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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

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6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #543 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zyy}\\&\text{Where }f(x,y,z)=\frac{{(3^{(x^2)}e^{(y)})}}{z^2}\end{align*}$ $\displaystyle \begin{align*}&\text{Perform the partial derivation, }f_{zyy}\\&\text{Where }f(x,y,z)=\frac{{(3^{(x^2)}e^{(y)})}}{z^2}\end{align*}$

Possible Answers:

${\frac{-{(2\cdot3^{(x^2)}e^{(y)})}}{z^3}}$ $\displaystyle {\frac{-{(2\cdot3^{(x^2)}e^{(y)})}}{z^3}}$

${\frac{-{(2\cdot3^{(3x^2)}e^{(3y)})}}{z^7}}$ $\displaystyle {\frac{-{(2\cdot3^{(3x^2)}e^{(3y)})}}{z^7}}$

${\frac{{(2\cdot3^{(x^2)}e^{(y)})}}{z^2} -\frac{ {(2\cdot3^{(x^2)}e^{(y)})}}{z^3}}$ $\displaystyle {\frac{{(2\cdot3^{(x^2)}e^{(y)})}}{z^2} -\frac{ {(2\cdot3^{(x^2)}e^{(y)})}}{z^3}}$

${\frac{-{(8\cdot3^{(3x^2)}e^{(3y)})}}{z^9}}$ $\displaystyle {\frac{-{(8\cdot3^{(3x^2)}e^{(3y)})}}{z^9}}$

Correct answer:

${\frac{-{(2\cdot3^{(x^2)}e^{(y)})}}{z^3}}$ $\displaystyle {\frac{-{(2\cdot3^{(x^2)}e^{(y)})}}{z^3}}$

Explanation:

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Example Question #541 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zxx}\\&\text{Where }f(x,y,z)=6x^2sin{(4z)}e^{(4y)}\end{align*}$ $\displaystyle \begin{align*}&\text{Perform the partial derivation, }f_{zxx}\\&\text{Where }f(x,y,z)=6x^2sin{(4z)}e^{(4y)}\end{align*}$

Possible Answers:

${55296x^3cos{(4z)}^3e^{(12y)}}$ $\displaystyle {55296x^3cos{(4z)}^3e^{(12y)}}$

${3456x^4cos{(4z)}sin{(4z)}^2e^{(12y)}}$ $\displaystyle {3456x^4cos{(4z)}sin{(4z)}^2e^{(12y)}}$

${48cos{(4z)}e^{(4y)}}$ $\displaystyle {48cos{(4z)}e^{(4y)}}$

${24xsin{(4z)}e^{(4y)} + 24x^2cos{(4z)}e^{(4y)}}$ $\displaystyle {24xsin{(4z)}e^{(4y)} + 24x^2cos{(4z)}e^{(4y)}}$

Correct answer:

${48cos{(4z)}e^{(4y)}}$ $\displaystyle {48cos{(4z)}e^{(4y)}}$

Explanation:

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Example Question #545 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zz}\\&\text{Where }f(x,y,z)=\frac{-{(y^2ln{(z)})}}{{(3x^2)}}\end{align*}$ $\displaystyle \begin{align*}&\text{Perform the partial derivation, }f_{zz}\\&\text{Where }f(x,y,z)=\frac{-{(y^2ln{(z)})}}{{(3x^2)}}\end{align*}$

Possible Answers:

${\frac{y^4}{{(9x^4z^2)}}}$ $\displaystyle {\frac{y^4}{{(9x^4z^2)}}}$

${\frac{-{(2y^2)}}{{(3x^2z)}}}$ $\displaystyle {\frac{-{(2y^2)}}{{(3x^2z)}}}$

${\frac{-y^4}{{(9x^4z^3)}}}$ $\displaystyle {\frac{-y^4}{{(9x^4z^3)}}}$

${\frac{y^2}{{(3x^2z^2)}}}$ $\displaystyle {\frac{y^2}{{(3x^2z^2)}}}$

Correct answer:

${\frac{y^2}{{(3x^2z^2)}}}$ $\displaystyle {\frac{y^2}{{(3x^2z^2)}}}$

Explanation:

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Example Question #546 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zyzy}\\&\text{Where }f(x,y,z)=\frac{-{(2ln{(4z)}e^{(3x)}e^{(y)})}}{9}\end{align*}$ $\displaystyle \begin{align*}&\text{Perform the partial derivation, }f_{zyzy}\\&\text{Where }f(x,y,z)=\frac{-{(2ln{(4z)}e^{(3x)}e^{(y)})}}{9}\end{align*}$

Possible Answers:

${\frac{{(16e^{(12x)}e^{(4y)})}}{{(6561z^6)}}}$ $\displaystyle {\frac{{(16e^{(12x)}e^{(4y)})}}{{(6561z^6)}}}$

