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Calculus 3 : Directional Derivatives

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Example Questions

Example Question #281 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=- 5xln(y^{2})ln(z) -\frac{ (z^{3}cos(x^{3}))}{y^{2}}\\&\text{In the direction of }\overrightarrow{v}=(0,19,-16).\end{align*}$

Possible Answers:

${\frac{(1.16z^{3}cos(x^{3}))}{y^{3}}-\frac{ (1.23z^{2}cos(x^{3}))}{y^{2}}-\frac{ (5.78xln(z))}{y}-\frac{ (2.05xln(y^{2}))}{z}}$

${\frac{(49.7z^{3}cos(x^{3}))}{y^{3}}-\frac{ (74.5z^{2}cos(x^{3}))}{y^{2}}- 124ln(y^{2})ln(z) -\frac{ (248xln(z))}{y}-\frac{ (124xln(y^{2}))}{z}+\frac{ (74.5x^{2}z^{3}sin(x^{3}))}{y^{2}}}$

${\frac{(1.92z^{2}cos(x^{3}))}{y^{2}}+\frac{ (1.52z^{3}cos(x^{3}))}{y^{3}}-\frac{ (7.6xln(z))}{y}+\frac{ (3.2xln(y^{2}))}{z}}$

${-\frac{ 248.0}{(yz)}-\frac{ (447x^{2}z^{2}sin(x^{3}))}{y^{3}}}$

Correct answer:

${\frac{(1.92z^{2}cos(x^{3}))}{y^{2}}+\frac{ (1.52z^{3}cos(x^{3}))}{y^{3}}-\frac{ (7.6xln(z))}{y}+\frac{ (3.2xln(y^{2}))}{z}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(0)^2+(19)^2+(-16)^2}=24.84\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{0}{24.84}=0\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{19}{24.84}=0.76\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{-16}{24.84}=-0.64\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0)(\frac{(3x^{2}z^{3}sin(x^{3}))}{y^{2}}- 5ln(y^{2})ln(z))+(0.76)(\frac{(2z^{3}cos(x^{3}))}{y^{3}}-\frac{ (10xln(z))}{y})+(-0.64)(-\frac{ (3z^{2}cos(x^{3}))}{y^{2}}-\frac{ (5xln(y^{2}))}{z})\\&D_{\overrightarrow{u}}(x,y,z)=\frac{(1.92z^{2}cos(x^{3}))}{y^{2}}+\frac{ (1.52z^{3}cos(x^{3}))}{y^{3}}-\frac{ (7.6xln(z))}{y}+\frac{ (3.2xln(y^{2}))}{z}\end{align*}$

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Example Question #282 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=-3^xcos(y)\\&\text{In the direction of }\overrightarrow{v}=(4,0,-6).\end{align*}$

Possible Answers:

${-0.604\cdot 3^xcos(y)}$

${7.21\cdot 3^xsin(y) - 7.92\cdot 3^xcos(y)}$

${-0.332\cdot 3^xcos(y)}$

Correct answer:

${-0.604\cdot 3^xcos(y)}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(4)^2+(0)^2+(-6)^2}=7.21\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{4}{7.21}=0.55\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{0}{7.21}=0\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{-6}{7.21}=-0.83\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0.55)(-1.1\cdot 3^xcos(y))+(0)(3^xsin(y))+(-0.83)(0.0)\\&D_{\overrightarrow{u}}(x,y,z)=-0.604\cdot 3^xcos(y)\end{align*}$

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Example Question #283 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=- 2^xln(y^{2}) - 3y^{4}z^{3}ln(x^{2})\\&\text{In the direction of }\overrightarrow{v}=(-2,10,0).\end{align*}$

Possible Answers:

${1.39\cdot 2^xln(y^{2}) -\frac{ (20\cdot 2^x)}{y}- 120y^{3}z^{3}ln(x^{2}) +\frac{ (12y^{4}z^{3})}{x}}$

${0.139\cdot 2^xln(y^{2}) -\frac{ (1.96\cdot 2^x)}{y}- 11.8y^{3}z^{3}ln(x^{2}) +\frac{ (1.2y^{4}z^{3})}{x}}$

