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Calculus 3 : Partial Derivatives

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Example Questions

Example Question #61 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=y^3sin{(z)} - 2z^2cos{(y^3)}\\&\text{In the direction of }\overrightarrow{v}=(-8,0,-3).\end{align*}$

Possible Answers:

${8.54y^3cos{(z)} - 34.2zcos{(y^3)} + 25.6y^2sin{(z)} + 51.2y^2z^2sin{(y^3)}}$

{0}

${12zcos{(y^3)} - 3y^3cos{(z)}}$

${1.4zcos{(y^3)} - 0.35y^3cos{(z)}}$

Correct answer:

${1.4zcos{(y^3)} - 0.35y^3cos{(z)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(-8)^2+(0)^2+(-3)^2}=8.54\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{-8}{8.54}=-0.94\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{0}{8.54}=0\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{-3}{8.54}=-0.35\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(-0.94)(0.0)+(0)(3y^2sin{(z)} + 6y^2z^2sin{(y^3)})+(-0.35)(y^3cos{(z)} - 4zcos{(y^3)})\\&D_{\overrightarrow{u}}(x,y,z)=1.4zcos{(y^3)} - 0.35y^3cos{(z)}\end{align*}$

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Example Question #62 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=-cos{(z^3)}e^{(y)}\\&\text{In the direction of }\overrightarrow{v}=(1,-11,0).\end{align*}$

Possible Answers:

${cos{(z^3)}e^{(y)}}$

${11cos{(z^3)}e^{(y)}}$

{0}

${33.2z^2sin{(z^3)}e^{(y)} - 11.1cos{(z^3)}e^{(y)}}$

Correct answer:

${cos{(z^3)}e^{(y)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(1)^2+(-11)^2+(0)^2}=11.05\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{1}{11.05}=0.09\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{-11}{11.05}=-1\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{0}{11.05}=0\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0.09)(0.0)+(-1)(-1cos{(z^3)}e^{(y)})+(0)(3z^2sin{(z^3)}e^{(y)})\\&D_{\overrightarrow{u}}(x,y,z)=cos{(z^3)}e^{(y)}\end{align*}$

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Example Question #63 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=x^4e^{(y^2)}\\&\text{In the direction of }\overrightarrow{v}=(0,-15,9).\end{align*}$

Possible Answers:

${-30x^4ye^{(y^2)}}$

${-1.72x^4ye^{(y^2)}}$

${70x^3e^{(y^2)} + 35x^4ye^{(y^2)}}$

{0}

Correct answer:

${-1.72x^4ye^{(y^2)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(0)^2+(-15)^2+(9)^2}=17.49\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{0}{17.49}=0\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{-15}{17.49}=-0.86\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{9}{17.49}=0.51\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0)(4x^3e^{(y^2)})+(-0.86)(2x^4ye^{(y^2)})+(0.51)(0.0)\\&D_{\overrightarrow{u}}(x,y,z)=-1.72x^4ye^{(y^2)}\end{align*}$

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Example Question #64 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=-cos{(x^3)}e^{(y)}e^{(z)}\\&\text{In the direction of }\overrightarrow{v}=(-1,0,5).\end{align*}$

Possible Answers:

${- 0.98cos{(x^3)}e^{(y)}e^{(z)} - 0.6x^2sin{(x^3)}e^{(y)}e^{(z)}}$

${15.3x^2sin{(x^3)}e^{(y)}e^{(z)}}$

${15.3x^2sin{(x^3)}e^{(y)}e^{(z)} - 10.2cos{(x^3)}e^{(y)}e^{(z)}}$

${- 5cos{(x^3)}e^{(y)}e^{(z)} - 3x^2sin{(x^3)}e^{(y)}e^{(z)}}$

Correct answer:

${- 0.98cos{(x^3)}e^{(y)}e^{(z)} - 0.6x^2sin{(x^3)}e^{(y)}e^{(z)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(-1)^2+(0)^2+(5)^2}=5.1\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{-1}{5.1}=-0.2\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{0}{5.1}=0\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{5}{5.1}=0.98\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(-0.2)(3x^2sin{(x^3)}e^{(y)}e^{(z)})+(0)(-1cos{(x^3)}e^{(y)}e^{(z)})+(0.98)(-1cos{(x^3)}e^{(y)}e^{(z)})\\&D_{\overrightarrow{u}}(x,y,z)=- 0.98cos{(x^3)}e^{(y)}e^{(z)} - 0.6x^2sin{(x^3)}e^{(y)}e^{(z)}\end{align*}$

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Example Question #65 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=-xy^2e^{(z)}\\&\text{In the direction of }\overrightarrow{v}=(0,-18,7).\end{align*}$

Possible Answers:

${- 19.3y^2e^{(z)} - 38.6xye^{(z)} - 19.3xy^2e^{(z)}}$

${1.86xye^{(z)} - 0.36xy^2e^{(z)}}$

${-38.6ye^{(z)}}$

${36xye^{(z)} - 7xy^2e^{(z)}}$

Correct answer:

