Question 1

A partial derivative is a specific example of a directional derivative.

Accepted Answer

A) True 
 B)False

Question 2

Let $z = f ( x , y ) = e ^ { ( x / y ) }$ .Find the instantaneous rate of change of f at the point $( 5,5 )$ in the direction of the point $( 8,11 )$ .

Accepted Answer

The instantaneous ra

Question 3

Given that $g \operatorname { gd } g ( 1,1,5 ) = 3 \vec { i } - 3 \vec { j } + \vec { k }$ , in which of the following directions does g increase the fastest?

Accepted Answer

A)  $7 \vec { i } - 5 \vec { j } + 2 \vec { k }$ 
B)  $\vec { i } - \vec { k }$ 
C)  $- 3 \vec { i } + 3 \vec { j } + 2 \vec { k }$ 
A)  $7 \vec { i } - 5 \vec { j } + 2 \vec { k }$ 
B)  $\vec { i } - \vec { k }$ 
C)  $- 3 \vec { i } + 3 \vec { j } + 2 \vec { k }$

Question 4

If $f ( x , y ) = \sin ( x ) + y ^ { 2 }$ x(u, v)= uv and y(u, v)= u + 3v.
If H(u, v)= f(x(u, v), y(u, v)), what is Hv(0,-2)? Give your answer to 4 decimal places.

Accepted Answer

Question 5

If $f ( x , y ) = \ln ( y )$ , then $\nabla f ( x , y ) = \frac { 1 } { y }$ .

Accepted Answer

A) True 
 B)False

Question 6

The quadratic Taylor approximations of $f ( x , y )$ at the points $( - 1 - 1 ) , ( 1,1 )$ , (2,0)and (0,2)are given respectively by: $\begin{array} { l } 
Q _ { 1 } ( x , y ) = - 3 + 3 ( x + 1 ) ^ { 2 } + 4 ( y + 1 ) ^ { 2 } \
Q _ { 2 } ( x , y ) = 1 - 3 ( x - 1 ) ^ { 2 } + 4 ( y - 1 ) ^ { 2 } \
Q _ { 3 } ( x , y ) = - 2 - 9 ( x - 2 ) - 6 ( x - 2 ) ^ { 2 } - 2 y ^ { 2 } \
Q _ { 4 } ( x , y ) = 8 + 3 x + 24 ( y - 2 ) + 22 ( y - 2 ) ^ { 2 }
\end{array}$ Determine fy
 
 $( 0,2 )$ .

Accepted Answer

Question 7

The tangent plane to the surface $z = x \sin ( x / y ) , y \neq 0$ at any point passes through the origin.

Accepted Answer

A) True 
 B)False

Question 8

Let $g ( x , y ) = \left( x ^ { 2 } + y ^ { 2 } \right) f ( x , y )$ , where $f ( x , y ) = \left\{ \begin{array} { l l } 
- | y | & | y | \leq | x | \
- | x | & | x | \leq | y |
\end{array} \right.$ g is differentiable at (0,0).

Accepted Answer

A) True 
 B)False

Question 9

The depth of a pond at the point with coordinates (x, y)is given by $h ( x , y ) = 5 x ^ { 2 } + 4 y ^ { 2 }$ .(Assume that x, y, and h are measured in feet.)If a boat at the point (-3, -5)is sailing in the direction of the vector $5 \vec { i } + \vec { j }$ ,
then at what rate is the depth changing?

Accepted Answer

The answer of The depth of a pond at the...

Question 10

Arrange the following quantities in ascending order.(The level curves of $z = f ( x , y )$ are shown in the figure.Assume that the scales on the x- and y-axes are the same.) $f _ { y } ( P ) , f _ { y } ( Q ) , f _ { x } ( R ) , - f _ { x } ( S )$

Accepted Answer

The answer of Arrange the following quantities in ascending order.(The...

Question 11

Let $f ( x , y , z ) = x ^ { 3 } + 3 x z + 4 y z + z ^ { 2 }$ .Find   , where $\vec { v }$ is a unit vector in the direction of $- \vec { i } + \vec { j } - \vec { k }$ .

Accepted Answer

The answer of Let \(f ( x , y ,...

Question 12

Find the equation of the tangent plane to the surface $3 x y z + x ^ { 3 } + y ^ { 3 } + z ^ { 3 } = 15$ at (1, -3, 2).