${\frac{{(2e^{(3x)}e^{(y)})}}{{(9z^2)}}}$ $\displaystyle {\frac{{(2e^{(3x)}e^{(y)})}}{{(9z^2)}}}$

${-\frac{ {(4e^{(3x)}e^{(y)})}}{{(9z)}} -\frac{ {(4ln{(4z)}e^{(3x)}e^{(y)})}}{9}}$ $\displaystyle {-\frac{ {(4e^{(3x)}e^{(y)})}}{{(9z)}} -\frac{ {(4ln{(4z)}e^{(3x)}e^{(y)})}}{9}}$

${\frac{{(16ln{(4z)}^2e^{(12x)}e^{(4y)})}}{{(6561z^2)}}}$ $\displaystyle {\frac{{(16ln{(4z)}^2e^{(12x)}e^{(4y)})}}{{(6561z^2)}}}$

Correct answer:

${\frac{{(2e^{(3x)}e^{(y)})}}{{(9z^2)}}}$ $\displaystyle {\frac{{(2e^{(3x)}e^{(y)})}}{{(9z^2)}}}$

Explanation:

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Example Question #547 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zzxz}\\&\text{Where }f(x,y,z)=\frac{-{(x^2cos{(y^2)}cos{(z)})}}{3}\end{align*}$ $\displaystyle \begin{align*}&\text{Perform the partial derivation, }f_{zzxz}\\&\text{Where }f(x,y,z)=\frac{-{(x^2cos{(y^2)}cos{(z)})}}{3}\end{align*}$

Possible Answers:

${x^2cos{(y^2)}sin{(z)} -\frac{ {(2xcos{(y^2)}cos{(z)})}}{3}}$ $\displaystyle {x^2cos{(y^2)}sin{(z)} -\frac{ {(2xcos{(y^2)}cos{(z)})}}{3}}$

${\frac{-{(2x^7cos{(y^2)}^4cos{(z)}sin{(z)}^3)}}{81}}$ $\displaystyle {\frac{-{(2x^7cos{(y^2)}^4cos{(z)}sin{(z)}^3)}}{81}}$

${\frac{-{(4x^6cos{(y^2)}^4cos{(z)}^2sin{(z)}^2)}}{81}}$ $\displaystyle {\frac{-{(4x^6cos{(y^2)}^4cos{(z)}^2sin{(z)}^2)}}{81}}$

${\frac{-{(2xcos{(y^2)}sin{(z)})}}{3}}$ $\displaystyle {\frac{-{(2xcos{(y^2)}sin{(z)})}}{3}}$

Correct answer:

${\frac{-{(2xcos{(y^2)}sin{(z)})}}{3}}$ $\displaystyle {\frac{-{(2xcos{(y^2)}sin{(z)})}}{3}}$

Explanation:

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Example Question #548 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{xy}\\&\text{Where }f(x,y,z)=\frac{{(2cos{(z)}ln{(x)})}}{{(9y)}}\end{align*}$ $\displaystyle \begin{align*}&\text{Perform the partial derivation, }f_{xy}\\&\text{Where }f(x,y,z)=\frac{{(2cos{(z)}ln{(x)})}}{{(9y)}}\end{align*}$

Possible Answers:

${\frac{-{(4cos{(z)}^2ln{(x)})}}{{(81xy^3)}}}$ $\displaystyle {\frac{-{(4cos{(z)}^2ln{(x)})}}{{(81xy^3)}}}$ ${\frac{-{(4cos{(z)}^2ln{(x)})}}{{(81xy^3)}}}$ $\displaystyle {\frac{-{(4cos{(z)}^2ln{(x)})}}{{(81xy^3)}}}$

${\frac{{(2cos{(z)})}}{{(9xy)}} -\frac{ {(2cos{(z)}ln{(x)})}}{{(9y^2)}}}$ $\displaystyle {\frac{{(2cos{(z)})}}{{(9xy)}} -\frac{ {(2cos{(z)}ln{(x)})}}{{(9y^2)}}}$

${\frac{-{(4cos{(z)}^2)}}{{(81x^2y^3)}}}$ $\displaystyle {\frac{-{(4cos{(z)}^2)}}{{(81x^2y^3)}}}$

${\frac{-{(2cos{(z)})}}{{(9xy^2)}}}$ $\displaystyle {\frac{-{(2cos{(z)})}}{{(9xy^2)}}}$

Correct answer:

