Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

Create An Account

All Calculus 3 Resources

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

Example Questions

Example Question #283 : 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:

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

Report an Error

Example Question #284 : 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:

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

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

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

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

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:

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

Report an Error

Example Question #285 : 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:

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

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

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

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

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:

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

Report an Error

Example Question #286 : 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:

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

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

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

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

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:

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

Report an Error

Example Question #287 : 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:

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

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

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

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

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:

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

Report an Error

Example Question #291 : Directional Derivatives

Determine the directional derivative of the following function in the direction of $\vec{v}=\left \langle 1, 3, 1\right \rangle$ :

f(x, y, z)=zx+xy+z^2y^2

Possible Answers:

$\frac{1}{\sqrt{11}}(4x+ z+y+8yz^2)$

4x+ z+y+6yz^2+2zy^2

$\frac{1}{\sqrt{11}}(4x+ z+y+6yz^2+2zy^2)$

$\frac{1}{\sqrt{11}}(x+ z+y+6yz^2+2zy^2)$

2x+ z+y+6yz^2+2zy^2

Correct answer:

$\frac{1}{\sqrt{11}}(4x+ z+y+6yz^2+2zy^2)$

Explanation:

The directional derivative is given by

$D_{\vec{u}}f(x, y, z)=f_x(x, y, z)a+f_y(x, y, z)b+f_c(x, y, z)c$ , where $\left \langle a, b, c\right \rangle$ is the unit directional vector.

Now, we must find the gradient of f, $\bigtriangledown f=\left \langle f_x, f_y, f_z\right \rangle$ :

f_x=z+y , f_y=x+2yz^2 , f_z=x+2zy^2

The partial derivatives were found by keeping the other variables constant, and using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} ax=a$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

Next, we must find the unit directional vector, which is given by

$\frac{\vec{v}}{\left | \vec{v}\right |}=\frac{\left \langle 1, 3, 1\right \rangle}{\sqrt{1^2+3^2+1^2}}=\left \langle \frac{1}{\sqrt{11}}, \frac{3}{\sqrt{11}}, \frac{1}{\sqrt{11}} \right \rangle$

Plugging all of this into the above formula, and after simplifying, we get

$\frac{1}{\sqrt{11}}(4x+ z+y+6yz^2+2zy^2)$

Report an Error

Example Question #292 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=y^{3}cos(z^{4})e^{(x^{4})}\\&\text{In the direction of }\overrightarrow{v}=(0,-15,-12).\end{align*}$

Possible Answers:

${48y^{3}z^{3}sin(z^{4})e^{(x^{4})} - 45y^{2}cos(z^{4})e^{(x^{4})}}$

${2.48y^{3}z^{3}sin(z^{4})e^{(x^{4})} - 2.34y^{2}cos(z^{4})e^{(x^{4})}}$

${57.6y^{2}cos(z^{4})e^{(x^{4})} + 76.8x^{3}y^{3}cos(z^{4})e^{(x^{4})} - 76.8y^{3}z^{3}sin(z^{4})e^{(x^{4})}}$

${-922x^{3}y^{2}z^{3}sin(z^{4})e^{(x^{4})}}$

Correct answer:

${2.48y^{3}z^{3}sin(z^{4})e^{(x^{4})} - 2.34y^{2}cos(z^{4})e^{(x^{4})}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(0)^2+(-15)^2+(-12)^2}=19.21\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{0}{19.21}=0\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{-15}{19.21}=-0.78\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{-12}{19.21}=-0.62\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0)(4x^{3}y^{3}cos(z^{4})e^{(x^{4})})+(-0.78)(3y^{2}cos(z^{4})e^{(x^{4})})+(-0.62)(-4y^{3}z^{3}sin(z^{4})e^{(x^{4})})\\&D_{\overrightarrow{u}}(x,y,z)=2.48y^{3}z^{3}sin(z^{4})e^{(x^{4})} - 2.34y^{2}cos(z^{4})e^{(x^{4})}\end{align*}$

Report an Error

Example Question #292 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=3^{(y^{4})}cos(x^{4})cos(z^{3}) - sin(x^{3})e^{(y^{2})}e^{(z^{2})}\\&\text{In the direction of }\overrightarrow{v}=(-12,0,6).\end{align*}$

Possible Answers:

${708\cdot 3^{(y^{4})}x^{3}y^{3}z^{2}sin(x^{4})sin(z^{3}) - 161x^{2}yzcos(x^{3})e^{(y^{2})}e^{(z^{2})}}$

${3.56\cdot 3^{(y^{4})}x^{3}cos(z^{3})sin(x^{4}) - 1.35\cdot 3^{(y^{4})}z^{2}cos(x^{4})sin(z^{3}) - 0.9zsin(x^{3})e^{(y^{2})}e^{(z^{2})} + 2.67x^{2}cos(x^{3})e^{(y^{2})}e^{(z^{2})}}$