Accepted Answer

A)  $- 15 x + 33 y + 3 z = - 108$ 
B)  $- 15 x + 33 y + 3 z = 108$ 
C)  $15 x + 33 y + 3 z = - 108$ 
D)  $- 15 x - 33 y + 3 z = - 108$ 
A)  $- 15 x + 33 y + 3 z = - 108$ 
B)  $- 15 x + 33 y + 3 z = 108$ 
C)  $15 x + 33 y + 3 z = - 108$ 
D)  $- 15 x - 33 y + 3 z = - 108$

Question 13

The quantity z can be expressed as a function of x and y as follows: z = f(x, y).Now x and y are themselves functions of r and $\theta$, as follows: $x = g ( r , \theta )$ and $y = h ( r , \theta )$ Suppose you know that g(1, $\pi$/2)= -1, and h(1,  $\pi$/2)= 1.In addition, you are told that $$\begin{array} { l } 
\frac { \partial f } { \partial x } ( - 1,1 ) = 1 , \quad \frac { \partial f } { \partial y } ( - 1,1 ) = 6 , \quad \frac { \partial g } { \partial r } \left( 1 , \frac { \pi } { 2 } \right) = 7 \
\frac { \partial g } { \partial \theta } \left( 1 , \frac { \pi } { 2 } \right) = 7 , \quad \frac { \partial h } { \partial r } \left( 1 , \frac { \pi } { 2 } \right) = 6 , \quad \frac { \partial h } { \partial \theta } \left( 1 , \frac { \pi } { 2 } \right) = 4
\end{array}$$ Find $\frac { \partial z } { \partial r } ( 1 , \pi / 2 )$

Accepted Answer

The answer of The quantity z can be expressed as...

Question 14

Find the following partial derivative: fxy if $f ( x , y ) = x ^ { 7 } y ^ { 8 }$ .

Accepted Answer

A)  $f _ { x y } = 8 x ^ { 7 } y ^ { 7 }$ 
B)  $f _ { x y } = 7 x ^ { 6 } y ^ { 8 }$ 
C)  $f _ { x y } = 8 x ^ { 6 } y ^ { 7 }$ 
D)  $f _ { x y } = 7 x ^ { 6 } y ^ { 7 }$ 
E)  $f _ { x y } = 56 x ^ { 6 } y ^ { 7 }$ 
A)  $f _ { x y } = 8 x ^ { 7 } y ^ { 7 }$ 
B)  $f _ { x y } = 7 x ^ { 6 } y ^ { 8 }$ 
C)  $f _ { x y } = 8 x ^ { 6 } y ^ { 7 }$ 
D)  $f _ { x y } = 7 x ^ { 6 } y ^ { 7 }$ 
E)  $f _ { x y } = 56 x ^ { 6 } y ^ { 7 }$

Question 15

Find $\frac { \partial } { \partial x } \left( \ln \left( x ^ { 5 } y + 5 \right) \right)$ .

Accepted Answer

A)  $\frac { 5 x ^ { 4 } y } { x ^ { 5 } y + 5 }$ 
B)  $\frac { 5 x ^ { 4 } } { x ^ { 5 } y + 5 }$ 
C)  $\frac { 5 x ^ { 4 } y } { x ^ { 5 } + 5 }$ 
D)  $\frac { x ^ { 4 } y } { x ^ { 5 } y + 5 }$ 
A)  $\frac { 5 x ^ { 4 } y } { x ^ { 5 } y + 5 }$ 
B)  $\frac { 5 x ^ { 4 } } { x ^ { 5 } y + 5 }$ 
C)  $\frac { 5 x ^ { 4 } y } { x ^ { 5 } + 5 }$ 
D)  $\frac { x ^ { 4 } y } { x ^ { 5 } y + 5 }$

Question 16

Suppose that the price P (in dollars)to purchase a used car is a function of C, its original cost (in dollars), and its age A (in years).So P = f(C,A). What is the sign of $\frac { \partial P } { \partial C } ?$

Accepted Answer

A) Positive 
B) Negative 
A) Positive 
B) Negative

Question 17

Let $\Gamma ( u , v , w )$ be a function of three variables with $F _ { u } + F _ { w } = - 1$ .Suppose that $G ( x , y ) = F ( x , x - y , x )$ .Simplify $G _ { x } + G _ { y }$ .

