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Calculus 3 : Calculus 3

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

Example Questions

Example Question #1171 : Partial Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=-e^{(x^2)}cos{(z)}sin{(y)}\\&\text{In the direction of }\overrightarrow{v}=(8,-8,1).\end{align*}$

Possible Answers:

${11.4e^{(x^2)}sin{(y)}sin{(z)} - 11.4e^{(x^2)}cos{(y)}cos{(z)} - 22.7xe^{(x^2)}cos{(z)}sin{(y)}}$

${22.7xe^{(x^2)}cos{(y)}sin{(z)}}$

${0.0081e^{(x^2)}sin{(y)}sin{(z)} - 0.49e^{(x^2)}cos{(y)}cos{(z)} - 0.98xe^{(x^2)}cos{(z)}sin{(y)}}$

${0.7e^{(x^2)}cos{(y)}cos{(z)} + 0.09e^{(x^2)}sin{(y)}sin{(z)} - 1.4xe^{(x^2)}cos{(z)}sin{(y)}}$

Correct answer:

${0.7e^{(x^2)}cos{(y)}cos{(z)} + 0.09e^{(x^2)}sin{(y)}sin{(z)} - 1.4xe^{(x^2)}cos{(z)}sin{(y)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(8)^2+(-8)^2+(1)^2}=11.36\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{8}{11.36}=0.7\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{-8}{11.36}=-0.7\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{1}{11.36}=0.09\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0.7)(-2xe^{(x^2)}cos{(z)}sin{(y)})+(-0.7)(-1e^{(x^2)}cos{(y)}cos{(z)})+(0.09)(e^{(x^2)}sin{(y)}sin{(z)})\\&D_{\overrightarrow{u}}(x,y,z)=0.7e^{(x^2)}cos{(y)}cos{(z)} + 0.09e^{(x^2)}sin{(y)}sin{(z)} - 1.4xe^{(x^2)}cos{(z)}sin{(y)}\end{align*}$

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Example Question #1172 : Partial Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=z^4sin{(x^2)}sin{(y)} - 4sin{(x)}sin{(y)}\\&\text{In the direction of }\overrightarrow{v}=(-9,-7,6).\end{align*}$

Possible Answers:

${12.9z^4sin{(x^2)}cos{(y)} - 51.5cos{(y)}sin{(x)} - 51.5cos{(x)}sin{(y)} + 51.5z^3sin{(x^2)}sin{(y)} + 25.8xz^4cos{(x^2)}sin{(y)}}$

${103xz^3cos{(x^2)}cos{(y)}}$

${2.8cos{(x)}sin{(y)} + 2.16cos{(y)}sin{(x)} - 0.54z^4sin{(x^2)}cos{(y)} + 1.88z^3sin{(x^2)}sin{(y)} - 1.4xz^4cos{(x^2)}sin{(y)}}$

${0.292z^4sin{(x^2)}cos{(y)} - 1.17cos{(y)}sin{(x)} - 1.96cos{(x)}sin{(y)} + 0.884z^3sin{(x^2)}sin{(y)} + 0.98xz^4cos{(x^2)}sin{(y)}}$

Correct answer:

${2.8cos{(x)}sin{(y)} + 2.16cos{(y)}sin{(x)} - 0.54z^4sin{(x^2)}cos{(y)} + 1.88z^3sin{(x^2)}sin{(y)} - 1.4xz^4cos{(x^2)}sin{(y)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(-9)^2+(-7)^2+(6)^2}=12.88\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{-9}{12.88}=-0.7\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{-7}{12.88}=-0.54\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{6}{12.88}=0.47\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(-0.7)(2xz^4cos{(x^2)}sin{(y)} - 4cos{(x)}sin{(y)})+(-0.54)(z^4sin{(x^2)}cos{(y)} - 4cos{(y)}sin{(x)})+(0.47)(4z^3sin{(x^2)}sin{(y)})\\&D_{\overrightarrow{u}}(x,y,z)=2.8cos{(x)}sin{(y)} + 2.16cos{(y)}sin{(x)} - 0.54z^4sin{(x^2)}cos{(y)} + 1.88z^3sin{(x^2)}sin{(y)} - 1.4xz^4cos{(x^2)}sin{(y)}\end{align*}$

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Example Question #108 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=2^x\cdot4^ycos{(z^3)} - 3x^2ln{(z^2)}\\&\text{In the direction of }\overrightarrow{v}=(-14,7,16).\end{align*}$

Possible Answers:

${0.408\cdot2^x\cdot4^ycos{(z^3)} - 2.38xln{(z^2)} -\frac{ {(3.02x^2)}}{z} - 1.51\cdot2^x\cdot4^yz^2sin{(z^3)}}$

${-64.5\cdot2^x\cdot4^yz^2sin{(z^3)}}$

${46.5\cdot2^x\cdot4^ycos{(z^3)} - 134xln{(z^2)} -\frac{ {(134x^2)}}{z} - 67.1\cdot2^x\cdot4^yz^2sin{(z^3)}}$

${3.78xln{(z^2)} -\frac{ {(4.26x^2)}}{z} - 0.00693\cdot2^x\cdot4^ycos{(z^3)} - 2.13\cdot2^x\cdot4^yz^2sin{(z^3)}}$

Correct answer:

${3.78xln{(z^2)} -\frac{ {(4.26x^2)}}{z} - 0.00693\cdot2^x\cdot4^ycos{(z^3)} - 2.13\cdot2^x\cdot4^yz^2sin{(z^3)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(-14)^2+(7)^2+(16)^2}=22.38\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{-14}{22.38}=-0.63\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{7}{22.38}=0.31\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{16}{22.38}=0.71\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(-0.63)(0.693\cdot2^x\cdot4^ycos{(z^3)} - 6xln{(z^2)})+(0.31)(1.39\cdot2^x\cdot4^ycos{(z^3)})+(0.71)(-\frac{ {(6x^2)}}{z} - 3\cdot2^x\cdot4^yz^2sin{(z^3)})\\&D_{\overrightarrow{u}}(x,y,z)=3.78xln{(z^2)} -\frac{ {(4.26x^2)}}{z} - 0.00693\cdot2^x\cdot4^ycos{(z^3)} - 2.13\cdot2^x\cdot4^yz^2sin{(z^3)}\end{align*}$

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Example Question #106 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=cos{(y^2)}sin{(x^3)}\\&\text{In the direction of }\overrightarrow{v}=(-4,-3,-13).\end{align*}$

Possible Answers:

{0}

${0.44ysin{(x^3)}sin{(y^2)} - 0.87x^2cos{(x^3)}cos{(y^2)}}$

${6ysin{(x^3)}sin{(y^2)} - 12x^2cos{(x^3)}cos{(y^2)}}$

${41.8x^2cos{(x^3)}cos{(y^2)} - 27.9ysin{(x^3)}sin{(y^2)}}$

Correct answer:

${0.44ysin{(x^3)}sin{(y^2)} - 0.87x^2cos{(x^3)}cos{(y^2)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(-4)^2+(-3)^2+(-13)^2}=13.93\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{-4}{13.93}=-0.29\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{-3}{13.93}=-0.22\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{-13}{13.93}=-0.93\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(-0.29)(3x^2cos{(x^3)}cos{(y^2)})+(-0.22)(-2ysin{(x^3)}sin{(y^2)})+(-0.93)(0.0)\\&D_{\overrightarrow{u}}(x,y,z)=0.44ysin{(x^3)}sin{(y^2)} - 0.87x^2cos{(x^3)}cos{(y^2)}\end{align*}$

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Example Question #110 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=ye^{(x)}e^{(z)} - 3y^3sin{(x^2)}ln{(z)}\\&\text{In the direction of }\overrightarrow{v}=(-18,11,7).\end{align*}$

Possible Answers:

${0.49e^{(x)}e^{(z)} -\frac{ {(0.93y^3sin{(x^2)})}}{z} - 4.41y^2sin{(x^2)}ln{(z)} - 0.5ye^{(x)}e^{(z)} + 4.86xy^3cos{(x^2)}ln{(z)}}$

${22.2e^{(x)}e^{(z)} -\frac{ {(400xy^2cos{(x^2)})}}{z}}$

${0.24e^{(x)}e^{(z)} -\frac{ {(0.288y^3sin{(x^2)})}}{z} - 2.16y^2sin{(x^2)}ln{(z)} + 0.752ye^{(x)}e^{(z)} - 3.94xy^3cos{(x^2)}ln{(z)}}$

${22.2e^{(x)}e^{(z)} -\frac{ {(66.7y^3sin{(x^2)})}}{z} - 200y^2sin{(x^2)}ln{(z)} + 44.5ye^{(x)}e^{(z)} - 133xy^3cos{(x^2)}ln{(z)}}$

Correct answer:

${0.49e^{(x)}e^{(z)} -\frac{ {(0.93y^3sin{(x^2)})}}{z} - 4.41y^2sin{(x^2)}ln{(z)} - 0.5ye^{(x)}e^{(z)} + 4.86xy^3cos{(x^2)}ln{(z)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(-18)^2+(11)^2+(7)^2}=22.23\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{-18}{22.23}=-0.81\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{11}{22.23}=0.49\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{7}{22.23}=0.31\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(-0.81)(ye^{(x)}e^{(z)} - 6xy^3cos{(x^2)}ln{(z)})+(0.49)(e^{(x)}e^{(z)} - 9y^2sin{(x^2)}ln{(z)})+(0.31)(ye^{(x)}e^{(z)} -\frac{ {(3y^3sin{(x^2)})}}{z})\\&D_{\overrightarrow{u}}(x,y,z)=0.49e^{(x)}e^{(z)} -\frac{ {(0.93y^3sin{(x^2)})}}{z} - 4.41y^2sin{(x^2)}ln{(z)} - 0.5ye^{(x)}e^{(z)} + 4.86xy^3cos{(x^2)}ln{(z)}\end{align*}$

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Example Question #111 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=-xsin{(z)}\\&\text{In the direction of }\overrightarrow{v}=(-13,19,-10).\end{align*}$

Possible Answers:

{- 0.27sin{(z)} - 0.16xcos{(z)}}

{- 25.1sin{(z)} - 25.1xcos{(z)}}

{0.52sin{(z)} + 0.4xcos{(z)}}

{0}

Correct answer:

{0.52sin{(z)} + 0.4xcos{(z)}}

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(-13)^2+(19)^2+(-10)^2}=25.1\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{-13}{25.1}=-0.52\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{19}{25.1}=0.76\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{-10}{25.1}=-0.4\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(-0.52)(-1sin{(z)})+(0.76)(0.0)+(-0.4)(-1xcos{(z)})\\&D_{\overrightarrow{u}}(x,y,z)=0.52sin{(z)} + 0.4xcos{(z)}\end{align*}$

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Example Question #112 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=ycos{(x^2)} - 2cos{(z^2)}sin{(y)}\\&\text{In the direction of }\overrightarrow{v}=(17,12,10).\end{align*}$

Possible Answers:

${0.52cos{(x^2)} - 1.04cos{(z^2)}cos{(y)} + 1.72zsin{(z^2)}sin{(y)} - 1.48xysin{(x^2)}}$

${0.27cos{(x^2)} - 0.541cos{(z^2)}cos{(y)} + 0.74zsin{(z^2)}sin{(y)} - 1.1xysin{(x^2)}}$

${23.1cos{(x^2)} - 46.2cos{(z^2)}cos{(y)} + 92.4zsin{(z^2)}sin{(y)} - 46.2xysin{(x^2)}}$

Correct answer:

${0.52cos{(x^2)} - 1.04cos{(z^2)}cos{(y)} + 1.72zsin{(z^2)}sin{(y)} - 1.48xysin{(x^2)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(17)^2+(12)^2+(10)^2}=23.09\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{17}{23.09}=0.74\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{12}{23.09}=0.52\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{10}{23.09}=0.43\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0.74)(-2xysin{(x^2)})+(0.52)(cos{(x^2)} - 2cos{(z^2)}cos{(y)})+(0.43)(4zsin{(z^2)}sin{(y)})\\&D_{\overrightarrow{u}}(x,y,z)=0.52cos{(x^2)} - 1.04cos{(z^2)}cos{(y)} + 1.72zsin{(z^2)}sin{(y)} - 1.48xysin{(x^2)}\end{align*}$

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Example Question #113 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=-y^3z^2sin{(x^3)}\\&\text{In the direction of }\overrightarrow{v}=(-17,12,20).\end{align*}$

Possible Answers:

${51x^2y^3z^2cos{(x^3)} - 40y^3zsin{(x^3)} - 36y^2z^2sin{(x^3)}}$

${1.77x^2y^3z^2cos{(x^3)} - 1.38y^3zsin{(x^3)} - 1.26y^2z^2sin{(x^3)}}$

${- 86.6y^2z^2sin{(x^3)} - 57.7y^3zsin{(x^3)} - 86.6x^2y^3z^2cos{(x^3)}}$

${-519x^2y^2zcos{(x^3)}}$

Correct answer:

${1.77x^2y^3z^2cos{(x^3)} - 1.38y^3zsin{(x^3)} - 1.26y^2z^2sin{(x^3)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(-17)^2+(12)^2+(20)^2}=28.86\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{-17}{28.86}=-0.59\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{12}{28.86}=0.42\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{20}{28.86}=0.69\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(-0.59)(-3x^2y^3z^2cos{(x^3)})+(0.42)(-3y^2z^2sin{(x^3)})+(0.69)(-2y^3zsin{(x^3)})\\&D_{\overrightarrow{u}}(x,y,z)=1.77x^2y^3z^2cos{(x^3)} - 1.38y^3zsin{(x^3)} - 1.26y^2z^2sin{(x^3)}\end{align*}$

