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Calculus 3 : Calculus 3

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Example Questions

Example Question #246 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=3^{(x^{3})}z^{4}sin(y)\\&\text{In the direction of }\overrightarrow{v}=(0,9,15).\end{align*}$

Possible Answers:

${17.5\cdot 3^{(x^{3})}z^{4}cos(y) + 70\cdot 3^{(x^{3})}z^{3}sin(y) + 57.6\cdot 3^{(x^{3})}x^{2}z^{4}sin(y)}$

${231\cdot 3^{(x^{3})}x^{2}z^{3}cos(y)}$

${0.26\cdot 3^{(x^{3})}z^{4}cos(y) + 2.96\cdot 3^{(x^{3})}z^{3}sin(y)}$

${0.51\cdot 3^{(x^{3})}z^{4}cos(y) + 3.44\cdot 3^{(x^{3})}z^{3}sin(y)}$

Correct answer:

${0.51\cdot 3^{(x^{3})}z^{4}cos(y) + 3.44\cdot 3^{(x^{3})}z^{3}sin(y)}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(0)^2+(9)^2+(15)^2}=17.49\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{0}{17.49}=0\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{9}{17.49}=0.51\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{15}{17.49}=0.86\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0)(3.3\cdot 3^{(x^{3})}x^{2}z^{4}sin(y))+(0.51)(3^{(x^{3})}z^{4}cos(y))+(0.86)(4\cdot 3^{(x^{3})}z^{3}sin(y))\\&D_{\overrightarrow{u}}(x,y,z)=0.51\cdot 3^{(x^{3})}z^{4}cos(y) + 3.44\cdot 3^{(x^{3})}z^{3}sin(y)\end{align*}$

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Example Question #247 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=\frac{ln(z^{4})}{x^{2}}- 2yz^{3}sin(x^{2})\\&\text{In the direction of }\overrightarrow{v}=(0,-14,-6).\end{align*}$

Possible Answers:

${\frac{60.9}{(x^{2}z)}-\frac{ (30.5ln(z^{4}))}{x^{3}}- 30.5z^{3}sin(x^{2}) - 91.4yz^{2}sin(x^{2}) - 60.9xyz^{3}cos(x^{2})}$

${-183xz^{2}cos(x^{2})}$

${1.84z^{3}sin(x^{2}) -\frac{ 1.56}{(x^{2}z)}+ 2.34yz^{2}sin(x^{2})}$

${\frac{0.608}{(x^{2}z)}- 1.69z^{3}sin(x^{2}) - 0.913yz^{2}sin(x^{2})}$

Correct answer:

${1.84z^{3}sin(x^{2}) -\frac{ 1.56}{(x^{2}z)}+ 2.34yz^{2}sin(x^{2})}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(0)^2+(-14)^2+(-6)^2}=15.23\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{0}{15.23}=0\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{-14}{15.23}=-0.92\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{-6}{15.23}=-0.39\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0)(-\frac{ (2ln(z^{4}))}{x^{3}}- 4xyz^{3}cos(x^{2}))+(-0.92)(-2z^{3}sin(x^{2}))+(-0.39)(\frac{4.0}{(x^{2}z)}- 6yz^{2}sin(x^{2}))\\&D_{\overrightarrow{u}}(x,y,z)=1.84z^{3}sin(x^{2}) -\frac{ 1.56}{(x^{2}z)}+ 2.34yz^{2}sin(x^{2})\end{align*}$

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Example Question #247 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=cos(z^{4})e^{(x^{3})}sin(y)\\&\text{In the direction of }\overrightarrow{v}=(-9,10,0).\end{align*}$

Possible Answers:

${0.548cos(z^{4})e^{(x^{3})}cos(y) + 1.35x^{2}cos(z^{4})e^{(x^{3})}sin(y)}$

${-161x^{2}z^{3}sin(z^{4})e^{(x^{3})}cos(y)}$

${0.74cos(z^{4})e^{(x^{3})}cos(y) - 2.01x^{2}cos(z^{4})e^{(x^{3})}sin(y)}$

${13.4cos(z^{4})e^{(x^{3})}cos(y) + 40.3x^{2}cos(z^{4})e^{(x^{3})}sin(y) - 53.8z^{3}sin(z^{4})e^{(x^{3})}sin(y)}$

Correct answer:

${0.74cos(z^{4})e^{(x^{3})}cos(y) - 2.01x^{2}cos(z^{4})e^{(x^{3})}sin(y)}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(-9)^2+(10)^2+(0)^2}=13.45\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{-9}{13.45}=-0.67\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{10}{13.45}=0.74\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{0}{13.45}=0\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(-0.67)(3x^{2}cos(z^{4})e^{(x^{3})}sin(y))+(0.74)(cos(z^{4})e^{(x^{3})}cos(y))+(0)(-4z^{3}sin(z^{4})e^{(x^{3})}sin(y))\\&D_{\overrightarrow{u}}(x,y,z)=0.74cos(z^{4})e^{(x^{3})}cos(y) - 2.01x^{2}cos(z^{4})e^{(x^{3})}sin(y)\end{align*}$

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Example Question #248 : Directional Derivatives

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

Possible Answers:

${19.2e^{(z)}cos(x)tan(y) + 19.2e^{(z)}sin(x)\cdot (tan(y)^{2} + 1) + 19.2e^{(z)}sin(x)tan(y) -\frac{ (13.4\cdot 3^ye^{(x)})}{z^{2}}+\frac{ (12.8\cdot 3^ye^{(x)})}{z^{3}}}$

${19.2e^{(z)}cos(x)\cdot (tan(y)^{2} + 1) +\frac{ (14.1\cdot 3^ye^{(x)})}{z^{3}}}$

${1.86e^{(z)}sin(x)\cdot (tan(y)^{2} + 1) - 2.34e^{(z)}cos(x)tan(y) +\frac{ (0.0989\cdot 3^ye^{(x)})}{z^{2}}}$

${1.83e^{(z)}cos(x)tan(y) + 1.15e^{(z)}sin(x)\cdot (tan(y)^{2} + 1) -\frac{ (1.03\cdot 3^ye^{(x)})}{z^{2}}}$

Correct answer:

${1.86e^{(z)}sin(x)\cdot (tan(y)^{2} + 1) - 2.34e^{(z)}cos(x)tan(y) +\frac{ (0.0989\cdot 3^ye^{(x)})}{z^{2}}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(-5)^2+(4)^2+(0)^2}=6.4\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{-5}{6.4}=-0.78\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{4}{6.4}=0.62\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{0}{6.4}=0\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(-0.78)(3e^{(z)}cos(x)tan(y) -\frac{ (1\cdot 3^ye^{(x)})}{z^{2}})+(0.62)(3e^{(z)}sin(x)\cdot (tan(y)^{2} + 1) -\frac{ (1.1\cdot 3^ye^{(x)})}{z^{2}})+(0)(3e^{(z)}sin(x)tan(y) +\frac{ (2\cdot 3^ye^{(x)})}{z^{3}})\\&D_{\overrightarrow{u}}(x,y,z)=1.86e^{(z)}sin(x)\cdot (tan(y)^{2} + 1) - 2.34e^{(z)}cos(x)tan(y) +\frac{ (0.0989\cdot 3^ye^{(x)})}{z^{2}}\end{align*}$

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Example Question #249 : Directional Derivatives

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

Possible Answers:

${3.46\cdot 3^{(y^{4})}x^{3}cos(x^{4})ln(z^{3}) + 0.602\cdot 3^{(y^{4})}y^{3}ln(z^{3})sin(x^{4}) + 6.92xcos(x^{2})tan(y^{2})tan(z^{2}) + 1.1ysin(x^{2})tan(z^{2})\cdot (tan(y^{2})^{2} + 1)}$

${\frac{(16.2\cdot 3^{(y^{4})}sin(x^{4}))}{z}+ 21.6\cdot 3^{(y^{4})}x^{3}cos(x^{4})ln(z^{3}) + 23.7\cdot 3^{(y^{4})}y^{3}ln(z^{3})sin(x^{4}) + 43.1xcos(x^{2})tan(y^{2})tan(z^{2}) + 43.1ysin(x^{2})tan(z^{2})\cdot (tan(y^{2})^{2} + 1) + 43.1zsin(x^{2})tan(y^{2})\cdot (tan(z^{2})^{2} + 1)}$

${- 3.72\cdot 3^{(y^{4})}x^{3}cos(x^{4})ln(z^{3}) - 1.63\cdot 3^{(y^{4})}y^{3}ln(z^{3})sin(x^{4}) - 7.44xcos(x^{2})tan(y^{2})tan(z^{2}) - 2.96ysin(x^{2})tan(z^{2})\cdot (tan(y^{2})^{2} + 1)}$

${\frac{(284\cdot 3^{(y^{4})}x^{3}y^{3}cos(x^{4}))}{z}+ 172xyzcos(x^{2})\cdot (tan(y^{2})^{2} + 1)\cdot (tan(z^{2})^{2} + 1)}$

Correct answer:

${- 3.72\cdot 3^{(y^{4})}x^{3}cos(x^{4})ln(z^{3}) - 1.63\cdot 3^{(y^{4})}y^{3}ln(z^{3})sin(x^{4}) - 7.44xcos(x^{2})tan(y^{2})tan(z^{2}) - 2.96ysin(x^{2})tan(z^{2})\cdot (tan(y^{2})^{2} + 1)}$

Explanation:

$\begin{align*}\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{-2}{5.39}=-0.37\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{0}{5.39}=0\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(-0.93)(4\cdot 3^{(y^{4})}x^{3}cos(x^{4})ln(z^{3}) + 8xcos(x^{2})tan(y^{2})tan(z^{2}))+(-0.37)(4.39\cdot 3^{(y^{4})}y^{3}ln(z^{3})sin(x^{4}) + 8ysin(x^{2})tan(z^{2})\cdot (tan(y^{2})^{2} + 1))+(0)(\frac{(3\cdot 3^{(y^{4})}sin(x^{4}))}{z}+ 8zsin(x^{2})tan(y^{2})\cdot (tan(z^{2})^{2} + 1))\\&D_{\overrightarrow{u}}(x,y,z)=- 3.72\cdot 3^{(y^{4})}x^{3}cos(x^{4})ln(z^{3}) - 1.63\cdot 3^{(y^{4})}y^{3}ln(z^{3})sin(x^{4}) - 7.44xcos(x^{2})tan(y^{2})tan(z^{2}) - 2.96ysin(x^{2})tan(z^{2})\cdot (tan(y^{2})^{2} + 1)\end{align*}$

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Example Question #251 : Directional Derivatives

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

Possible Answers:

${\frac{(124xzsin(z^{2}))}{y}+\frac{ (1244yz^{3}e^{(y^{2})})}{x}}$

${\frac{(155z^{4}e^{(y^{2})})}{x}-\frac{ (31x^{2}cos(z^{2}))}{y}- 31xcos(z^{2})ln(y^{2}) + 310z^{3}ln(x^{2})e^{(y^{2})} + 155yz^{4}ln(x^{2})e^{(y^{2})} + 31x^{2}zln(y^{2})sin(z^{2})}$

${1.35z^{3}ln(x^{2})e^{(y^{2})} -\frac{ (1.88x^{2}cos(z^{2}))}{y}+ 9.41yz^{4}ln(x^{2})e^{(y^{2})} + 0.135x^{2}zln(y^{2})sin(z^{2})}$

${\frac{(1.94x^{2}cos(z^{2}))}{y}- 5.2z^{3}ln(x^{2})e^{(y^{2})} - 9.7yz^{4}ln(x^{2})e^{(y^{2})} - 0.52x^{2}zln(y^{2})sin(z^{2})}$

Correct answer:

${\frac{(1.94x^{2}cos(z^{2}))}{y}- 5.2z^{3}ln(x^{2})e^{(y^{2})} - 9.7yz^{4}ln(x^{2})e^{(y^{2})} - 0.52x^{2}zln(y^{2})sin(z^{2})}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(0)^2+(-15)^2+(-4)^2}=15.52\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{0}{15.52}=0\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{-15}{15.52}=-0.97\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{-4}{15.52}=-0.26\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0)(\frac{(10z^{4}e^{(y^{2})})}{x}- 2xcos(z^{2})ln(y^{2}))+(-0.97)(10yz^{4}ln(x^{2})e^{(y^{2})} -\frac{ (2x^{2}cos(z^{2}))}{y})+(-0.26)(20z^{3}ln(x^{2})e^{(y^{2})} + 2x^{2}zln(y^{2})sin(z^{2}))\\&D_{\overrightarrow{u}}(x,y,z)=\frac{(1.94x^{2}cos(z^{2}))}{y}- 5.2z^{3}ln(x^{2})e^{(y^{2})} - 9.7yz^{4}ln(x^{2})e^{(y^{2})} - 0.52x^{2}zln(y^{2})sin(z^{2})\end{align*}$

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Example Question #252 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=x^{4}sin(y^{3})sin(z^{4})\\&\text{In the direction of }\overrightarrow{v}=(18,9,0).\end{align*}$

Possible Answers:

${80.5x^{3}sin(y^{3})sin(z^{4}) + 60.4x^{4}y^{2}cos(y^{3})sin(z^{4}) + 80.5x^{4}z^{3}cos(z^{4})sin(y^{3})}$

${72x^{3}sin(y^{3})sin(z^{4}) + 27x^{4}y^{2}cos(y^{3})sin(z^{4})}$

${966x^{3}y^{2}z^{3}cos(y^{3})cos(z^{4})}$

${3.56x^{3}sin(y^{3})sin(z^{4}) + 1.35x^{4}y^{2}cos(y^{3})sin(z^{4})}$

Correct answer:

${3.56x^{3}sin(y^{3})sin(z^{4}) + 1.35x^{4}y^{2}cos(y^{3})sin(z^{4})}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(18)^2+(9)^2+(0)^2}=20.12\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{18}{20.12}=0.89\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{9}{20.12}=0.45\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{0}{20.12}=0\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0.89)(4x^{3}sin(y^{3})sin(z^{4}))+(0.45)(3x^{4}y^{2}cos(y^{3})sin(z^{4}))+(0)(4x^{4}z^{3}cos(z^{4})sin(y^{3}))\\&D_{\overrightarrow{u}}(x,y,z)=3.56x^{3}sin(y^{3})sin(z^{4}) + 1.35x^{4}y^{2}cos(y^{3})sin(z^{4})\end{align*}$

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Example Question #253 : Directional Derivatives

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

Possible Answers:

${- 21.7y^{2}zsin(y^{3})e^{(z^{2})} -\frac{ (28.9xycos(x^{2})cos(y^{2}))}{z^{3}}}$

${3.61cos(y^{3})e^{(z^{2})} -\frac{ (7.22sin(x^{2})sin(y^{2}))}{z^{3}}+\frac{ (7.22xcos(x^{2})sin(y^{2}))}{z^{2}}+\frac{ (7.22ycos(y^{2})sin(x^{2}))}{z^{2}}- 10.8xy^{2}sin(y^{3})e^{(z^{2})} + 7.22xzcos(y^{3})e^{(z^{2})}}$

${3cos(y^{3})e^{(z^{2})} -\frac{ (4sin(x^{2})sin(y^{2}))}{z^{3}}+\frac{ (6xcos(x^{2})sin(y^{2}))}{z^{2}}+ 4xzcos(y^{3})e^{(z^{2})}}$

${0.83cos(y^{3})e^{(z^{2})} -\frac{ (1.1sin(x^{2})sin(y^{2}))}{z^{3}}+\frac{ (1.66xcos(x^{2})sin(y^{2}))}{z^{2}}+ 1.1xzcos(y^{3})e^{(z^{2})}}$

Correct answer:

${0.83cos(y^{3})e^{(z^{2})} -\frac{ (1.1sin(x^{2})sin(y^{2}))}{z^{3}}+\frac{ (1.66xcos(x^{2})sin(y^{2}))}{z^{2}}+ 1.1xzcos(y^{3})e^{(z^{2})}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(3)^2+(0)^2+(2)^2}=3.61\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{3}{3.61}=0.83\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{0}{3.61}=0\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{2}{3.61}=0.55\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0.83)(cos(y^{3})e^{(z^{2})} +\frac{ (2xcos(x^{2})sin(y^{2}))}{z^{2}})+(0)(\frac{(2ycos(y^{2})sin(x^{2}))}{z^{2}}- 3xy^{2}sin(y^{3})e^{(z^{2})})+(0.55)(2xzcos(y^{3})e^{(z^{2})} -\frac{ (2sin(x^{2})sin(y^{2}))}{z^{3}})\\&D_{\overrightarrow{u}}(x,y,z)=0.83cos(y^{3})e^{(z^{2})} -\frac{ (1.1sin(x^{2})sin(y^{2}))}{z^{3}}+\frac{ (1.66xcos(x^{2})sin(y^{2}))}{z^{2}}+ 1.1xzcos(y^{3})e^{(z^{2})}\end{align*}$