${1.86xye^{(z)} - 0.36xy^2e^{(z)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(0)^2+(-18)^2+(7)^2}=19.31\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{0}{19.31}=0\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{-18}{19.31}=-0.93\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{7}{19.31}=0.36\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0)(-1y^2e^{(z)})+(-0.93)(-2xye^{(z)})+(0.36)(-1xy^2e^{(z)})\\&D_{\overrightarrow{u}}(x,y,z)=1.86xye^{(z)} - 0.36xy^2e^{(z)}\end{align*}$

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Example Question #66 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=-3^{(x^2)}\cdot3^{(y^2)}cos{(z^3)}\\&\text{In the direction of }\overrightarrow{v}=(0,-17,-15).\end{align*}$

Possible Answers:

${328\cdot3^{(x^2)}\cdot3^{(y^2)}xyz^2sin{(z^3)}}$

${1.65\cdot3^{(x^2)}\cdot3^{(y^2)}ycos{(z^3)} - 1.98\cdot3^{(x^2)}\cdot3^{(y^2)}z^2sin{(z^3)}}$

${68\cdot3^{(x^2)}\cdot3^{(y^2)}z^2sin{(z^3)} - 49.8\cdot3^{(x^2)}\cdot3^{(y^2)}ycos{(z^3)} - 49.8\cdot3^{(x^2)}\cdot3^{(y^2)}xcos{(z^3)}}$

${37.4\cdot3^{(x^2)}\cdot3^{(y^2)}ycos{(z^3)} - 45\cdot3^{(x^2)}\cdot3^{(y^2)}z^2sin{(z^3)}}$

Correct answer:

${1.65\cdot3^{(x^2)}\cdot3^{(y^2)}ycos{(z^3)} - 1.98\cdot3^{(x^2)}\cdot3^{(y^2)}z^2sin{(z^3)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(0)^2+(-17)^2+(-15)^2}=22.67\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{0}{22.67}=0\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{-17}{22.67}=-0.75\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{-15}{22.67}=-0.66\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0)(-2.2\cdot3^{(x^2)}\cdot3^{(y^2)}xcos{(z^3)})+(-0.75)(-2.2\cdot3^{(x^2)}\cdot3^{(y^2)}ycos{(z^3)})+(-0.66)(3\cdot3^{(x^2)}\cdot3^{(y^2)}z^2sin{(z^3)})\\&D_{\overrightarrow{u}}(x,y,z)=1.65\cdot3^{(x^2)}\cdot3^{(y^2)}ycos{(z^3)} - 1.98\cdot3^{(x^2)}\cdot3^{(y^2)}z^2sin{(z^3)}\end{align*}$

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Example Question #67 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=- 4y^3sin{(z)} - sin{(z^3)}sin{(y)}\\&\text{In the direction of }\overrightarrow{v}=(20,0,-16).\end{align*}$

Possible Answers:

${2.48y^3cos{(z)} + 1.86z^2cos{(z^3)}sin{(y)}}$

${- 102y^3cos{(z)} - 307y^2sin{(z)} - 25.6sin{(z^3)}cos{(y)} - 76.8z^2cos{(z^3)}sin{(y)}}$

${64y^3cos{(z)} + 48z^2cos{(z^3)}sin{(y)}}$

Correct answer:

${2.48y^3cos{(z)} + 1.86z^2cos{(z^3)}sin{(y)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(20)^2+(0)^2+(-16)^2}=25.61\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{20}{25.61}=0.78\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{0}{25.61}=0\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{-16}{25.61}=-0.62\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0.78)(0.0)+(0)(- 12y^2sin{(z)} - sin{(z^3)}cos{(y)})+(-0.62)(- 4y^3cos{(z)} - 3z^2cos{(z^3)}sin{(y)})\\&D_{\overrightarrow{u}}(x,y,z)=2.48y^3cos{(z)} + 1.86z^2cos{(z^3)}sin{(y)}\end{align*}$

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Example Question #71 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=e^{(x^2)}e^{(y)}\\&\text{In the direction of }\overrightarrow{v}=(20,0,8).\end{align*}$

Possible Answers:

${21.5e^{(x^2)}e^{(y)} + 43.1xe^{(x^2)}e^{(y)}}$

${40xe^{(x^2)}e^{(y)}}$

${1.86xe^{(x^2)}e^{(y)}}$

Correct answer:

${1.86xe^{(x^2)}e^{(y)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(20)^2+(0)^2+(8)^2}=21.54\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{20}{21.54}=0.93\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{0}{21.54}=0\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{8}{21.54}=0.37\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0.93)(2xe^{(x^2)}e^{(y)})+(0)(e^{(x^2)}e^{(y)})+(0.37)(0.0)\\&D_{\overrightarrow{u}}(x,y,z)=1.86xe^{(x^2)}e^{(y)}\end{align*}$