Accepted Answer

A)  $G _ { x } + G _ { y } = 0$ . 
B)  $G _ { x } + G _ { y } = - 1$ . 
C)  $G _ { x } + G _ { y } = 1$ . 
D)  $G _ { x } + G _ { y } = - 2$ . 
A)  $G _ { x } + G _ { y } = 0$ . 
B)  $G _ { x } + G _ { y } = - 1$ . 
C)  $G _ { x } + G _ { y } = 1$ . 
D)  $G _ { x } + G _ { y } = - 2$ .

Question 18

Consider the level curves shown for the function z = f(x, y).   Determine the sign of $f _ { y x } ( - 1 , - 5 )$

Accepted Answer

A) Positive 
B) Negative 
A) Positive 
B) Negative

Question 19

Find the directional derivative of $f ( x , y , z ) = 3 x y z ^ { 2 } - 4 x z$ at the point (3, 3, 2), in the direction of the vector $\vec { v } = 2 \vec { i } + 3 \vec { j } - \vec { k }$

Accepted Answer

The directional deri

Question 20

If $f ( x , y ) = \sin ( x ) + y ^ { 2 }$ x(u, v)= uv and y(u, v)= u + 4v.
If H(u, v)= f(x(u, v), y(u, v)), what is H(0,-1)? Give your answer to 4 decimal places.

Accepted Answer

The answer of If \(f ( x , y )...

A partial derivative is a specific example of a directional derivative.

Let $z = f ( x , y ) = e ^ { ( x / y ) }$ .Find the instantaneous rate of change of f at the point $( 5,5 )$ in the direction of the point $( 8,11 )$ .

Given that $g \operatorname { gd } g ( 1,1,5 ) = 3 \vec { i } - 3 \vec { j } + \vec { k }$ , in which of the following directions does g increase the fastest?

If $f ( x , y ) = \sin ( x ) + y ^ { 2 }$ x(u, v)= uv and y(u, v)= u + 3v. If H(u, v)= f(x(u, v), y(u, v)), what is H_v(0,-2)? Give your answer to 4 decimal places.

If $f ( x , y ) = \ln ( y )$ , then $\nabla f ( x , y ) = \frac { 1 } { y }$ .

The quadratic Taylor approximations of $f ( x , y )$ at the points $( - 1 - 1 ) , ( 1,1 )$ , (2,0)and (0,2)are given respectively by: (x,y)=-3+3(x+1+4(y+1 (x,y)=1-3(x-1+4(y-1 (x,y)=-2-9(x-2)-6(x-2-2 (x,y)=8+3x+24(y-2)+22(y-2 Determine f_y $( 0,2 )$ .

The tangent plane to the surface $z = x \sin ( x / y ) , y \neq 0$ at any point passes through the origin.

Let $g ( x , y ) = \left( x ^ { 2 } + y ^ { 2 } \right) f ( x , y )$ , where $f ( x , y ) = \left\{ \begin{array} { l l } - | y | & | y | \leq | x | \\- | x | & | x | \leq | y |\end{array} \right.$ g is differentiable at (0,0).

Find the equation of the tangent plane to the surface $3 x y z + x ^ { 3 } + y ^ { 3 } + z ^ { 3 } = 15$ at (1, -3, 2).

Find the following partial derivative: f_xy if $f ( x , y ) = x ^ { 7 } y ^ { 8 }$ .

Find $\frac { \partial } { \partial x } \left( \ln \left( x ^ { 5 } y + 5 \right) \right)$ .

Suppose that the price P (in dollars)to purchase a used car is a function of C, its original cost (in dollars), and its age A (in years).So P = f(C,A). What is the sign of $\frac { \partial P } { \partial C } ?$

Let $\Gamma ( u , v , w )$ be a function of three variables with $F _ { u } + F _ { w } = - 1$ .Suppose that $G ( x , y ) = F ( x , x - y , x )$ .Simplify $G _ { x } + G _ { y }$ .

Consider the level curves shown for the function z = f(x, y). $Consider the level curves shown for the function z = f(x, y). Determine the sign of f _ { y x } ( - 1 , - 5 )$ Determine the sign of $f _ { y x } ( - 1 , - 5 )$

Find the directional derivative of $f ( x , y , z ) = 3 x y z ^ { 2 } - 4 x z$ at the point (3, 3, 2), in the direction of the vector $\vec { v } = 2 \vec { i } + 3 \vec { j } - \vec { k }$

If $f ( x , y ) = \sin ( x ) + y ^ { 2 }$ x(u, v)= uv and y(u, v)= u + 4v. If H(u, v)= f(x(u, v), y(u, v)), what is H(0,-1)? Give your answer to 4 decimal places.

