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AP Calculus AB : Relationship between the concavity of ƒ and the sign of ƒ''

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Example Questions

Example Question #1 : Second Derivatives

What is the concavity of f(x)=6x^4-7x^3+8x^2-9x^2+18x at x=1?

Possible Answers:

Concave Horizontal

Concave Diagonal

Concave Up

No Concavity

Concave Down

Correct answer:

Concave Up

Explanation:

First, find the double derivative of the function given.

You should get f''(x)=72x^2-42x-2 .

Next, plug in x=1 to get f''(1)=28 .

The double derivative is a positive number, which means the concavity is Concave Up.

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Example Question #1 : Second Derivatives

Find the second derivative.

y= 3x^3

Possible Answers:

y=9x^2

y=18

y=18x

y=9x

y=18x^2

Correct answer:

y=18x

Explanation:

We need to find the second derivative. To do this we will use power rule twice. We will take the first derivative by power rule, then the second derivative by power rule once again. Power rule says that we take the exponent of the “x” value and bring it to the front. Then we subtract one from the exponent. $n(x^{n-1})$

Therefore, the first derivative is 9x^2 and the second derivative is 18x by power rule.

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Example Question #1 : Second Derivatives

$\begin{align*}&\text{For the point }x=4\text{, find the concavity of the function}\\&y=-20tan(3x)\end{align*}$

Possible Answers:

$\text{The function is concave downward.}$

$\text{The function is concave upward.}$

Correct answer:

$\text{The function is concave upward.}$

Explanation:

$\begin{align*}&\text{To find the concavity of a function, we'll wish to first find}\\&\text{its second derivative, so let's do it in steps. For the function}\\&\text{we're given,}\text{ we'll wish to use the following:}\\&d[tan(ax)]=\frac{adx}{cos^2(ax)}=(atan^2(ax)+a)dx\\&\text{Our first derivative is thus:}\\&- 60tan(3x)^{2} - 60\\&\text{And our second is:}\\&-120tan(3x)\cdot (3tan(3x)^{2} + 3)\\&\text{Now if we plug in our x-value we find}\\&-120tan(3(4))\cdot (3tan(3(4))^{2} + 3)=321.46\\&\text{The function is concave upward.}\end{align*}$

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Example Question #2 : Second Derivatives

$\begin{align*}&\text{For the point }x=-4\text{, find the concavity of the function}\\&y=13cos(17x)\end{align*}$

Possible Answers:

$\text{The function is concave downward.}$

$\text{The function is concave upward.}$

Correct answer:

$\text{The function is concave downward.}$

Explanation:

$\begin{align*}&\text{To find the concavity of a function, we'll wish to first find}\\&\text{its second derivative, so let's do it in steps. For the function}\\&\text{we're given,}\text{ we'll wish to use the following:}\\&d[cos(ax)]=-asin(ax)dx\\&d[sin(ax)]=acos(ax)dx\\&\text{Our first derivative is thus:}\\&-221sin(17x)\\&\text{And our second is:}\\&-3757cos(17x)\\&\text{Now if we plug in our x-value we find}\\&-3757cos(17(-4))=-1653.62\\&\text{The function is concave downward.}\end{align*}$

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Example Question #3 : Second Derivatives

$\begin{align*}&\text{Find the concavity of the function }\\&f(x)=-12tan(13x)\\&\text{When }x=4\end{align*}$

Possible Answers:

$\text{The function is concave upward.}$

$\text{The function is concave downward.}$

Correct answer:

$\text{The function is concave upward.}$

Explanation:

$\begin{align*}&\text{To find the concavity of a function, we'll wish to first find}\\&\text{its second derivative, so let's do it in steps. For the function}\\&\text{we're given,}\text{ we'll wish to use the following:}\\&d[tan(ax)]=\frac{adx}{cos^2(ax)}=(atan^2(ax)+a)dx\\&\text{Our first derivative is thus:}\\&- 156tan(13x)^{2} - 156\\&\text{And our second is:}\\&-312tan(13x)\cdot (13tan(13x)^{2} + 13)\\&\text{Now if we plug in our x-value we find}\\&-312tan(13(4))\cdot (13tan(13(4))^{2} + 13)=9.24\cdot10^{5}\\&\text{The function is concave upward.}\end{align*}$

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Example Question #4 : Second Derivatives

$\begin{align*}&\text{Find the concavity of the function }\\&f(x)=17e^{(6x)}\\&\text{When }x=3\end{align*}$

Possible Answers:

$\text{The function is concave upward.}$

$\text{The function is concave downward.}$

Correct answer:

$\text{The function is concave upward.}$

Explanation:

$\begin{align*}&\text{To find the concavity of a function, we'll wish to first find}\\&\text{its second derivative, so let's do it in steps. For the function}\\&\text{we're given,}\text{ we'll wish to use the following:}\\&d[e^{ax}]=ae^{ax}dx\\&\text{Our first derivative is thus:}\\&102e^{(6x)}\\&\text{And our second is:}\\&612e^{(6x)}\\&\text{Now if we plug in our x-value we find}\\&612e^{(6(3))}=4.02\cdot10^{10}\\&\text{The function is concave upward.}\end{align*}$

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Example Question #5 : Second Derivatives

$\begin{align*}&\text{Determine the concavity of the function}\\&-cos(15x)\\&\text{At the point }x=6\end{align*}$

Possible Answers:

$\text{The function is concave downward.}$

$\text{The function is concave upward.}$

Correct answer:

$\text{The function is concave downward.}$

Explanation:

$\begin{align*}&\text{To find the concavity of a function, we'll wish to first find}\\&\text{its second derivative, so let's do it in steps. For the function}\\&\text{we're given,}\text{ we'll wish to use the following:}\\&d[cos(ax)]=-asin(ax)dx\\&d[sin(ax)]=acos(ax)dx\\&\text{Our first derivative is thus:}\\&15sin(15x)\\&\text{And our second is:}\\&225cos(15x)\\&\text{Now if we plug in our x-value we find}\\&225cos(15(6))=-100.82\\&\text{The function is concave downward.}\end{align*}$

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Example Question #6 : Second Derivatives

$\begin{align*}&\text{Determine the concavity of the function}\\&\frac{6}{x^{10}}\\&\text{At the point }x=-1\end{align*}$

Possible Answers:

$\text{The function is concave downward.}$

$\text{The function is concave upward.}$

Correct answer:

$\text{The function is concave upward.}$

Explanation:

$\begin{align*}&\text{To find the concavity of a function, we'll wish to first find}\\&\text{its second derivative, so let's do it in steps. For the function}\\&\text{we're given,}\text{ we'll wish to use the following:}\\&d[x^a]=ax^{a-1}dx\\&\text{Our first derivative is thus:}\\&-\frac{60}{x^{11}}\\&\text{And our second is:}\\&\frac{660}{x^{12}}\\&\text{Now if we plug in our x-value we find}\\&\frac{660}{(-1)^{12}}=660.00\\&\text{The function is concave upward.}\end{align*}$

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Example Question #7 : Second Derivatives

$\begin{align*}&\text{Determine the concavity of the function}\\&-9e^{(-6x)}\\&\text{At the point }x=-8\end{align*}$

Possible Answers:

$\text{The function is concave downward.}$

$\text{The function is concave upward.}$

Correct answer:

$\text{The function is concave downward.}$

Explanation:

$\begin{align*}&\text{To find the concavity of a function, we'll wish to first find}\\&\text{its second derivative, so let's do it in steps. For the function}\\&\text{we're given,}\text{ we'll wish to use the following:}\\&d[e^{ax}]=ae^{ax}dx\\&\text{Our first derivative is thus:}\\&54e^{(-6x)}\\&\text{And our second is:}\\&-324e^{(-6x)}\\&\text{Now if we plug in our x-value we find}\\&-324e^{(-6(-8))}=-2.27\cdot10^{23}\\&\text{The function is concave downward.}\end{align*}$

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Example Question #8 : Second Derivatives

$\begin{align*}&\text{Determine the concavity of the function}\\&\frac{19}{5^{(5x)}}\\&\text{At the point }x=-5\end{align*}$

Possible Answers:

$\text{The function is concave upward.}$

$\text{The function is concave downward.}$

Correct answer:

$\text{The function is concave upward.}$

Explanation:

$\begin{align*}&\text{To find the concavity of a function, we'll wish to first find}\\&\text{its second derivative, so let's do it in steps. For the function}\\&\text{we're given,}\text{ we'll wish to use the following:}\\&d[b^{ax}]=ab^{ax}ln(b)sx\\&\text{Our first derivative is thus:}\\&-\frac{(95ln(5))}{5^{(5x)}}\\&\text{And our second is:}\\&\frac{(475ln(5)^{2})}{5^{(5x)}}\\&\text{Now if we plug in our x-value we find}\\&\frac{(475ln(5)^{2})}{5^{(5(-5))}}=3.67\cdot10^{20}\\&\text{The function is concave upward.}\end{align*}$

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AP Calculus AB : Relationship between the concavity of ƒ and the sign of ƒ''

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #1 : Second Derivatives

Example Question #1 : Second Derivatives

Example Question #1 : Second Derivatives

Example Question #2 : Second Derivatives

Example Question #3 : Second Derivatives

Example Question #4 : Second Derivatives

Example Question #5 : Second Derivatives

Example Question #6 : Second Derivatives

Example Question #7 : Second Derivatives

Example Question #8 : Second Derivatives

All AP Calculus AB Resources

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