${\frac{-{(2cos{(z)})}}{{(9xy^2)}}}$ $\displaystyle {\frac{-{(2cos{(z)})}}{{(9xy^2)}}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[ln(u)]=\frac{du}{u}\\&\text{Solve step-by-step:}\\&f(x,y,z)=\frac{{(2cos{(z)}ln{(x)})}}{{(9y)}}\\&f_{x}=\frac{{(2cos{(z)})}}{{(9xy)}}\\&f_{xy}=\frac{-{(2cos{(z)})}}{{(9xy^2)}}\end{align*}$ $\displaystyle \begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[ln(u)]=\frac{du}{u}\\&\text{Solve step-by-step:}\\&f(x,y,z)=\frac{{(2cos{(z)}ln{(x)})}}{{(9y)}}\\&f_{x}=\frac{{(2cos{(z)})}}{{(9xy)}}\\&f_{xy}=\frac{-{(2cos{(z)})}}{{(9xy^2)}}\end{align*}$ $\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[ln(u)]=\frac{du}{u}\\&\text{Solve step-by-step:}\\&f(x,y,z)=\frac{{(2cos{(z)}ln{(x)})}}{{(9y)}}\\&f_{x}=\frac{{(2cos{(z)})}}{{(9xy)}}\\&f_{xy}=\frac{-{(2cos{(z)})}}{{(9xy^2)}}\end{align*}$ $\displaystyle \begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[ln(u)]=\frac{du}{u}\\&\text{Solve step-by-step:}\\&f(x,y,z)=\frac{{(2cos{(z)}ln{(x)})}}{{(9y)}}\\&f_{x}=\frac{{(2cos{(z)})}}{{(9xy)}}\\&f_{xy}=\frac{-{(2cos{(z)})}}{{(9xy^2)}}\end{align*}$

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Example Question #549 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{xyx}\\&\text{Where }f(x,y,z)=\frac{-{(2\cdot2^{(x^2)}\cdot2^ye^{(4z)})}}{5}\end{align*}$ $\displaystyle \begin{align*}&\text{Perform the partial derivation, }f_{xyx}\\&\text{Where }f(x,y,z)=\frac{-{(2\cdot2^{(x^2)}\cdot2^ye^{(4z)})}}{5}\end{align*}$

Possible Answers:

${\frac{ {(2\cdot2^{(x^2)}\cdot2^ye^{(4z)}ln{(2)})}}{5} +\frac{ {(8\cdot2^{(x^2)}\cdot2^yxe^{(4z)}ln{(2)})}}{5}}$ $\displaystyle {\frac{ {(2\cdot2^{(x^2)}\cdot2^ye^{(4z)}ln{(2)})}}{5} +\frac{ {(8\cdot2^{(x^2)}\cdot2^yxe^{(4z)}ln{(2)})}}{5}}$

${-\frac{ {(2\cdot2^{(x^2)}\cdot2^ye^{(4z)}ln{(2)})}}{5} -\frac{ {(8\cdot2^{(x^2)}\cdot2^yxe^{(4z)}ln{(2)})}}{5}}$ $\displaystyle {-\frac{ {(2\cdot2^{(x^2)}\cdot2^ye^{(4z)}ln{(2)})}}{5} -\frac{ {(8\cdot2^{(x^2)}\cdot2^yxe^{(4z)}ln{(2)})}}{5}}$

${-\frac{ {(4\cdot2^{(x^2)}\cdot2^ye^{(4z)}ln{(2)}^2)}}{5} -\frac{ {(8\cdot2^{(x^2)}\cdot2^yx^2e^{(4z)}ln{(2)}^3)}}{5}}$ $\displaystyle {-\frac{ {(4\cdot2^{(x^2)}\cdot2^ye^{(4z)}ln{(2)}^2)}}{5} -\frac{ {(8\cdot2^{(x^2)}\cdot2^yx^2e^{(4z)}ln{(2)}^3)}}{5}}$

${\frac{-{(32\cdot2^{(3y)}\cdot2^{(3x^2)}x^2e^{(12z)}ln{(2)}^3)}}{125}}$ $\displaystyle {\frac{-{(32\cdot2^{(3y)}\cdot2^{(3x^2)}x^2e^{(12z)}ln{(2)}^3)}}{125}}$

Correct answer:

Explanation:

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Example Question #550 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{xyzx}\\&\text{Where }f(x,y,z)=\frac{{(9x^2z^2ln{(y)})}}{2}\end{align*}$ $\displaystyle \begin{align*}&\text{Perform the partial derivation, }f_{xyzx}\\&\text{Where }f(x,y,z)=\frac{{(9x^2z^2ln{(y)})}}{2}\end{align*}$

Possible Answers:

${\frac{{(18z)}}{y}}$ $\displaystyle {\frac{{(18z)}}{y}}$

${\frac{{(6561x^6z^7ln{(y)}^3)}}{{(2y)}}}$ $\displaystyle {\frac{{(6561x^6z^7ln{(y)}^3)}}{{(2y)}}}$

