=x+{x}^{2}-{x}^{3}\)
\(f(x)={x}^{2}+x+1\)

Gosi nzaghachi

a. Increasing over \(x>-\frac{1}{2},\) decreasing over \(x<-\frac{1}{2}\) b. Minimum at \(x=-\frac{1}{2}\) c. Concave up for all \(x\) d. No inflection points
\(f(x)={x}^{3}+{x}^{4}\)
[T] \(f(x)=\text{sin}(\pi x)-\text{cos}(\pi x)\) over \(x=[-1,1]\)

Gosi nzaghachi

a. Increases over \(-\frac{1}{4}\frac{3}{4}\) and \(x<-\frac{1}{4}\) b. Minimum at \(x=-\frac{1}{4},\) maximum at \(x=\frac{3}{4}\) c. Concave up for \(-\frac{3}{4}\frac{1}{4}\) d. Inflection points at \(x=-\frac{3}{4},x=\frac{1}{4}\)
[T] \(f(x)=x+\text{sin}(2x)\) over \(x=[-\frac{\pi }{2},\frac{\pi }{2}]\)
[T] \(f(x)=\text{sin}\ x+\text{tan}\ x\) over \((-\frac{\pi }{2},\frac{\pi }{2})\)

Gosi nzaghachi

a. Increasing for all \(x\) b. No local minimum or maximum c. Concave up for \(x>0,\) concave down for \(x<0\) d. Inflection point at \(x=0\)
[T] \(f(x)={(x-2)}^{2}{(x-4)}^{2}\)
[T] \(f(x)=\frac{1}{1-x},x\ne 1\)

Gosi nzaghachi

a. Increasing for all \(x\) where defined b. No local minima or maxima c. Concave up for \(x<1;\) concave down for \(x>1\) d. No inflection points in domain
[T] \(f(x)=\frac{\text{sin}\ x}{x}\) over \(x=\) \([2\pi ,0)\cup (0,2\pi ]\)
\(f(x)=\text{sin}(x){e}^{x}\) over \(x=[\text{-}\pi ,\pi ]\)

Gosi nzaghachi

a. Increasing over \(-\frac{\pi }{4}\frac{3\pi }{4},x<-\frac{\pi }{4}\) b. Minimum at \(x=-\frac{\pi }{4},\) maximum at \(x=\frac{3\pi }{4}\) c. Concave up for \(-\frac{\pi }{2}\frac{\pi }{2}\) d. Infection points at \(x=\text{\pm }\frac{\pi }{2}\)
\(f(x)=(\text{ln}\ x)\sqrt{x},x>0\)
\(f(x)=\frac{1}{4}\sqrt{x}+\frac{1}{x},x>0\)

Gosi nzaghachi

a. Increasing over \(x>4,\) decreasing over \(08\sqrt[3]{2}\) d. Inflection point at \(x=8\sqrt[3]{2}\)
\(f(x)=\frac{{e}^{x}}{x},x\ne 0\)

Symbols used here

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

\infty

infinity
Not a number: "grows without bound" in limits and intervals.

x \in A,\ A \subseteq B

element of, subset
x belongs to A; every element of A is in B.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\sup,\ \inf

supremum, infimum
Least upper bound, greatest lower bound.

\pi

pi
Ratio of a circle's circumference to its diameter, 3.14159…

Euler's number
2.71828…, the base whose exponential is its own derivative.

\sin,\ \cos,\ \tan

sine, cosine, tangent
Ratios of sides in a right triangle; coordinates on the unit circle.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\sum_{k=1}^{n} a_k

summation
Add a_k for k = 1 up to n.

\lim_{x \to a} f(x)

limit
The value f(x) approaches as x approaches a.

f'(x),\ \frac{dy}{dx}

derivative
Instantaneous rate of change; slope of the graph.

\int f(x)\,dx,\ \int_a^b

integral
Antiderivative (indefinite) or signed area from a to b (definite).

y',\ y''

first and second derivative of y
Prime notation for derivatives with respect to x (or t).

C,\ C_1,\ C_2

arbitrary constants
Constants of integration fixed by initial conditions.

How to: Derivatives and the Shape of a Graph

Explain how the sign of the first derivative affects the shape of a function’s graph.
State the first derivative test for critical points.
Use concavity and inflection points to explain how the sign of the second derivative affects the shape of a function’s graph.
Explain the concavity test for a function over an open interval.
Explain the relationship between a function and its first and second derivatives.
State the second derivative test for local extrema.
If a continuous function
The function has a local extremum at the critical point

Questions people ask

What is a derivative in one sentence?

The slope of the graph at a point — the rate at which the output is changing there. Speed is the derivative of position.

What is an integral in one sentence?

The accumulated total of a rate — the area under the curve. Distance travelled is the integral of speed.

Why are derivatives and integrals opposites?

That is the fundamental theorem of calculus: accumulating a rate and then measuring how fast the accumulation grows gets you back the rate. Integration undoes differentiation up to a constant.

When do I use substitution and when integration by parts?

Substitution when part of the integrand is the derivative of another part (u and du both present). Parts when the integrand is a product of two unrelated kinds of function — a polynomial times an exponential, log or trig function.

Jiri gị onwe gị

Parts of this page are adapted from OpenStax Calculus Volume 1 (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Oge Calculus

Limits Derivatives Integrals Definite integrals Taylor series Series and sums Maxima and minima The chain rule Implicit differentiation Related rates and optimisation Integration techniques: substitution, parts, partial fractions Applications of integration: area, volume, arc length Infinite series and convergence tests

Interval	Test Point	Sign of \({f}^{″}(x)=6x-12\) at Test Point	Conclusion
\((\text{-}\infty ,2)\)	\(x=0\)	\(-\)	\(f\) is concave down
\((2,\infty )\)	\(x=3\)	\(+\)	\(f\) is concave up.

\(x\)	\({f}^{″}(x)\)	Conclusion
\(\text{-}\sqrt{3}\)	\(-30\sqrt{3}\)	Local maximum
\(0\)	\(0\)	Second derivative test is inconclusive
\(\sqrt{3}\)	\(30\sqrt{3}\)	Local minimum