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Example Question #254 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=2e^{(z^{2})}e^{(y)} +\frac{ (cos(z)ln(x))}{y}\\&\text{In the direction of }\overrightarrow{v}=(-13,0,16).\end{align*}$

Possible Answers:

${\frac{(0.397cos(z))}{(xy)}-\frac{ (0.608ln(x)sin(z))}{y}+ 2.43ze^{(z^{2})}e^{(y)}}$

${3.12ze^{(z^{2})}e^{(y)} -\frac{ (0.63cos(z))}{(xy)}-\frac{ (0.78ln(x)sin(z))}{y}}$

${\frac{(20.6sin(z))}{(xy^{2})}}$

${41.2e^{(z^{2})}e^{(y)} -\frac{ (20.6ln(x)sin(z))}{y}+\frac{ (20.6cos(z))}{(xy)}+ 82.5ze^{(z^{2})}e^{(y)} -\frac{ (20.6cos(z)ln(x))}{y^{2}}}$

Correct answer:

${3.12ze^{(z^{2})}e^{(y)} -\frac{ (0.63cos(z))}{(xy)}-\frac{ (0.78ln(x)sin(z))}{y}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(-13)^2+(0)^2+(16)^2}=20.62\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{-13}{20.62}=-0.63\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{0}{20.62}=0\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{16}{20.62}=0.78\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(-0.63)(\frac{cos(z)}{(xy)})+(0)(2e^{(z^{2})}e^{(y)} -\frac{ (cos(z)ln(x))}{y^{2}})+(0.78)(4ze^{(z^{2})}e^{(y)} -\frac{ (ln(x)sin(z))}{y})\\&D_{\overrightarrow{u}}(x,y,z)=3.12ze^{(z^{2})}e^{(y)} -\frac{ (0.63cos(z))}{(xy)}-\frac{ (0.78ln(x)sin(z))}{y}\end{align*}$

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Example Question #255 : Directional Derivatives

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

Possible Answers:

${155xe^{(x^{2})}e^{(y)}sin(z) -\frac{ (15.5e^{(x)})}{(y^{2}z)}}$

${4.85e^{(x^{2})}e^{(y)}cos(z) -\frac{ (0.26e^{(x)}ln(z))}{y}+\frac{ (0.97e^{(x)}ln(z))}{y^{2}}+ 2.6xe^{(x^{2})}e^{(y)}cos(z)}$

${75e^{(x^{2})}e^{(y)}cos(z) -\frac{ (4e^{(x)}ln(z))}{y}+\frac{ (15e^{(x)}ln(z))}{y^{2}}+ 40xe^{(x^{2})}e^{(y)}cos(z)}$

${77.6e^{(x^{2})}e^{(y)}sin(z) - 77.6e^{(x^{2})}e^{(y)}cos(z) +\frac{ (15.5e^{(x)})}{(yz)}+\frac{ (15.5e^{(x)}ln(z))}{y}-\frac{ (15.5e^{(x)}ln(z))}{y^{2}}- 155xe^{(x^{2})}e^{(y)}cos(z)}$

Correct answer:

${4.85e^{(x^{2})}e^{(y)}cos(z) -\frac{ (0.26e^{(x)}ln(z))}{y}+\frac{ (0.97e^{(x)}ln(z))}{y^{2}}+ 2.6xe^{(x^{2})}e^{(y)}cos(z)}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(-4)^2+(-15)^2+(0)^2}=15.52\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{-4}{15.52}=-0.26\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{-15}{15.52}=-0.97\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{0}{15.52}=0\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(-0.26)(\frac{(e^{(x)}ln(z))}{y}- 10xe^{(x^{2})}e^{(y)}cos(z))+(-0.97)(- 5e^{(x^{2})}e^{(y)}cos(z) -\frac{ (e^{(x)}ln(z))}{y^{2}})+(0)(5e^{(x^{2})}e^{(y)}sin(z) +\frac{ e^{(x)}}{(yz)})\\&D_{\overrightarrow{u}}(x,y,z)=4.85e^{(x^{2})}e^{(y)}cos(z) -\frac{ (0.26e^{(x)}ln(z))}{y}+\frac{ (0.97e^{(x)}ln(z))}{y^{2}}+ 2.6xe^{(x^{2})}e^{(y)}cos(z)\end{align*}$

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