A Library of Functions

Key Concept: the Derivative

Short-Cuts to Differentiation

Using the Derivative

Key Concept- the Definite Integral

Constructing Antiderivatives

Integration

Using the Definite Integral

Sequences and Series

Approximating Functions Using Series

Differential Equations

Functions of Several Variables

A Fundamental Tool- Vectors

Optimization- Local and Global Extrema

Integrating Functions of Several Variables

Parameterization and Vector Fields

Line Integrals

Flux Integrals and Divergence

The Curl and Stokes Theorem

Parameters, Coordinates, Integrals

Filters

Exam 14: Differentiating Functions of Several Variables

A partial derivative is a specific example of a directional derivative.

Let z=f(x,y)=e(x/y)z = f ( x , y ) = e ^ { ( x / y ) }z=f(x,y)=e(x/y) .Find the instantaneous rate of change of f at the point (5,5)( 5,5 )(5,5) in the direction of the point (8,11)( 8,11 )(8,11) .

Given that ggd⁡g(1,1,5)=3i⃗−3j⃗+k⃗g \operatorname { gd } g ( 1,1,5 ) = 3 \vec { i } - 3 \vec { j } + \vec { k }ggdg(1,1,5)=3i−3j​+k , in which of the following directions does g increase the fastest?

If f(x,y)=sin⁡(x)+y2f ( x , y ) = \sin ( x ) + y ^ { 2 }f(x,y)=sin(x)+y2 x(u, v)= uv and y(u, v)= u + 3v. If H(u, v)= f(x(u, v), y(u, v)), what is Hv(0,-2)? Give your answer to 4 decimal places.

If f(x,y)=ln⁡(y)f ( x , y ) = \ln ( y )f(x,y)=ln(y) , then ∇f(x,y)=1y\nabla f ( x , y ) = \frac { 1 } { y }∇f(x,y)=y1​ .

The quadratic Taylor approximations of f(x,y)f ( x , y )f(x,y) at the points (−1−1),(1,1)( - 1 - 1 ) , ( 1,1 )(−1−1),(1,1) , (2,0)and (0,2)are given respectively by: (x,y)=-3+3(x+1+4(y+1 (x,y)=1-3(x-1+4(y-1 (x,y)=-2-9(x-2)-6(x-2-2 (x,y)=8+3x+24(y-2)+22(y-2 Determine fy (0,2)( 0,2 )(0,2) .

The tangent plane to the surface z=xsin⁡(x/y),y≠0z = x \sin ( x / y ) , y \neq 0z=xsin(x/y),y=0 at any point passes through the origin.

Let f(x,y,z)=x3+3xz+4yz+z2f ( x , y , z ) = x ^ { 3 } + 3 x z + 4 y z + z ^ { 2 }f(x,y,z)=x3+3xz+4yz+z2 .Find , where v⃗\vec { v }v is a unit vector in the direction of −i⃗+j⃗−k⃗- \vec { i } + \vec { j } - \vec { k }−i+j​−k .

Find the equation of the tangent plane to the surface 3xyz+x3+y3+z3=153 x y z + x ^ { 3 } + y ^ { 3 } + z ^ { 3 } = 153xyz+x3+y3+z3=15 at (1, -3, 2).

Find the following partial derivative: fxy if f(x,y)=x7y8f ( x , y ) = x ^ { 7 } y ^ { 8 }f(x,y)=x7y8 .

Find ∂∂x(ln⁡(x5y+5))\frac { \partial } { \partial x } \left( \ln \left( x ^ { 5 } y + 5 \right) \right)∂x∂​(ln(x5y+5)) .

Suppose that the price P (in dollars)to purchase a used car is a function of C, its original cost (in dollars), and its age A (in years).So P = f(C,A). What is the sign of ∂P∂C?\frac { \partial P } { \partial C } ?∂C∂P​?

Let Γ(u,v,w)\Gamma ( u , v , w )Γ(u,v,w) be a function of three variables with Fu+Fw=−1F _ { u } + F _ { w } = - 1Fu​+Fw​=−1 .Suppose that G(x,y)=F(x,x−y,x)G ( x , y ) = F ( x , x - y , x )G(x,y)=F(x,x−y,x) .Simplify Gx+GyG _ { x } + G _ { y }Gx​+Gy​ .

Consider the level curves shown for the function z = f(x, y). Determine the sign of fyx(−1,−5)f _ { y x } ( - 1 , - 5 )fyx​(−1,−5)

Find the directional derivative of f(x,y,z)=3xyz2−4xzf ( x , y , z ) = 3 x y z ^ { 2 } - 4 x zf(x,y,z)=3xyz2−4xz at the point (3, 3, 2), in the direction of the vector v⃗=2i⃗+3j⃗−k⃗\vec { v } = 2 \vec { i } + 3 \vec { j } - \vec { k }v=2i+3j​−k

If f(x,y)=sin⁡(x)+y2f ( x , y ) = \sin ( x ) + y ^ { 2 }f(x,y)=sin(x)+y2 x(u, v)= uv and y(u, v)= u + 4v. If H(u, v)= f(x(u, v), y(u, v)), what is H(0,-1)? Give your answer to 4 decimal places.

A Library of Functions

Key Concept: the Derivative

Short-Cuts to Differentiation

Using the Derivative

Key Concept- the Definite Integral

Constructing Antiderivatives

Integration

Using the Definite Integral

Sequences and Series

Approximating Functions Using Series

Differential Equations

Functions of Several Variables

A Fundamental Tool- Vectors

Optimization- Local and Global Extrema

Integrating Functions of Several Variables

Parameterization and Vector Fields

Line Integrals

Flux Integrals and Divergence

The Curl and Stokes Theorem

Parameters, Coordinates, Integrals

Filters

Let $z = f ( x , y ) = e ^ { ( x / y ) }$ .Find the instantaneous rate of change of f at the point $( 5,5 )$ in the direction of the point $( 8,11 )$ .

Given that $g \operatorname { gd } g ( 1,1,5 ) = 3 \vec { i } - 3 \vec { j } + \vec { k }$ , in which of the following directions does g increase the fastest?

If $f ( x , y ) = \sin ( x ) + y ^ { 2 }$ x(u, v)= uv and y(u, v)= u + 3v. If H(u, v)= f(x(u, v), y(u, v)), what is H_v(0,-2)? Give your answer to 4 decimal places.

If $f ( x , y ) = \ln ( y )$ , then $\nabla f ( x , y ) = \frac { 1 } { y }$ .

The quadratic Taylor approximations of $f ( x , y )$ at the points $( - 1 - 1 ) , ( 1,1 )$ , (2,0)and (0,2)are given respectively by: (x,y)=-3+3(x+1+4(y+1 (x,y)=1-3(x-1+4(y-1 (x,y)=-2-9(x-2)-6(x-2-2 (x,y)=8+3x+24(y-2)+22(y-2 Determine f_y $( 0,2 )$ .

The tangent plane to the surface $z = x \sin ( x / y ) , y \neq 0$ at any point passes through the origin.

Find the equation of the tangent plane to the surface $3 x y z + x ^ { 3 } + y ^ { 3 } + z ^ { 3 } = 15$ at (1, -3, 2).

Find the following partial derivative: f_xy if $f ( x , y ) = x ^ { 7 } y ^ { 8 }$ .

Find $\frac { \partial } { \partial x } \left( \ln \left( x ^ { 5 } y + 5 \right) \right)$ .

Suppose that the price P (in dollars)to purchase a used car is a function of C, its original cost (in dollars), and its age A (in years).So P = f(C,A). What is the sign of $\frac { \partial P } { \partial C } ?$

Let $\Gamma ( u , v , w )$ be a function of three variables with $F _ { u } + F _ { w } = - 1$ .Suppose that $G ( x , y ) = F ( x , x - y , x )$ .Simplify $G _ { x } + G _ { y }$ .

Consider the level curves shown for the function z = f(x, y). $Consider the level curves shown for the function z = f(x, y). Determine the sign of f _ { y x } ( - 1 , - 5 )$ Determine the sign of $f _ { y x } ( - 1 , - 5 )$

Find the directional derivative of $f ( x , y , z ) = 3 x y z ^ { 2 } - 4 x z$ at the point (3, 3, 2), in the direction of the vector $\vec { v } = 2 \vec { i } + 3 \vec { j } - \vec { k }$

If $f ( x , y ) = \sin ( x ) + y ^ { 2 }$ x(u, v)= uv and y(u, v)= u + 4v. If H(u, v)= f(x(u, v), y(u, v)), what is H(0,-1)? Give your answer to 4 decimal places.