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Example Question #114 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=3^xsin{(y^3)}e^{(z)} -\frac{ {(x^2y)}}{z}\\&\text{In the direction of }\overrightarrow{v}=(5,18,-6).\end{align*}$

Possible Answers:

${2.76\cdot3^xy^2cos{(y^3)}e^{(z)} -\frac{ {(0.5xy)}}{z} -\frac{ {(0.31x^2y)}}{z^2} - 0.0353\cdot3^xsin{(y^3)}e^{(z)} -\frac{ {(0.92x^2)}}{z}}$

${\frac{{(39.2x)}}{z^2} + 64.7\cdot3^xy^2cos{(y^3)}e^{(z)}}$

${54\cdot3^xy^2cos{(y^3)}e^{(z)} -\frac{ {(10xy)}}{z} -\frac{ {(6x^2y)}}{z^2} - 0.507\cdot3^xsin{(y^3)}e^{(z)} -\frac{ {(18x^2)}}{z}}$

${\frac{{(19.6x^2y)}}{z^2} -\frac{ {(39.2xy)}}{z} -\frac{ {(19.6x^2)}}{z} + 41.2\cdot3^xsin{(y^3)}e^{(z)} + 58.9\cdot3^xy^2cos{(y^3)}e^{(z)}}$

Correct answer:

${2.76\cdot3^xy^2cos{(y^3)}e^{(z)} -\frac{ {(0.5xy)}}{z} -\frac{ {(0.31x^2y)}}{z^2} - 0.0353\cdot3^xsin{(y^3)}e^{(z)} -\frac{ {(0.92x^2)}}{z}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(5)^2+(18)^2+(-6)^2}=19.62\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{5}{19.62}=0.25\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{18}{19.62}=0.92\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{-6}{19.62}=-0.31\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0.25)(1.1\cdot3^xsin{(y^3)}e^{(z)} -\frac{ {(2xy)}}{z})+(0.92)(3\cdot3^xy^2cos{(y^3)}e^{(z)} -\frac{ {(1x^2)}}{z})+(-0.31)(\frac{{(x^2y)}}{z^2} + 3^xsin{(y^3)}e^{(z)})\\&D_{\overrightarrow{u}}(x,y,z)=2.76\cdot3^xy^2cos{(y^3)}e^{(z)} -\frac{ {(0.5xy)}}{z} -\frac{ {(0.31x^2y)}}{z^2} - 0.0353\cdot3^xsin{(y^3)}e^{(z)} -\frac{ {(0.92x^2)}}{z}\end{align*}$

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Example Question #115 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=2^yz^4e^{(x^2)}\\&\text{In the direction of }\overrightarrow{v}=(10,3,2).\end{align*}$

Possible Answers:

${58.9\cdot2^yxz^3e^{(x^2)}}$

${42.5\cdot2^yz^3e^{(x^2)} + 7.37\cdot2^yz^4e^{(x^2)} + 21.3\cdot2^yxz^4e^{(x^2)}}$

${0.76\cdot2^yz^3e^{(x^2)} + 0.194\cdot2^yz^4e^{(x^2)} + 1.88\cdot2^yxz^4e^{(x^2)}}$

${0.144\cdot2^yz^3e^{(x^2)} + 0.0543\cdot2^yz^4e^{(x^2)} + 1.77\cdot2^yxz^4e^{(x^2)}}$

Correct answer:

${0.76\cdot2^yz^3e^{(x^2)} + 0.194\cdot2^yz^4e^{(x^2)} + 1.88\cdot2^yxz^4e^{(x^2)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(10)^2+(3)^2+(2)^2}=10.63\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{10}{10.63}=0.94\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{3}{10.63}=0.28\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{2}{10.63}=0.19\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0.94)(2\cdot2^yxz^4e^{(x^2)})+(0.28)(0.693\cdot2^yz^4e^{(x^2)})+(0.19)(4\cdot2^yz^3e^{(x^2)})\\&D_{\overrightarrow{u}}(x,y,z)=0.76\cdot2^yz^3e^{(x^2)} + 0.194\cdot2^yz^4e^{(x^2)} + 1.88\cdot2^yxz^4e^{(x^2)}\end{align*}$

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I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

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Calculus 3 : Calculus 3

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