${18xz^2ln{(y)} + 9x^2zln{(y)} +\frac{ {(9x^2z^2)}}{{(2y)}}}$ $\displaystyle {18xz^2ln{(y)} + 9x^2zln{(y)} +\frac{ {(9x^2z^2)}}{{(2y)}}}$

${\frac{{(26244x^3z^6ln{(y)})}}{y^3}}$ $\displaystyle {\frac{{(26244x^3z^6ln{(y)})}}{y^3}}$

Correct answer:

${\frac{{(18z)}}{y}}$ $\displaystyle {\frac{{(18z)}}{y}}$

Explanation:

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Example Question #551 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{xyxy}\\&\text{Where }f(x,y,z)=\frac{{(2e^{(4x)}e^{(z^2)})}}{{(7y^2)}}\end{align*}$ $\displaystyle \begin{align*}&\text{Perform the partial derivation, }f_{xyxy}\\&\text{Where }f(x,y,z)=\frac{{(2e^{(4x)}e^{(z^2)})}}{{(7y^2)}}\end{align*}$

Possible Answers:

${\frac{{(1572864e^{(16x)}e^{(4z^2)})}}{{(2401y^{12})}}}$ $\displaystyle {\frac{{(1572864e^{(16x)}e^{(4z^2)})}}{{(2401y^{12})}}}$

${\frac{{(192e^{(4x)}e^{(z^2)})}}{{(7y^4)}}}$ $\displaystyle {\frac{{(192e^{(4x)}e^{(z^2)})}}{{(7y^4)}}}$

${\frac{{(1024e^{(16x)}e^{(4z^2)})}}{{(2401y^{10})}}}$ $\displaystyle {\frac{{(1024e^{(16x)}e^{(4z^2)})}}{{(2401y^{10})}}}$

${\frac{{(16e^{(4x)}e^{(z^2)})}}{{(7y^2)}} -\frac{ {(8e^{(4x)}e^{(z^2)})}}{{(7y^3)}}}$ $\displaystyle {\frac{{(16e^{(4x)}e^{(z^2)})}}{{(7y^2)}} -\frac{ {(8e^{(4x)}e^{(z^2)})}}{{(7y^3)}}}$

Correct answer:

${\frac{{(192e^{(4x)}e^{(z^2)})}}{{(7y^4)}}}$ $\displaystyle {\frac{{(192e^{(4x)}e^{(z^2)})}}{{(7y^4)}}}$

Explanation:

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Example Question #552 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zyzx}\\&\text{Where }f(x,y,z)=\frac{{(2^{(4x)}z^2e^{(4y)})}}{5}\end{align*}$ $\displaystyle \begin{align*}&\text{Perform the partial derivation, }f_{zyzx}\\&\text{Where }f(x,y,z)=\frac{{(2^{(4x)}z^2e^{(4y)})}}{5}\end{align*}$

Possible Answers:

${\frac{{(4096\cdot2^{(16x)}z^2e^{(16y)}ln{(2)})}}{625}}$ $\displaystyle {\frac{{(4096\cdot2^{(16x)}z^2e^{(16y)}ln{(2)})}}{625}}$

${\frac{{(64\cdot2^{(16x)}z^6e^{(16y)}ln{(2)})}}{625}}$ $\displaystyle {\frac{{(64\cdot2^{(16x)}z^6e^{(16y)}ln{(2)})}}{625}}$

${\frac{{(4\cdot2^{(4x)}ze^{(4y)})}}{5} +\frac{ {(4\cdot2^{(4x)}z^2e^{(4y)})}}{5} +\frac{ {(4\cdot2^{(4x)}z^2e^{(4y)}ln{(2)})}}{5}}$ $\displaystyle {\frac{{(4\cdot2^{(4x)}ze^{(4y)})}}{5} +\frac{ {(4\cdot2^{(4x)}z^2e^{(4y)})}}{5} +\frac{ {(4\cdot2^{(4x)}z^2e^{(4y)}ln{(2)})}}{5}}$

${\frac{{(32\cdot2^{(4x)}e^{(4y)}ln{(2)})}}{5}}$ $\displaystyle {\frac{{(32\cdot2^{(4x)}e^{(4y)}ln{(2)})}}{5}}$

Correct answer:

${\frac{{(32\cdot2^{(4x)}e^{(4y)}ln{(2)})}}{5}}$ $\displaystyle {\frac{{(32\cdot2^{(4x)}e^{(4y)}ln{(2)})}}{5}}$

Explanation:

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #543 : Partial Derivatives

Example Question #541 : Partial Derivatives

Example Question #545 : Partial Derivatives

Example Question #546 : Partial Derivatives

Example Question #547 : Partial Derivatives

Example Question #548 : Partial Derivatives

Example Question #549 : Partial Derivatives

Example Question #550 : Partial Derivatives

Example Question #551 : Partial Derivatives

Example Question #552 : Partial Derivatives

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