${-\frac{(734y^{3}z^{2})}{x}}$

${-\frac{ (20.4\cdot 2^x)}{y}- 7.07\cdot 2^xln(y^{2}) - 122y^{3}z^{3}ln(x^{2}) - 91.8y^{4}z^{2}ln(x^{2}) -\frac{ (61.2y^{4}z^{3})}{x}}$

Correct answer:

${0.139\cdot 2^xln(y^{2}) -\frac{ (1.96\cdot 2^x)}{y}- 11.8y^{3}z^{3}ln(x^{2}) +\frac{ (1.2y^{4}z^{3})}{x}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(-2)^2+(10)^2+(0)^2}=10.2\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{-2}{10.2}=-0.2\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{10}{10.2}=0.98\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{0}{10.2}=0\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(-0.2)(- 0.693\cdot 2^xln(y^{2}) -\frac{ (6y^{4}z^{3})}{x})+(0.98)(-\frac{ (2\cdot 2^x)}{y}- 12y^{3}z^{3}ln(x^{2}))+(0)(-9y^{4}z^{2}ln(x^{2}))\\&D_{\overrightarrow{u}}(x,y,z)=0.139\cdot 2^xln(y^{2}) -\frac{ (1.96\cdot 2^x)}{y}- 11.8y^{3}z^{3}ln(x^{2}) +\frac{ (1.2y^{4}z^{3})}{x}\end{align*}$

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Example Question #284 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=2^{(y^{2})}xsin(z^{4}) - 4y^{2}ln(z)\\&\text{In the direction of }\overrightarrow{v}=(-13,-19,0).\end{align*}$

Possible Answers:

${128\cdot 2^{(y^{2})}yz^{3}cos(z^{4})}$

${23\cdot 2^{(y^{2})}sin(z^{4}) -\frac{ (92.1y^{2})}{z}- 184yln(z) + 31.9\cdot 2^{(y^{2})}xysin(z^{4}) + 92.1\cdot 2^{(y^{2})}xz^{3}cos(z^{4})}$

${6.64yln(z) - 0.56\cdot 2^{(y^{2})}sin(z^{4}) - 1.15\cdot 2^{(y^{2})}xysin(z^{4})}$

${152yln(z) - 13\cdot 2^{(y^{2})}sin(z^{4}) - 26.3\cdot 2^{(y^{2})}xysin(z^{4})}$

Correct answer:

${6.64yln(z) - 0.56\cdot 2^{(y^{2})}sin(z^{4}) - 1.15\cdot 2^{(y^{2})}xysin(z^{4})}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(-13)^2+(-19)^2+(0)^2}=23.02\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{-13}{23.02}=-0.56\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{-19}{23.02}=-0.83\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{0}{23.02}=0\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(-0.56)(2^{(y^{2})}sin(z^{4}))+(-0.83)(1.39\cdot 2^{(y^{2})}xysin(z^{4}) - 8yln(z))+(0)(4\cdot 2^{(y^{2})}xz^{3}cos(z^{4}) -\frac{ (4y^{2})}{z})\\&D_{\overrightarrow{u}}(x,y,z)=6.64yln(z) - 0.56\cdot 2^{(y^{2})}sin(z^{4}) - 1.15\cdot 2^{(y^{2})}xysin(z^{4})\end{align*}$

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Example Question #285 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=- 2cos(z^{3})tan(y^{2}) - y^{4}ln(x^{3})ln(z^{4})\\&\text{In the direction of }\overrightarrow{v}=(0,-14,-11).\end{align*}$

Possible Answers:

${2.31z^{2}tan(y^{2})sin(z^{3}) - 2.5ycos(z^{3})\cdot (tan(y^{2})^{2} + 1) - 2.5y^{3}ln(x^{3})ln(z^{4}) -\frac{ (1.54y^{4}ln(x^{3}))}{z}}$

${\frac{(2.48y^{4}ln(x^{3}))}{z}+ 3.16ycos(z^{3})\cdot (tan(y^{2})^{2} + 1) + 3.16y^{3}ln(x^{3})ln(z^{4}) - 3.72z^{2}tan(y^{2})sin(z^{3})}$

${107z^{2}tan(y^{2})sin(z^{3}) -\frac{ (53.4y^{4}ln(z^{4}))}{x}- 71.2ycos(z^{3})\cdot (tan(y^{2})^{2} + 1) - 71.2y^{3}ln(x^{3})ln(z^{4}) -\frac{ (71.2y^{4}ln(x^{3}))}{z}}$

${-\frac{(854y^{3})}{(xz)}}$

Correct answer:

${\frac{(2.48y^{4}ln(x^{3}))}{z}+ 3.16ycos(z^{3})\cdot (tan(y^{2})^{2} + 1) + 3.16y^{3}ln(x^{3})ln(z^{4}) - 3.72z^{2}tan(y^{2})sin(z^{3})}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(0)^2+(-14)^2+(-11)^2}=17.8\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{0}{17.8}=0\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{-14}{17.8}=-0.79\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{-11}{17.8}=-0.62\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0)(-\frac{(3y^{4}ln(z^{4}))}{x})+(-0.79)(- 4ycos(z^{3})\cdot (tan(y^{2})^{2} + 1) - 4y^{3}ln(x^{3})ln(z^{4}))+(-0.62)(6z^{2}tan(y^{2})sin(z^{3}) -\frac{ (4y^{4}ln(x^{3}))}{z})\\&D_{\overrightarrow{u}}(x,y,z)=\frac{(2.48y^{4}ln(x^{3}))}{z}+ 3.16ycos(z^{3})\cdot (tan(y^{2})^{2} + 1) + 3.16y^{3}ln(x^{3})ln(z^{4}) - 3.72z^{2}tan(y^{2})sin(z^{3})\end{align*}$

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Example Question #286 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=-2^{(y^{2})}sin(z^{4})e^{(x^{2})}\\&\text{In the direction of }\overrightarrow{v}=(13,0,-1).\end{align*}$

Possible Answers:

${0.32\cdot 2^{(y^{2})}z^{3}cos(z^{4})e^{(x^{2})} - 2\cdot 2^{(y^{2})}xsin(z^{4})e^{(x^{2})}}$

${-145\cdot 2^{(y^{2})}xyz^{3}cos(z^{4})e^{(x^{2})}}$

${4\cdot 2^{(y^{2})}z^{3}cos(z^{4})e^{(x^{2})} - 26\cdot 2^{(y^{2})}xsin(z^{4})e^{(x^{2})}}$

${- 52.2\cdot 2^{(y^{2})}z^{3}cos(z^{4})e^{(x^{2})} - 26.1\cdot 2^{(y^{2})}xsin(z^{4})e^{(x^{2})} - 18.1\cdot 2^{(y^{2})}ysin(z^{4})e^{(x^{2})}}$

Correct answer:

${0.32\cdot 2^{(y^{2})}z^{3}cos(z^{4})e^{(x^{2})} - 2\cdot 2^{(y^{2})}xsin(z^{4})e^{(x^{2})}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(13)^2+(0)^2+(-1)^2}=13.04\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{13}{13.04}=1\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{0}{13.04}=0\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{-1}{13.04}=-0.08\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(1)(-2\cdot 2^{(y^{2})}xsin(z^{4})e^{(x^{2})})+(0)(-1.39\cdot 2^{(y^{2})}ysin(z^{4})e^{(x^{2})})+(-0.08)(-4\cdot 2^{(y^{2})}z^{3}cos(z^{4})e^{(x^{2})})\\&D_{\overrightarrow{u}}(x,y,z)=0.32\cdot 2^{(y^{2})}z^{3}cos(z^{4})e^{(x^{2})} - 2\cdot 2^{(y^{2})}xsin(z^{4})e^{(x^{2})}\end{align*}$

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Example Question #287 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=- z^{4}ln(x^{2})e^{(y)} - cos(y^{3})e^{(x^{4})}cos(z)\\&\text{In the direction of }\overrightarrow{v}=(-5,0,14).\end{align*}$

Possible Answers:

${14.9cos(y^{3})e^{(x^{4})}sin(z) - 14.9z^{4}ln(x^{2})e^{(y)} - 59.5z^{3}ln(x^{2})e^{(y)} -\frac{ (29.7z^{4}e^{(y)})}{x}- 59.5x^{3}cos(y^{3})e^{(x^{4})}cos(z) + 44.6y^{2}sin(y^{3})e^{(x^{4})}cos(z)}$

${0.94cos(y^{3})e^{(x^{4})}sin(z) - 3.76z^{3}ln(x^{2})e^{(y)} +\frac{ (0.68z^{4}e^{(y)})}{x}+ 1.36x^{3}cos(y^{3})e^{(x^{4})}cos(z)}$

${0.884cos(y^{3})e^{(x^{4})}sin(z) - 3.53z^{3}ln(x^{2})e^{(y)} -\frac{ (0.231z^{4}e^{(y)})}{x}- 0.462x^{3}cos(y^{3})e^{(x^{4})}cos(z)}$

${-\frac{ (119z^{3}e^{(y)})}{x}- 178x^{3}y^{2}sin(y^{3})e^{(x^{4})}sin(z)}$

Correct answer:

${0.94cos(y^{3})e^{(x^{4})}sin(z) - 3.76z^{3}ln(x^{2})e^{(y)} +\frac{ (0.68z^{4}e^{(y)})}{x}+ 1.36x^{3}cos(y^{3})e^{(x^{4})}cos(z)}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(-5)^2+(0)^2+(14)^2}=14.87\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{-5}{14.87}=-0.34\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{0}{14.87}=0\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{14}{14.87}=0.94\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(-0.34)(-\frac{ (2z^{4}e^{(y)})}{x}- 4x^{3}cos(y^{3})e^{(x^{4})}cos(z))+(0)(3y^{2}sin(y^{3})e^{(x^{4})}cos(z) - z^{4}ln(x^{2})e^{(y)})+(0.94)(cos(y^{3})e^{(x^{4})}sin(z) - 4z^{3}ln(x^{2})e^{(y)})\\&D_{\overrightarrow{u}}(x,y,z)=0.94cos(y^{3})e^{(x^{4})}sin(z) - 3.76z^{3}ln(x^{2})e^{(y)} +\frac{ (0.68z^{4}e^{(y)})}{x}+ 1.36x^{3}cos(y^{3})e^{(x^{4})}cos(z)\end{align*}$

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Example Question #288 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=- 4y^{3}cos(z^{3}) -\frac{ (cos(x^{2})ln(z^{2}))}{y}\\&\text{In the direction of }\overrightarrow{v}=(6,0,11).\end{align*}$

Possible Answers:

${132y^{3}z^{2}sin(z^{3}) -\frac{ (22cos(x^{2}))}{(yz)}+\frac{ (12xln(z^{2})sin(x^{2}))}{y}}$

${150y^{3}z^{2}sin(z^{3}) -\frac{ (25.1cos(x^{2}))}{(yz)}- 150y^{2}cos(z^{3}) +\frac{ (12.5cos(x^{2})ln(z^{2}))}{y^{2}}+\frac{ (25.1xln(z^{2})sin(x^{2}))}{y}}$

${10.6y^{3}z^{2}sin(z^{3}) -\frac{ (1.76cos(x^{2}))}{(yz)}+\frac{ (0.96xln(z^{2})sin(x^{2}))}{y}}$

${-\frac{(50.1xsin(x^{2}))}{(y^{2}z)}}$

Correct answer:

${10.6y^{3}z^{2}sin(z^{3}) -\frac{ (1.76cos(x^{2}))}{(yz)}+\frac{ (0.96xln(z^{2})sin(x^{2}))}{y}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(6)^2+(0)^2+(11)^2}=12.53\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{6}{12.53}=0.48\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{0}{12.53}=0\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{11}{12.53}=0.88\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0.48)(\frac{(2xln(z^{2})sin(x^{2}))}{y})+(0)(\frac{(cos(x^{2})ln(z^{2}))}{y^{2}}- 12y^{2}cos(z^{3}))+(0.88)(12y^{3}z^{2}sin(z^{3}) -\frac{ (2cos(x^{2}))}{(yz)})\\&D_{\overrightarrow{u}}(x,y,z)=10.6y^{3}z^{2}sin(z^{3}) -\frac{ (1.76cos(x^{2}))}{(yz)}+\frac{ (0.96xln(z^{2})sin(x^{2}))}{y}\end{align*}$

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Example Question #289 : Directional Derivatives

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

Possible Answers:

${-\frac{ (21.6xe^{(x^{2})}cos(z))}{y^{2}}- 21.6xzsin(x^{2})sin(z^{2})sin(y)}$

${\frac{(0.274e^{(x^{2})}cos(z))}{y}-\frac{ (1.73e^{(x^{2})}sin(z))}{y^{2}}- 0.865cos(x^{2})cos(z^{2})sin(y) - 0.274zcos(x^{2})sin(z^{2})cos(y)}$

${\frac{(10.8e^{(x^{2})}cos(z))}{y}-\frac{ (10.8e^{(x^{2})}sin(z))}{y^{2}}- 5.39cos(x^{2})cos(z^{2})sin(y) - 10.8xcos(z^{2})sin(x^{2})cos(y) - 10.8zcos(x^{2})sin(z^{2})cos(y) +\frac{ (21.6xe^{(x^{2})}sin(z))}{y}}$

${0.74zcos(x^{2})sin(z^{2})cos(y) -\frac{ (1.86e^{(x^{2})}sin(z))}{y^{2}}- 0.93cos(x^{2})cos(z^{2})sin(y) -\frac{ (0.74e^{(x^{2})}cos(z))}{y}}$

Correct answer:

${0.74zcos(x^{2})sin(z^{2})cos(y) -\frac{ (1.86e^{(x^{2})}sin(z))}{y^{2}}- 0.93cos(x^{2})cos(z^{2})sin(y) -\frac{ (0.74e^{(x^{2})}cos(z))}{y}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(0)^2+(5)^2+(-2)^2}=5.39\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{0}{5.39}=0\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{5}{5.39}=0.93\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{-2}{5.39}=-0.37\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0)(\frac{(4xe^{(x^{2})}sin(z))}{y}- 2xcos(z^{2})sin(x^{2})cos(y))+(0.93)(-\frac{ (2e^{(x^{2})}sin(z))}{y^{2}}- cos(x^{2})cos(z^{2})sin(y))+(-0.37)(\frac{(2e^{(x^{2})}cos(z))}{y}- 2zcos(x^{2})sin(z^{2})cos(y))\\&D_{\overrightarrow{u}}(x,y,z)=0.74zcos(x^{2})sin(z^{2})cos(y) -\frac{ (1.86e^{(x^{2})}sin(z))}{y^{2}}- 0.93cos(x^{2})cos(z^{2})sin(y) -\frac{ (0.74e^{(x^{2})}cos(z))}{y}\end{align*}$

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Example Question #290 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=-2^{(y^{4})}cos(x^{3})ln(z)\\&\text{In the direction of }\overrightarrow{v}=(0,-5,15).\end{align*}$

Possible Answers:

${\frac{(132\cdot 2^{(y^{4})}x^{2}y^{3}sin(x^{3}))}{z}}$

${0.887\cdot 2^{(y^{4})}y^{3}cos(x^{3})ln(z) -\frac{ (0.95\cdot 2^{(y^{4})}cos(x^{3}))}{z}}$

${13.9\cdot 2^{(y^{4})}y^{3}cos(x^{3})ln(z) -\frac{ (15\cdot 2^{(y^{4})}cos(x^{3}))}{z}}$

${47.4\cdot 2^{(y^{4})}x^{2}sin(x^{3})ln(z) - 43.8\cdot 2^{(y^{4})}y^{3}cos(x^{3})ln(z) -\frac{ (15.8\cdot 2^{(y^{4})}cos(x^{3}))}{z}}$

Correct answer:

${0.887\cdot 2^{(y^{4})}y^{3}cos(x^{3})ln(z) -\frac{ (0.95\cdot 2^{(y^{4})}cos(x^{3}))}{z}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(0)^2+(-5)^2+(15)^2}=15.81\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{0}{15.81}=0\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{-5}{15.81}=-0.32\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{15}{15.81}=0.95\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0)(3\cdot 2^{(y^{4})}x^{2}sin(x^{3})ln(z))+(-0.32)(-2.77\cdot 2^{(y^{4})}y^{3}cos(x^{3})ln(z))+(0.95)(-\frac{(1\cdot 2^{(y^{4})}cos(x^{3}))}{z})\\&D_{\overrightarrow{u}}(x,y,z)=0.887\cdot 2^{(y^{4})}y^{3}cos(x^{3})ln(z) -\frac{ (0.95\cdot 2^{(y^{4})}cos(x^{3}))}{z}\end{align*}$

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Calculus 3 : Directional Derivatives

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