${48\cdot 3^{(y^{4})}x^{3}cos(z^{3})sin(x^{4}) - 18\cdot 3^{(y^{4})}z^{2}cos(x^{4})sin(z^{3}) - 12zsin(x^{3})e^{(y^{2})}e^{(z^{2})} + 36x^{2}cos(x^{3})e^{(y^{2})}e^{(z^{2})}}$

${59\cdot 3^{(y^{4})}y^{3}cos(x^{4})cos(z^{3}) - 53.7\cdot 3^{(y^{4})}x^{3}cos(z^{3})sin(x^{4}) - 40.3\cdot 3^{(y^{4})}z^{2}cos(x^{4})sin(z^{3}) - 26.8ysin(x^{3})e^{(y^{2})}e^{(z^{2})} - 26.8zsin(x^{3})e^{(y^{2})}e^{(z^{2})} - 40.3x^{2}cos(x^{3})e^{(y^{2})}e^{(z^{2})}}$

Correct answer:

${3.56\cdot 3^{(y^{4})}x^{3}cos(z^{3})sin(x^{4}) - 1.35\cdot 3^{(y^{4})}z^{2}cos(x^{4})sin(z^{3}) - 0.9zsin(x^{3})e^{(y^{2})}e^{(z^{2})} + 2.67x^{2}cos(x^{3})e^{(y^{2})}e^{(z^{2})}}$

Explanation:

$\begin{align*}\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{0}{13.42}=0\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{6}{13.42}=0.45\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(-0.89)(- 4\cdot 3^{(y^{4})}x^{3}cos(z^{3})sin(x^{4}) - 3x^{2}cos(x^{3})e^{(y^{2})}e^{(z^{2})})+(0)(4.39\cdot 3^{(y^{4})}y^{3}cos(x^{4})cos(z^{3}) - 2ysin(x^{3})e^{(y^{2})}e^{(z^{2})})+(0.45)(- 3\cdot 3^{(y^{4})}z^{2}cos(x^{4})sin(z^{3}) - 2zsin(x^{3})e^{(y^{2})}e^{(z^{2})})\\&D_{\overrightarrow{u}}(x,y,z)=3.56\cdot 3^{(y^{4})}x^{3}cos(z^{3})sin(x^{4}) - 1.35\cdot 3^{(y^{4})}z^{2}cos(x^{4})sin(z^{3}) - 0.9zsin(x^{3})e^{(y^{2})}e^{(z^{2})} + 2.67x^{2}cos(x^{3})e^{(y^{2})}e^{(z^{2})}\end{align*}$

Report an Error

Example Question #291 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=5ln(x^{2})ln(y^{2}) + 2^{(z^{2})}xy^{2}\\&\text{In the direction of }\overrightarrow{v}=(18,20,0).\end{align*}$

Possible Answers:

${\frac{(4.49ln(y^{2}))}{x}+\frac{ (5.48ln(x^{2}))}{y}+ 0.449\cdot 2^{(z^{2})}y^{2} + 1.1\cdot 2^{(z^{2})}xy}$

${74.6\cdot 2^{(z^{2})}yz}$

${\frac{(269ln(y^{2}))}{x}+\frac{ (269ln(x^{2}))}{y}+ 26.9\cdot 2^{(z^{2})}y^{2} + 53.8\cdot 2^{(z^{2})}xy + 37.3\cdot 2^{(z^{2})}xy^{2}z}$

${\frac{(6.7ln(y^{2}))}{x}+\frac{ (7.4ln(x^{2}))}{y}+ 0.67\cdot 2^{(z^{2})}y^{2} + 1.48\cdot 2^{(z^{2})}xy}$

Correct answer:

${\frac{(6.7ln(y^{2}))}{x}+\frac{ (7.4ln(x^{2}))}{y}+ 0.67\cdot 2^{(z^{2})}y^{2} + 1.48\cdot 2^{(z^{2})}xy}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(18)^2+(20)^2+(0)^2}=26.91\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{18}{26.91}=0.67\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{20}{26.91}=0.74\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{0}{26.91}=0\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0.67)(\frac{(10ln(y^{2}))}{x}+ 2^{(z^{2})}y^{2})+(0.74)(\frac{(10ln(x^{2}))}{y}+ 2\cdot 2^{(z^{2})}xy)+(0)(1.39\cdot 2^{(z^{2})}xy^{2}z)\\&D_{\overrightarrow{u}}(x,y,z)=\frac{(6.7ln(y^{2}))}{x}+\frac{ (7.4ln(x^{2}))}{y}+ 0.67\cdot 2^{(z^{2})}y^{2} + 1.48\cdot 2^{(z^{2})}xy\end{align*}$

Report an Error

Example Question #294 : Directional Derivatives

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

Possible Answers:

${\frac{(75.8zcos(z^{2}))}{(x^{3}y)}+\frac{ (114x^{2}y^{2}cos(x^{3})sin(y^{3}))}{z^{2}}}$