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Example Question #1131 : Partial Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=- xye^{(z^2)} - cos{(z^2)}sin{(x^3)}e^{(y)}\\&\text{In the direction of }\overrightarrow{v}=(0,-2,20).\end{align*}$

Possible Answers:

${0.1xe^{(z^2)} + 0.1cos{(z^2)}sin{(x^3)}e^{(y)} - 2xyze^{(z^2)} + 2zsin{(x^3)}sin{(z^2)}e^{(y)}}$

${2xe^{(z^2)} + 2cos{(z^2)}sin{(x^3)}e^{(y)} - 40xyze^{(z^2)} + 40zsin{(x^3)}sin{(z^2)}e^{(y)}}$

${121x^2zcos{(x^3)}sin{(z^2)}e^{(y)} - 40.2ze^{(z^2)}}$

${40.2zsin{(x^3)}sin{(z^2)}e^{(y)} - 20.1ye^{(z^2)} - 20.1cos{(z^2)}sin{(x^3)}e^{(y)} - 40.2xyze^{(z^2)} - 20.1xe^{(z^2)} - 60.3x^2cos{(x^3)}cos{(z^2)}e^{(y)}}$

Correct answer:

${0.1xe^{(z^2)} + 0.1cos{(z^2)}sin{(x^3)}e^{(y)} - 2xyze^{(z^2)} + 2zsin{(x^3)}sin{(z^2)}e^{(y)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(0)^2+(-2)^2+(20)^2}=20.1\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{0}{20.1}=0\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{-2}{20.1}=-0.1\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{20}{20.1}=1\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0)(- ye^{(z^2)} - 3x^2cos{(x^3)}cos{(z^2)}e^{(y)})+(-0.1)(- xe^{(z^2)} - cos{(z^2)}sin{(x^3)}e^{(y)})+(1)(2zsin{(x^3)}sin{(z^2)}e^{(y)} - 2xyze^{(z^2)})\\&D_{\overrightarrow{u}}(x,y,z)=0.1xe^{(z^2)} + 0.1cos{(z^2)}sin{(x^3)}e^{(y)} - 2xyze^{(z^2)} + 2zsin{(x^3)}sin{(z^2)}e^{(y)}\end{align*}$

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Example Question #73 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=- 3x^2cos{(z)}sin{(y)} - x^2sin{(z^2)}sin{(y)}\\&\text{In the direction of }\overrightarrow{v}=(-9,5,0).\end{align*}$

Possible Answers:

${61.8xcos{(y)}sin{(z)} - 41.2xzcos{(z^2)}cos{(y)}}$

${18xsin{(z^2)}sin{(y)} - 5x^2sin{(z^2)}cos{(y)} + 54xcos{(z)}sin{(y)} - 15x^2cos{(y)}cos{(z)}}$

${1.74xsin{(z^2)}sin{(y)} - 0.49x^2sin{(z^2)}cos{(y)} + 5.22xcos{(z)}sin{(y)} - 1.47x^2cos{(y)}cos{(z)}}$

${30.9x^2sin{(y)}sin{(z)} - 20.6xsin{(z^2)}sin{(y)} - 10.3x^2sin{(z^2)}cos{(y)} - 61.8xcos{(z)}sin{(y)} - 30.9x^2cos{(y)}cos{(z)} - 20.6x^2zcos{(z^2)}sin{(y)}}$

Correct answer:

${1.74xsin{(z^2)}sin{(y)} - 0.49x^2sin{(z^2)}cos{(y)} + 5.22xcos{(z)}sin{(y)} - 1.47x^2cos{(y)}cos{(z)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(-9)^2+(5)^2+(0)^2}=10.3\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{-9}{10.3}=-0.87\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{5}{10.3}=0.49\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{0}{10.3}=0\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(-0.87)(- 2xsin{(z^2)}sin{(y)} - 6xcos{(z)}sin{(y)})+(0.49)(- x^2sin{(z^2)}cos{(y)} - 3x^2cos{(y)}cos{(z)})+(0)(3x^2sin{(y)}sin{(z)} - 2x^2zcos{(z^2)}sin{(y)})\\&D_{\overrightarrow{u}}(x,y,z)=1.74xsin{(z^2)}sin{(y)} - 0.49x^2sin{(z^2)}cos{(y)} + 5.22xcos{(z)}sin{(y)} - 1.47x^2cos{(y)}cos{(z)}\end{align*}$

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Calculus 3 : Partial Derivatives

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Example Questions

Example Question #61 : Directional Derivatives

Example Question #62 : Directional Derivatives

Example Question #63 : Directional Derivatives

Example Question #64 : Directional Derivatives

Example Question #65 : Directional Derivatives

Example Question #66 : Directional Derivatives

Example Question #67 : Directional Derivatives

Example Question #71 : Directional Derivatives

Example Question #1131 : Partial Derivatives

Example Question #73 : Directional Derivatives

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