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Calculus 2 : Calculus II

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9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #221 : First And Second Derivatives Of Functions

$\begin{align*}&\text{Find the derivative of the function }\\&f(x)=19cos(2x^{2})\end{align*}$

Possible Answers:

19cos(4x)

$19sin(2x^{2})$

-4x

$-76xsin(2x^{2})$

Correct answer:

$-76xsin(2x^{2})$

Explanation:

$\begin{align*}&\text{The function }f(x)=19cos(2x^{2})\text{, can be seen as a function nested within another.}\\&\text{Notice the }-2x^{2}\text{ within the }19cos()\\&\text{We'll wish to use the chain rule:}\\&(f\circ g )'=(f'\circ g )\cdot g'\\&\text{Along with the following:}\\&d[cos(u)]=-sin(u)du\\&u=-2x^{2}\\&du=-4x\\&f'(x)=(-4x)(19sin(2x^{2}))=-76xsin(2x^{2})\end{align*}$

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Example Question #222 : First And Second Derivatives Of Functions

$\begin{align*}&\text{Find the 2nd derivative of the function }\\&f(x)=-14e^{(16x)}\end{align*}$

Possible Answers:

$-224e^{(16x)}$

$-3584e^{(16x)}$

$802816e^{(32x)}$

$50176e^{(32x)}$

Correct answer:

$-3584e^{(16x)}$

Explanation:

$\text{To find the second derivative of a function, one need only be}\\&\text{cautious in performing the steps of derivation twice. For the}\\&\text{function we're given,}\text{we'll wish to use the following:}\\&d[e^{ax}]=ae^{ax}dx\\&\text{Our first derivative is thus:}\\&-224e^{(16x)}\\&\text{And our second is:}\\&-3584e^{(16x)}$

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Example Question #223 : First And Second Derivatives Of Functions

$\begin{align*}&\text{Take the derivative twice with respect to }x\text{ of the function}\\&y=-5x^{4}\end{align*}$

Possible Answers:

$400x^{6}$

$-60x^{2}$

$-15x^{2}$

$1200x^{5}$

Correct answer:

$-60x^{2}$

Explanation:

$\begin{align*}&\text{To find the second derivative of a function, one need only be}\\&\text{cautious in performing the steps of derivation twice. For the}\\&\text{function we're given,}\text{ we'll wish to use the following:}\\&d[x^a]=ax^{a-1}dx\\&\text{Our first derivative is thus:}\\&-20x^{3}\\&\text{And our second is:}\\&-60x^{2}\end{align*}$

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Example Question #229 : First And Second Derivatives Of Functions

$\begin{align*}&\text{Take the derivative twice with respect to }x\text{ of the function}\\&y=20ln(10x)\end{align*}$

Possible Answers:

$-\frac{20}{x^{2}}$

$-\frac{2}{x^{2}}$

$\frac{400}{x^{2}}$

$-\frac{400}{x^{3}}$

Correct answer:

$-\frac{20}{x^{2}}$

Explanation:

$\begin{align*}&\text{To find the second derivative of a function, one need only be}\\&\text{cautious in performing the steps of derivation twice. For the}\\&\text{function we're given,}\text{ we'll wish to use the following:}\\&d[ln(ax)]=\frac{dx}{x}\\&d[x^a]=ax^{a-1}dx\\&\text{Our first derivative is thus:}\\&\frac{20}{x}\\&\text{And our second is:}\\&-\frac{20}{x^{2}}\end{align*}$

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Example Question #224 : First And Second Derivatives Of Functions

$\begin{align*}&\text{Find the 2nd derivative of the function }\\&f(x)=-16x^{18}\end{align*}$

Possible Answers:

$-4896x^{16}$

$1410048x^{33}$

$-272x^{16}$

$82944x^{34}$

Correct answer:

$-4896x^{16}$

Explanation:

$\begin{align*}&\text{To find the second derivative of a function, one need only be}\\&\text{cautious in performing the steps of derivation twice. For the}\\&\text{function we're given,}\text{ we'll wish to use the following:}\\&d[x^a]=ax^{a-1}dx\\&\text{Our first derivative is thus:}\\&-288x^{17}\\&\text{And our second is:}\\&-4896x^{16}\end{align*}$

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Example Question #1551 : Calculus Ii

$\begin{align*}&\text{Calculate the second derivative of the function}\\&17e^{(19x)}\end{align*}$

Possible Answers:

$323e^{(19x)}$

$1982251e^{(38x)}$

$6137e^{(19x)}$

$104329e^{(38x)}$

Correct answer:

$6137e^{(19x)}$

Explanation:

$\begin{align*}&\text{To find the second derivative of a function, one need only be}\\&\text{cautious in performing the steps of derivation twice. For the}\\&\text{function we're given,}\text{ we'll wish to use the following:}\\&d[e^{ax}]=ae^{ax}dx\\&\text{Our first derivative is thus:}\\&323e^{(19x)}\\&\text{And our second is:}\\&6137e^{(19x)}\end{align*}$

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Example Question #1552 : Calculus Ii

$\begin{align*}&\text{Calculate the second derivative of the function}\\&-3e^{(-13x)}\end{align*}$

Possible Answers:

$39e^{(-13x)}$

$-19773e^{(-26x)}$

$1521e^{(-26x)}$

$-507e^{(-13x)}$

Correct answer:

$-507e^{(-13x)}$

Explanation:

$\begin{align*}&\text{To find the second derivative of a function, one need only be}\\&\text{cautious in performing the steps of derivation twice. For the}\\&\text{function we're given,}\text{ we'll wish to use the following:}\\&d[e^{ax}]=ae^{ax}dx\\&\text{Our first derivative is thus:}\\&39e^{(-13x)}\\&\text{And our second is:}\\&-507e^{(-13x)}\end{align*}$

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Example Question #1553 : Calculus Ii

$\begin{align*}&\text{Calculate the second derivative of the function}\\&13cos(11x)\end{align*}$

Possible Answers:

-143cos(11x)

$20449sin(11x)^{2}$

224939cos(11x)sin(11x)

-1573cos(11x)

Correct answer:

-1573cos(11x)

Explanation:

$\begin{align*}&\text{To find the second derivative of a function, one need only be}\\&\text{cautious in performing the steps of derivation twice. For the}\\&\text{function we're given,}\text{ we'll wish to use the following:}\\&d[cos(ax)]=-asin(ax)dx\\&d[sin(ax)]=acos(ax)dx\\&\text{Our first derivative is thus:}\\&-143sin(11x)\\&\text{And our second is:}\\&-1573cos(11x)\end{align*}$

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Example Question #1554 : Calculus Ii

$\begin{align*}&\text{Calculate the second derivative of the function}\\&3e^{(6x)}\end{align*}$

Possible Answers:

$1944e^{(12x)}$

$18e^{(6x)}$

$108e^{(6x)}$

$324e^{(12x)}$

Correct answer:

$108e^{(6x)}$

Explanation:

$\begin{align*}&\text{To find the second derivative of a function, one need only be}\\&\text{cautious in performing the steps of derivation twice. For the}\\&\text{function we're given,}\text{ we'll wish to use the following:}\\&d[e^{ax}]=ae^{ax}dx\\&\text{Our first derivative is thus:}\\&18e^{(6x)}\\&\text{And our second is:}\\&108e^{(6x)}\end{align*}$

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Example Question #235 : First And Second Derivatives Of Functions

$\begin{align*}&\text{Take the derivative twice with respect to }x\text{ of the function}\\&y=19tan(11x)\end{align*}$

Possible Answers:

$(209tan(11x)^{2} + 209)^{2}$

$418tan(11x)\cdot (11tan(11x)^{2} + 11)\cdot (209tan(11x)^{2} + 209)$

$38tan(11x)\cdot (11tan(11x)^{2} + 11)$

$418tan(11x)\cdot (11tan(11x)^{2} + 11)$

Correct answer:

$418tan(11x)\cdot (11tan(11x)^{2} + 11)$

Explanation:

$\begin{align*}&\text{To find the second derivative of a function, one need only be}\\&\text{cautious in performing the steps of derivation twice. For the}\\&\text{function we're given,}\text{ we'll wish to use the following:}\\&d[tan(ax)]=\frac{adx}{cos^2(ax)}=(atan^2(ax)+a)dx\\&\text{Our first derivative is thus:}\\&209tan(11x)^{2} + 209\\&\text{And our second is:}\\&418tan(11x)\cdot (11tan(11x)^{2} + 11)\end{align*}$

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Calculus 2 : Calculus II

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #221 : First And Second Derivatives Of Functions

Example Question #222 : First And Second Derivatives Of Functions

Example Question #223 : First And Second Derivatives Of Functions

Example Question #229 : First And Second Derivatives Of Functions

Example Question #224 : First And Second Derivatives Of Functions

Example Question #1551 : Calculus Ii

Example Question #1552 : Calculus Ii

Example Question #1553 : Calculus Ii

Example Question #1554 : Calculus Ii

Example Question #235 : First And Second Derivatives Of Functions

All Calculus 2 Resources

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