${\frac{(1.9cos(y^{3})sin(x^{3}))}{z^{2}}+\frac{ (0.64ln(y^{3})sin(z^{2}))}{x^{3}}+\frac{ (1.9zcos(z^{2})ln(y^{3}))}{x^{2}}+\frac{ (1.92x^{2}cos(x^{3})cos(y^{3}))}{z}}$

${\frac{(12.6ln(y^{3})sin(z^{2}))}{x^{3}}-\frac{ (12.6cos(y^{3})sin(x^{3}))}{z^{2}}-\frac{ (19sin(z^{2}))}{(x^{2}y)}-\frac{ (12.6zcos(z^{2})ln(y^{3}))}{x^{2}}+\frac{ (37.9x^{2}cos(x^{3})cos(y^{3}))}{z}-\frac{ (37.9y^{2}sin(x^{3})sin(y^{3}))}{z}}$

${\frac{(12cos(y^{3})sin(x^{3}))}{z^{2}}+\frac{ (4ln(y^{3})sin(z^{2}))}{x^{3}}+\frac{ (12zcos(z^{2})ln(y^{3}))}{x^{2}}+\frac{ (12x^{2}cos(x^{3})cos(y^{3}))}{z}}$

Correct answer:

${\frac{(1.9cos(y^{3})sin(x^{3}))}{z^{2}}+\frac{ (0.64ln(y^{3})sin(z^{2}))}{x^{3}}+\frac{ (1.9zcos(z^{2})ln(y^{3}))}{x^{2}}+\frac{ (1.92x^{2}cos(x^{3})cos(y^{3}))}{z}}$

Explanation:

$\begin{align*}\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{0}{6.32}=0\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{-6}{6.32}=-0.95\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0.32)(\frac{(2ln(y^{3})sin(z^{2}))}{x^{3}}+\frac{ (6x^{2}cos(x^{3})cos(y^{3}))}{z})+(0)(-\frac{ (3sin(z^{2}))}{(x^{2}y)}-\frac{ (6y^{2}sin(x^{3})sin(y^{3}))}{z})+(-0.95)(-\frac{ (2cos(y^{3})sin(x^{3}))}{z^{2}}-\frac{ (2zcos(z^{2})ln(y^{3}))}{x^{2}})\\&D_{\overrightarrow{u}}(x,y,z)=\frac{(1.9cos(y^{3})sin(x^{3}))}{z^{2}}+\frac{ (0.64ln(y^{3})sin(z^{2}))}{x^{3}}+\frac{ (1.9zcos(z^{2})ln(y^{3}))}{x^{2}}+\frac{ (1.92x^{2}cos(x^{3})cos(y^{3}))}{z}\end{align*}$

Report an Error

View Calculus 3 Tutors

Magdy
Certified Tutor

Cairo University, Egypt, Bachelor of Science, Electrical Engineering. New Mexico State University-Main Campus, Doctor of Phil...

View Calculus 3 Tutors

Anabel
Certified Tutor

UDELAR, Diploma, Chemistry. IPA, Bachelor of Education, Mathematics.

View Calculus 3 Tutors

Vikash
Certified Tutor

national institute of technology Jamshedpur , Bachelor, Civil engineering .

All Calculus 3 Resources

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

Popular Subjects

ISEE Tutors in Denver, MCAT Tutors in Phoenix, English Tutors in Washington DC, ISEE Tutors in San Diego, GRE Tutors in New York City, SAT Tutors in Miami, Math Tutors in Dallas Fort Worth, English Tutors in Houston, French Tutors in Miami, Calculus Tutors in Los Angeles

Popular Courses & Classes

ISEE Courses & Classes in Denver, LSAT Courses & Classes in Miami, SSAT Courses & Classes in Washington DC, GRE Courses & Classes in Philadelphia, MCAT Courses & Classes in San Diego, ACT Courses & Classes in Phoenix, GMAT Courses & Classes in San Francisco-Bay Area, GRE Courses & Classes in Miami, SAT Courses & Classes in Dallas Fort Worth, GRE Courses & Classes in Houston

Popular Test Prep

MCAT Test Prep in Atlanta, GRE Test Prep in Houston, MCAT Test Prep in Phoenix, GMAT Test Prep in Boston, GRE Test Prep in New York City, ACT Test Prep in Phoenix, GMAT Test Prep in Dallas Fort Worth, GRE Test Prep in Los Angeles, SAT Test Prep in Washington DC, SSAT Test Prep in Chicago

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

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

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #283 : Directional Derivatives

Example Question #284 : Directional Derivatives

Example Question #285 : Directional Derivatives

Example Question #286 : Directional Derivatives

Example Question #287 : Directional Derivatives

Example Question #291 : Directional Derivatives

Example Question #292 : Directional Derivatives

Example Question #292 : Directional Derivatives

Example Question #291 : Directional Derivatives

Example Question #294 : Directional Derivatives

All Calculus 3 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: