Question 1

Describe the level curves of the function $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + 3 x - 4 y + 73$ .

Accepted Answer

A) Concentric circles 
B) Non-concentric circles 
C) Concentric ellipses (not circles) 
D) Non-concentric ellipses (not circles) 
E) Parabolas with the same vertex
F)Parabolas with the same focus
G)Hyperbolas with the same vertices
H)Hyperbolas with the same foci 
A) Concentric circles 
B) Non-concentric circles 
C) Concentric ellipses (not circles) 
D) Non-concentric ellipses (not circles) 
E) Parabolas with the same vertex
F)Parabolas with the same focus
G)Hyperbolas with the same vertices
H)Hyperbolas with the same foci

Question 2

Find the differential of $z = 3 x + y ^ { 2 }$

Accepted Answer

A)  $3 d x + 2 y d y$ 
B)  $3 x d x + 2 y ^ { 2 } d y$ 
C)  $2 d x + 3 y d y$ 
D)  $2 x d x + 3 y ^ { 2 } d y$ 
E)  $3 y d x + 2 x d y$ 
F) $2 y d x + 3 x d y$

G) $2 d x + 3 d y$

H) $3 d x + 2 d y$ 
A)  $3 d x + 2 y d y$ 
B)  $3 x d x + 2 y ^ { 2 } d y$ 
C)  $2 d x + 3 y d y$ 
D)  $2 x d x + 3 y ^ { 2 } d y$ 
E)  $3 y d x + 2 x d y$ 
F) $2 y d x + 3 x d y$

G) $2 d x + 3 d y$

H) $3 d x + 2 d y$

Question 3

Let $f ( x , y ) = \ln ( x - 2 y )$ .(a) Evaluate $f ( 3,1 )$ .(b) Sketch the domain of $f$ .(c) What is the range of the function $f$ ?

Accepted Answer

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

Question 4

Find an equation of the tangent plane to the surface given by $\mathbf { r } ( u , v ) = ( u + v ) \mathbf { i } + u \cos v \mathbf { j } + v \sin u \mathbf { k }$ at the point $( 1,1,0 )$ .

Accepted Answer

The answer of Find an equation of the tangent plane...

Question 5

Find the maximum and minimum values of $f ( x , y ) = 2 x + 3 y + 4$ on the circle $x ^ { 2 } + y ^ { 2 } = 1$ .

Accepted Answer

Maximum va

Question 6

Find the directional derivative of $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } - z$ at the point $( 1,3,5 )$ in the direction of $\mathbf { a } = 2 \mathbf { i } - \mathbf { j } + 4 \mathbf { k }$

Accepted Answer

The answer of Find the directional derivative of \(f (...

Question 7

If $g ( x , y ) = \left\{ \begin{array} { l l } 
\frac { x ^ { 4 } + y ^ { 2 } } { x ^ { 2 } + y ^ { 4 } } & \text { if } ( x , y ) \neq ( 0,0 ) \
0 & \text { if } ( x , y ) = ( 0,0 )
\end{array} \right.$ find $\frac { \partial g } { \partial x } ( 1,0 )$ and $\frac { \partial g } { \partial y } ( 1,0 )$ .

Accepted Answer

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

Question 8

Let $e ^ { x } = 3 \sin y$ . Use implicit differentiation to find $d y / d x$ when $( x , y ) = ( 1,0 )$ .

Accepted Answer

A)  $- \frac { 1 } { 3 } e$ 
B)  $\frac { 1 } { 3 } e$ 
C)  $- \frac { 1 } { 2 } e$ 
D)  $\frac { 1 } { 2 } e$ 
E)  $- e$ 
F) $e$

G) $- 2 e$

H) $2e$ 
A)  $- \frac { 1 } { 3 } e$ 
B)  $\frac { 1 } { 3 } e$ 
C)  $- \frac { 1 } { 2 } e$ 
D)  $\frac { 1 } { 2 } e$ 
E)  $- e$ 
F) $e$

G) $- 2 e$

H) $2e$

Question 9

Evaluate $\lim _ { ( x , y ) \rightarrow ( 1,2 ) } \frac { 2 x ^ { 2 } + y ^ { 2 } } { x y }$

Accepted Answer

A) 1 
B) 2 
C) 3 
D) 4 
E) 8
F)9
G)16
H)Does not exist 
A) 1 
B) 2 
C) 3 
D) 4 
E) 8
F)9
G)16
H)Does not exist

Question 10

Evaluate $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } x \sin y$

Accepted Answer

A) 0 
B) 1 
C) 2 
D) 3 
E) 4
F)5
G)6
H)Does not exist 
A) 0 
B) 1 
C) 2 
D) 3 
E) 4
F)5
G)6
H)Does not exist

Question 11

Suppose that $z = x ^ { 3 } y ^ { 2 }$ , where both $x$ and $y$ are changing with time. At a certain instant when $x = 1$ and $y = 2$ , $x$ is decreasing at the rate of 2 units/s and $y$ is increasing at the rate of 3 units/s. How fast is $Z$ changing at this instant? Is $Z$ increasing or decreasing?

Accepted Answer

The answer of Suppose that \(z = x ^ {...

Question 12

Let $z = x y + x ^ { 2 } y$ , and let $x$ and $y$ be functions of $S$ and $t$ with $x ( 0,0 ) = 1 , y ( 0,0 ) = 2$ , $\delta x / \delta s = 3$ , and $\delta y / \delta s = 4$ at $( s , t ) = ( 0,0 )$ . Find $\delta z /  \delta s $ when $( s , t ) = ( 0,0 )$ .

Accepted Answer

A) 21 
B) 22 
C) 23 
D) 24 
E) 25
F)26
G)27
H)28 
A) 21 
B) 22 
C) 23 
D) 24 
E) 25
F)26
G)27
H)28

Question 13

Show that at $\left( \frac { 1 } { \sqrt { 2 } } , 0 \right)$ , the equation $x ^ { 2 } + \frac { 1 } { 2 } y ^ { 2 } + \frac { 1 } { 3 } z ^ { 2 } = 1$ defines $Z$ implicitly as a function of $x$ and $y$ , and then compute $\frac { \delta z } { \delta x } \left( \frac { 1 } { \sqrt { 2 } } , 0 , \sqrt { \frac { 3 } { 2 } } \right)$ and $\frac { \delta z } { \delta y } \left( \frac { 1 } { \sqrt { 2 } } , 0 , \sqrt { \frac { 3 } { 2 } } \right)$ .

Accepted Answer

The answer of Show that at \(\left( \frac { 1...

Question 14

Find an equation of the tangent plant to the surface $z = \tan ^ { - 1 } ( y / x )$ at the point $\left( - 2,2 , - \frac { \pi } { 4 } \right)$ .

Accepted Answer

A)  $x + y + z = \pi$ 
B)  $x + y + 4 z + \pi = 0$ 
C)  $4 x + 4 y + z + \pi = 0$ 
D)  $2 x - 2 y + z = \frac { \pi } { 4 }$ 
E)  $2 x - 2 y + 4 z = \pi$ 
F) $4 x - 2 y - z + \pi = 0$

G) $x + 2 y + 4 z = 2 - \pi$

H) $x + 2 y - 4 z = 2 + \pi$ 
A)  $x + y + z = \pi$ 
B)  $x + y + 4 z + \pi = 0$ 
C)  $4 x + 4 y + z + \pi = 0$ 
D)  $2 x - 2 y + z = \frac { \pi } { 4 }$ 
E)  $2 x - 2 y + 4 z = \pi$ 
F) $4 x - 2 y - z + \pi = 0$

G) $x + 2 y + 4 z = 2 - \pi$

H) $x + 2 y - 4 z = 2 + \pi$

Question 15

One side of a rectangle is increasing at 4 ft/min and another at 7 ft/min. At the time when the first side is 24 ft long and the second is 32 ft long, find
(a) how fast the area is changing.(b) how fast the diagonal is changing.

Accepted Answer

(a) 296 ft

Question 16

If all the third-order partial derivatives of $f ( x , y )$ are continuous, what is the largest number of them that can be distinct?

Accepted Answer

A) 3 
B) 4 
C) 5 
D) 6 
E) 7
F)8
G)9
H)10 
A) 3 
B) 4 
C) 5 
D) 6 
E) 7
F)8
G)9
H)10

Question 17

Find the domain of the function $f ( x , y , z ) = \frac { 1 } { \sqrt { 3 x + y - z } }$ .

Accepted Answer

A) All points above $3 x + y - z = 0$ 
B) All points below $3 x + y - z = 0$ 
C) All points above $3 x + y - z = 1$ 
D) All points below $3 x + y - z = 1$ 
E) All points below $3 x + y - z = 0$ but not above the plane $3 x + y - z = 1$ 
F)All points above $3 x + y - z = 1$ but not above the plane $3 x + y - z = 0$ 
G)All points below $3 x + y - z = 1$ or between $3 x + y - z = 0$ and $3 x + y - z = 1$ 
H)All points in the $x y$ -plane 
A) All points above $3 x + y - z = 0$ 
B) All points below $3 x + y - z = 0$ 
C) All points above $3 x + y - z = 1$ 
D) All points below $3 x + y - z = 1$ 
E) All points below $3 x + y - z = 0$ but not above the plane $3 x + y - z = 1$ 
F)All points above $3 x + y - z = 1$ but not above the plane $3 x + y - z = 0$ 
G)All points below $3 x + y - z = 1$ or between $3 x + y - z = 0$ and $3 x + y - z = 1$ 
H)All points in the $x y$ -plane

Question 18

If $z = f ( x , y )$ has continuous partial derivatives, $x = s + 2 t$ , and $y = s - 2 t$ , show that $\frac { \delta \mathcal { d } } { \delta s} \frac { \delta z } { \delta t } = 2 \left( \frac { \delta z } { \delta x } \right) ^ { 2 } - 2 \left( \frac { \delta z } { \delta y } \right) ^ { 2 }$

Accepted Answer

To show that \frac { \delta \mathcal { d

Question 19

Let $f ( x , y ) = 3 x + 2 y$ . Find the gradient vector $\nabla f$ .

Accepted Answer

A)  $3 x \mathbf { i } + 2 y \mathbf { j }$ 
B)  $3 x ^ { 2 } \mathbf { i } + 2 y ^ { 2 } \mathbf { j }$ 
C) 
$2 x \mathbf { i } + 3 y \mathbf { j }$ 
D)  $2 x ^ { 2 } \mathbf { i } + 3 y ^ { 2 } \mathbf { j }$ 
E)  $3 y \mathbf { i } + 2 x \mathbf { j }$  
F) $2 y \mathbf { i } + 3 x \mathbf { j }$ 
 
G) $2 \mathbf { i } + 3 \mathbf { j }$

H) $3 \mathbf { i } + 2 \mathbf { j }$ 
A)  $3 x \mathbf { i } + 2 y \mathbf { j }$ 
B)  $3 x ^ { 2 } \mathbf { i } + 2 y ^ { 2 } \mathbf { j }$ 
C) 
$2 x \mathbf { i } + 3 y \mathbf { j }$ 
D)  $2 x ^ { 2 } \mathbf { i } + 3 y ^ { 2 } \mathbf { j }$ 
E)  $3 y \mathbf { i } + 2 x \mathbf { j }$  
F) $2 y \mathbf { i } + 3 x \mathbf { j }$ 
 
G) $2 \mathbf { i } + 3 \mathbf { j }$

H) $3 \mathbf { i } + 2 \mathbf { j }$

Question 20

If $f ( x , y ) = 2 x ^ { 2 } + 4 y ^ { 2 } - x y$ , find the gradient at the point $( 2,1 )$ . Also find the rate of change of $f ( x , y )$ in the direction $\theta = \frac { \pi } { 3 }$ at $( 2,1 )$ .

Accepted Answer

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

Describe the level curves of the function $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + 3 x - 4 y + 73$ .

Find the differential of $z = 3 x + y ^ { 2 }$

Let $f ( x , y ) = \ln ( x - 2 y )$ .(a) Evaluate $f ( 3,1 )$ .(b) Sketch the domain of $f$ .(c) What is the range of the function $f$ ?

Find an equation of the tangent plane to the surface given by $\mathbf { r } ( u , v ) = ( u + v ) \mathbf { i } + u \cos v \mathbf { j } + v \sin u \mathbf { k }$ at the point $( 1,1,0 )$ .

Find the maximum and minimum values of $f ( x , y ) = 2 x + 3 y + 4$ on the circle $x ^ { 2 } + y ^ { 2 } = 1$ .

Find the directional derivative of $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } - z$ at the point $( 1,3,5 )$ in the direction of $\mathbf { a } = 2 \mathbf { i } - \mathbf { j } + 4 \mathbf { k }$

Let $e ^ { x } = 3 \sin y$ . Use implicit differentiation to find $d y / d x$ when $( x , y ) = ( 1,0 )$ .

Evaluate $\lim _ { ( x , y ) \rightarrow ( 1,2 ) } \frac { 2 x ^ { 2 } + y ^ { 2 } } { x y }$

Evaluate $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } x \sin y$

Let $z = x y + x ^ { 2 } y$ , and let $x$ and $y$ be functions of $S$ and $t$ with $x ( 0,0 ) = 1 , y ( 0,0 ) = 2$ , $\delta x / \delta s = 3$ , and $\delta y / \delta s = 4$ at $( s , t ) = ( 0,0 )$ . Find $\delta z / \delta s$ when $( s , t ) = ( 0,0 )$ .

Find an equation of the tangent plant to the surface $z = \tan ^ { - 1 } ( y / x )$ at the point $\left( - 2,2 , - \frac { \pi } { 4 } \right)$ .

One side of a rectangle is increasing at 4 ft/min and another at 7 ft/min. At the time when the first side is 24 ft long and the second is 32 ft long, find (a) how fast the area is changing.(b) how fast the diagonal is changing.

If all the third-order partial derivatives of $f ( x , y )$ are continuous, what is the largest number of them that can be distinct?

Find the domain of the function $f ( x , y , z ) = \frac { 1 } { \sqrt { 3 x + y - z } }$ .

If $z = f ( x , y )$ has continuous partial derivatives, $x = s + 2 t$ , and $y = s - 2 t$ , show that $\frac { \delta \mathcal { d } } { \delta s} \frac { \delta z } { \delta t } = 2 \left( \frac { \delta z } { \delta x } \right) ^ { 2 } - 2 \left( \frac { \delta z } { \delta y } \right) ^ { 2 }$

Let $f ( x , y ) = 3 x + 2 y$ . Find the gradient vector $\nabla f$ .

If $f ( x , y ) = 2 x ^ { 2 } + 4 y ^ { 2 } - x y$ , find the gradient at the point $( 2,1 )$ . Also find the rate of change of $f ( x , y )$ in the direction $\theta = \frac { \pi } { 3 }$ at $( 2,1 )$ .

Functions and Models

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Exam 11: Partial Derivatives

Describe the level curves of the function f(x,y)=x2+y2+3x−4y+73f ( x , y ) = x ^ { 2 } + y ^ { 2 } + 3 x - 4 y + 73f(x,y)=x2+y2+3x−4y+73 .

Find the differential of z=3x+y2z = 3 x + y ^ { 2 }z=3x+y2

Let f(x,y)=ln⁡(x−2y)f ( x , y ) = \ln ( x - 2 y )f(x,y)=ln(x−2y) .(a) Evaluate f(3,1)f ( 3,1 )f(3,1) .(b) Sketch the domain of fff .(c) What is the range of the function fff ?

Find an equation of the tangent plane to the surface given by r(u,v)=(u+v)i+ucos⁡vj+vsin⁡uk\mathbf { r } ( u , v ) = ( u + v ) \mathbf { i } + u \cos v \mathbf { j } + v \sin u \mathbf { k }r(u,v)=(u+v)i+ucosvj+vsinuk at the point (1,1,0)( 1,1,0 )(1,1,0) .

Find the maximum and minimum values of f(x,y)=2x+3y+4f ( x , y ) = 2 x + 3 y + 4f(x,y)=2x+3y+4 on the circle x2+y2=1x ^ { 2 } + y ^ { 2 } = 1x2+y2=1 .

Find the directional derivative of f(x,y,z)=x2+y2−zf ( x , y , z ) = x ^ { 2 } + y ^ { 2 } - zf(x,y,z)=x2+y2−z at the point (1,3,5)( 1,3,5 )(1,3,5) in the direction of a=2i−j+4k\mathbf { a } = 2 \mathbf { i } - \mathbf { j } + 4 \mathbf { k }a=2i−j+4k

Let ex=3sin⁡ye ^ { x } = 3 \sin yex=3siny . Use implicit differentiation to find dy/dxd y / d xdy/dx when (x,y)=(1,0)( x , y ) = ( 1,0 )(x,y)=(1,0) .

Evaluate lim⁡(x,y)→(1,2)2x2+y2xy\lim _ { ( x , y ) \rightarrow ( 1,2 ) } \frac { 2 x ^ { 2 } + y ^ { 2 } } { x y }lim(x,y)→(1,2)​xy2x2+y2​

Evaluate lim⁡(x,y)→(0,0)xsin⁡y\lim _ { ( x , y ) \rightarrow ( 0,0 ) } x \sin ylim(x,y)→(0,0)​xsiny

Find an equation of the tangent plant to the surface z=tan⁡−1(y/x)z = \tan ^ { - 1 } ( y / x )z=tan−1(y/x) at the point (−2,2,−π4)\left( - 2,2 , - \frac { \pi } { 4 } \right)(−2,2,−4π​) .

One side of a rectangle is increasing at 4 ft/min and another at 7 ft/min. At the time when the first side is 24 ft long and the second is 32 ft long, find (a) how fast the area is changing.(b) how fast the diagonal is changing.

If all the third-order partial derivatives of f(x,y)f ( x , y )f(x,y) are continuous, what is the largest number of them that can be distinct?

Find the domain of the function f(x,y,z)=13x+y−zf ( x , y , z ) = \frac { 1 } { \sqrt { 3 x + y - z } }f(x,y,z)=3x+y−z​1​ .

Let f(x,y)=3x+2yf ( x , y ) = 3 x + 2 yf(x,y)=3x+2y . Find the gradient vector ∇f\nabla f∇f .

If f(x,y)=2x2+4y2−xyf ( x , y ) = 2 x ^ { 2 } + 4 y ^ { 2 } - x yf(x,y)=2x2+4y2−xy , find the gradient at the point (2,1)( 2,1 )(2,1) . Also find the rate of change of f(x,y)f ( x , y )f(x,y) in the direction θ=π3\theta = \frac { \pi } { 3 }θ=3π​ at (2,1)( 2,1 )(2,1) .

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Describe the level curves of the function $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + 3 x - 4 y + 73$ .

Find the differential of $z = 3 x + y ^ { 2 }$

Let $f ( x , y ) = \ln ( x - 2 y )$ .(a) Evaluate $f ( 3,1 )$ .(b) Sketch the domain of $f$ .(c) What is the range of the function $f$ ?

Find an equation of the tangent plane to the surface given by $\mathbf { r } ( u , v ) = ( u + v ) \mathbf { i } + u \cos v \mathbf { j } + v \sin u \mathbf { k }$ at the point $( 1,1,0 )$ .

Find the maximum and minimum values of $f ( x , y ) = 2 x + 3 y + 4$ on the circle $x ^ { 2 } + y ^ { 2 } = 1$ .

Find the directional derivative of $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } - z$ at the point $( 1,3,5 )$ in the direction of $\mathbf { a } = 2 \mathbf { i } - \mathbf { j } + 4 \mathbf { k }$

Let $e ^ { x } = 3 \sin y$ . Use implicit differentiation to find $d y / d x$ when $( x , y ) = ( 1,0 )$ .

Evaluate $\lim _ { ( x , y ) \rightarrow ( 1,2 ) } \frac { 2 x ^ { 2 } + y ^ { 2 } } { x y }$

Evaluate $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } x \sin y$

Find an equation of the tangent plant to the surface $z = \tan ^ { - 1 } ( y / x )$ at the point $\left( - 2,2 , - \frac { \pi } { 4 } \right)$ .

If all the third-order partial derivatives of $f ( x , y )$ are continuous, what is the largest number of them that can be distinct?

Find the domain of the function $f ( x , y , z ) = \frac { 1 } { \sqrt { 3 x + y - z } }$ .

Let $f ( x , y ) = 3 x + 2 y$ . Find the gradient vector $\nabla f$ .

If $f ( x , y ) = 2 x ^ { 2 } + 4 y ^ { 2 } - x y$ , find the gradient at the point $( 2,1 )$ . Also find the rate of change of $f ( x , y )$ in the direction $\theta = \frac { \pi } { 3 }$ at $( 2,1 )$ .