Question 1

Find the range of the function $f ( x , y ) = \sqrt { x - y ^ { 2 } }$ .

Accepted Answer

A)  $( 0 , \infty )$ 
B)  $[ 0 , \infty )$ 
C)  $( - \infty , 0 )$ 
D)  $( - \infty , \infty )$ 
E)  $( 1 , \infty )$
F) $[ 1 , \infty )$

G) $( \sqrt { 2 } , \infty )$ 
 
H) $[ \sqrt { 2 } , \infty )$ 
A)  $( 0 , \infty )$ 
B)  $[ 0 , \infty )$ 
C)  $( - \infty , 0 )$ 
D)  $( - \infty , \infty )$ 
E)  $( 1 , \infty )$
F) $[ 1 , \infty )$

G) $( \sqrt { 2 } , \infty )$ 
 
H) $[ \sqrt { 2 } , \infty )$

Question 2

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

Accepted Answer

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

Question 3

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

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 4

Let $z = \sqrt { 1 + 2 x + x y }$ , and let $x$ and $y$ be functions of $t$ with $x ( 1 ) = 2 , y ( 1 ) = 2 , x ^ { \prime } ( 1 ) = 3$ , and $y ^ { \prime } ( 1 ) = - 6$ . Find $d z / d t$ when $t = 1$ .

Accepted Answer

A) 0 
B)  $- 2$ 
C) 4 
D)  $- 4$ 
E) 2
F)1
G) $- 1$  
H) $\frac { 2 } { 3 }$ 
A) 0 
B)  $- 2$ 
C) 4 
D)  $- 4$ 
E) 2
F)1
G) $- 1$  
H) $\frac { 2 } { 3 }$

Question 5

For each of the following functions, find the critical point, if there is one, and determine if it is a local maximum, local minimum, saddle point, or otherwise.(a) $f ( x , y ) = 4 x ^ { 2 } + 3 x - 5 y ^ { 2 } - 8 y + 7$ (b) $f ( x , y ) = - 2 x ^ { 2 } + 7 x - 7 y ^ { 2 } + 4 y + 9$ (c) $f ( x , y ) = 3 x ^ { 2 } - 5 y + 4 y ^ { 2 } + 12 x - 11$ (d) $f ( x , y ) = 2 y ^ { 2 } + 7 x + 17 y + 12$

Accepted Answer

(a) @#IMG-DLM& is a saddle poi

Question 6

What is the direction of maximal decrease for $f ( x , y , z ) = x y + y z + x z$ at $( 1,1,1 )$ ?

Accepted Answer

Any vector

Question 7

If $w = x ^ { 2 } z + x y ^ { 2 } - y z ^ { 2 }$ , find $\frac { \delta w } { \delta x } , \frac { \delta w } { \delta y } \text {, and } \frac { \delta w } { \delta z }$ .

Accepted Answer

The answer of If \(w = x ^ { 2...

Question 8

Find the local maximum and minimum values and saddle points of the function $f ( x , y ) = 4 x y - x ^ { 4 } - y ^ { 4 } + \frac { 1 } { 16 }$ .

Accepted Answer

@#IMG-DLM& is a saddle point;

Question 9

Find the differential of $z = \frac { 1 } { x + y ^ { 2 } }$ at the point $( x , y ) = ( 1,1 )$ .

Accepted Answer

A)  $- d x - d y$ 
B)  $- \frac { 1 } { 4 } d x - \frac { 1 } { 2 } d y$ 
C) $- d x - \frac { 1 } { 2 } d y$ 
D) $- \frac { 1 } { 2 } d x - d y$ 
E)  $d x + d y$ 
F) $\frac { 1 } { 4 } d x + \frac { 1 } { 2 } d y$

G) $d x + \frac { 1 } { 2 } d y$

H) $\frac { 1 } { 2 } d x + d y$ 
A)  $- d x - d y$ 
B)  $- \frac { 1 } { 4 } d x - \frac { 1 } { 2 } d y$ 
C) $- d x - \frac { 1 } { 2 } d y$ 
D) $- \frac { 1 } { 2 } d x - d y$ 
E)  $d x + d y$ 
F) $\frac { 1 } { 4 } d x + \frac { 1 } { 2 } d y$

G) $d x + \frac { 1 } { 2 } d y$

H) $\frac { 1 } { 2 } d x + d y$

Question 10

Find the directional derivative of the function $f ( x , y , z ) = x e ^ { x y / z }$ at the point $P$ $( 3,0,1 )$ in the direction from $P$ toward the point $( 2,2,3 )$ ..

Accepted Answer

A)  $\frac { 17 } { 3 }$ 
B)  $6$ 
C)  $\frac { 19 } { 3 }$ 
D)  $\frac { 20 } { 3 }$ 
E) 7
F) $\frac { 22 } { 3 }$  
G) $\frac { 23 } { 3 }$ 
H)8 
A)  $\frac { 17 } { 3 }$ 
B)  $6$ 
C)  $\frac { 19 } { 3 }$ 
D)  $\frac { 20 } { 3 }$ 
E) 7
F) $\frac { 22 } { 3 }$  
G) $\frac { 23 } { 3 }$ 
H)8

Question 11

Find the minimum value of the function $f ( x , y , z ) = 2 x + y - 2 z$ subject to the constraint that $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ .

Accepted Answer

A)  $- 4$ 
B)  $- 5$ 
C)  $- 6$ 
D)  $- 7$ 
E) 4 
F)5
G)6
H)7 
A)  $- 4$ 
B)  $- 5$ 
C)  $- 6$ 
D)  $- 7$ 
E) 4 
F)5
G)6
H)7

Question 12

Compare the minimum value of $Z$ and sketch a portion of the graph of $z = 3 x ^ { 2 } + 6 x + 2 y ^ { 2 } - 8 y$ near its lowest point.

Accepted Answer

@#IMG-DLM& is the ab

Question 13

Let $f ( x , y ) = \int _ { 0 } ^ { x y } \sin \left( t ^ { 2 } \right) d t$ . Find $f _ { x } \left( 1 , \sqrt { \frac { \pi } { 2 } } \right)$ .

Accepted Answer

A) 0 
B)  $\sqrt { \frac { \pi } { 2 } }$ 
C)  $\frac { \pi } { 2 }$ 
D) 3 
E) 4
F) $- 1$ 
G) $- \sqrt { \frac { \pi } { 2 } }$

H) $- 3$ 
A) 0 
B)  $\sqrt { \frac { \pi } { 2 } }$ 
C)  $\frac { \pi } { 2 }$ 
D) 3 
E) 4
F) $- 1$ 
G) $- \sqrt { \frac { \pi } { 2 } }$

H) $- 3$

Question 14

Let $f ( x , y , z ) = 3 \sqrt { x ^ { 2 } + 4 y ^ { 2 } + z ^ { 2 } }$ . Find $f _ { z } ( 2,1,1 )$ .

Accepted Answer

A) 0 
B) 1 
C)  $\frac { 1 } { 2 }$ 
D) 3 
E) 4
F) $- 1$ 
G) $- \frac { 1 } { 2 }$
H) $- 3$ 
A) 0 
B) 1 
C)  $\frac { 1 } { 2 }$ 
D) 3 
E) 4
F) $- 1$ 
G) $- \frac { 1 } { 2 }$
H) $- 3$

Question 15

Consider $f ( x , y , z ) = \frac { x ^ { 3 } y ^ { 2 } z + 1 } { x ^ { 2 } + y ^ { 2 } - z ^ { 2 } }$ . Where $f$ is continuous?

Accepted Answer

Continuous everywher

Question 16

Find the second directional derivative of $f ( x , y ) = x ^ { 2 } e ^ { y }$ at the point $( 2,0 )$ in the direction $(3 , - 4 )$ .

Accepted Answer

A)  $\frac { 14 } { 5 }$ 
B)  $\frac { 4 } { 5 }$ 
C)  $\frac { 14 } { 25 }$ 
D) 14 
E)  $- \frac { 14 } { 5 }$
F) $- \frac { 4 } { 5 }$

G) $- \frac { 14 } { 25 }$ 
H) $- 14$ 
A)  $\frac { 14 } { 5 }$ 
B)  $\frac { 4 } { 5 }$ 
C)  $\frac { 14 } { 25 }$ 
D) 14 
E)  $- \frac { 14 } { 5 }$
F) $- \frac { 4 } { 5 }$

G) $- \frac { 14 } { 25 }$ 
H) $- 14$

Question 17

Let $f ( x , y , z ) = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } }$ .(a) Evaluate $f ( 1 , - 1,2 )$ .(b) Sketch the domain of $f$ .(c) What is the range of the function $f$ ?

Accepted Answer

(a) @#IMG-DLM& (b) @#IMG-DLM&

Question 18

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

Accepted Answer

A) 0 
B) 1 
C) 2 
D) 3 
E) 4
F)5
G)6
H)7 
A) 0 
B) 1 
C) 2 
D) 3 
E) 4
F)5
G)6
H)7

Question 19

Find the domain of the function $f ( x , y ) = \ln \left( x - y ^ { 2 } \right)$ .

Accepted Answer

A) All points on or to the left of $x = y ^ { 2 }$ 
B) All points on or to the right of $x = y ^ { 2 }$ 
C) All points to the left of $x = y ^ { 2 }$ 
D) All points to the right of $x = y ^ { 2 }$ 
E) All points on or to the left of $x = 0$ 
F)All points on or to the right of $x = 0$ 
 
G)All points to the left of $x = 0$

H)All points in the $x y$ -plane 
A) All points on or to the left of $x = y ^ { 2 }$ 
B) All points on or to the right of $x = y ^ { 2 }$ 
C) All points to the left of $x = y ^ { 2 }$ 
D) All points to the right of $x = y ^ { 2 }$ 
E) All points on or to the left of $x = 0$ 
F)All points on or to the right of $x = 0$ 
 
G)All points to the left of $x = 0$

H)All points in the $x y$ -plane

Question 20

Let $f ( x , y ) = x e ^ { y / x }$ . Find the value of the partial derivative $f _ { x } ( 2,4 )$ .

Accepted Answer

A)  $- 2 e ^ { 2 }$ 
B)  $- e ^ { 2 }$ 
C) 0 
D)  $e / 2$ 
E)  $e ^ { 2 } / 2$ 
F) $e ^ { 2 }$ 
G) $2 e ^ { 2 }$  
H) $4 e ^ { 2 }$ 
A)  $- 2 e ^ { 2 }$ 
B)  $- e ^ { 2 }$ 
C) 0 
D)  $e / 2$ 
E)  $e ^ { 2 } / 2$ 
F) $e ^ { 2 }$ 
G) $2 e ^ { 2 }$  
H) $4 e ^ { 2 }$

Find the range of the function $f ( x , y ) = \sqrt { x - y ^ { 2 } }$ .

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

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

Let $z = \sqrt { 1 + 2 x + x y }$ , and let $x$ and $y$ be functions of $t$ with $x ( 1 ) = 2 , y ( 1 ) = 2 , x ^ { \prime } ( 1 ) = 3$ , and $y ^ { \prime } ( 1 ) = - 6$ . Find $d z / d t$ when $t = 1$ .

What is the direction of maximal decrease for $f ( x , y , z ) = x y + y z + x z$ at $( 1,1,1 )$ ?

If $w = x ^ { 2 } z + x y ^ { 2 } - y z ^ { 2 }$ , find $\frac { \delta w } { \delta x } , \frac { \delta w } { \delta y } \text {, and } \frac { \delta w } { \delta z }$ .

Find the local maximum and minimum values and saddle points of the function $f ( x , y ) = 4 x y - x ^ { 4 } - y ^ { 4 } + \frac { 1 } { 16 }$ .

Find the differential of $z = \frac { 1 } { x + y ^ { 2 } }$ at the point $( x , y ) = ( 1,1 )$ .

Find the directional derivative of the function $f ( x , y , z ) = x e ^ { x y / z }$ at the point $P$ $( 3,0,1 )$ in the direction from $P$ toward the point $( 2,2,3 )$ ..

Find the minimum value of the function $f ( x , y , z ) = 2 x + y - 2 z$ subject to the constraint that $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ .

Compare the minimum value of $Z$ and sketch a portion of the graph of $z = 3 x ^ { 2 } + 6 x + 2 y ^ { 2 } - 8 y$ near its lowest point. $Compare the minimum value of Z and sketch a portion of the graph of z = 3 x ^ { 2 } + 6 x + 2 y ^ { 2 } - 8 y near its lowest point.$

Let $f ( x , y ) = \int _ { 0 } ^ { x y } \sin \left( t ^ { 2 } \right) d t$ . Find $f _ { x } \left( 1 , \sqrt { \frac { \pi } { 2 } } \right)$ .

Let $f ( x , y , z ) = 3 \sqrt { x ^ { 2 } + 4 y ^ { 2 } + z ^ { 2 } }$ . Find $f _ { z } ( 2,1,1 )$ .

Consider $f ( x , y , z ) = \frac { x ^ { 3 } y ^ { 2 } z + 1 } { x ^ { 2 } + y ^ { 2 } - z ^ { 2 } }$ . Where $f$ is continuous?

Find the second directional derivative of $f ( x , y ) = x ^ { 2 } e ^ { y }$ at the point $( 2,0 )$ in the direction $(3 , - 4 )$ .

Let $f ( x , y , z ) = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } }$ .(a) Evaluate $f ( 1 , - 1,2 )$ .(b) Sketch the domain of $f$ .(c) What is the range of the function $f$ ?

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

Find the domain of the function $f ( x , y ) = \ln \left( x - y ^ { 2 } \right)$ .

Let $f ( x , y ) = x e ^ { y / x }$ . Find the value of the partial derivative $f _ { x } ( 2,4 )$ .

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Multiple Integrals

Vector Calculus

Filters

Exam 11: Partial Derivatives

Find the range of the function f(x,y)=x−y2f ( x , y ) = \sqrt { x - y ^ { 2 } }f(x,y)=x−y2​ .

Find the directional derivative of f(x,y)=x2−3xy+2y2f ( x , y ) = x ^ { 2 } - 3 x y + 2 y ^ { 2 }f(x,y)=x2−3xy+2y2 at (−1,2)( - 1,2 )(−1,2) in the direction of i+3j\mathbf { i } + \sqrt { 3 } \mathbf { j }i+3​j

Evaluate lim⁡(x,y)→(1,2)(x2+2xy)\lim _ { ( x , y ) \rightarrow ( 1,2 ) } \left( x ^ { 2 } + 2 x y \right)lim(x,y)→(1,2)​(x2+2xy)

What is the direction of maximal decrease for f(x,y,z)=xy+yz+xzf ( x , y , z ) = x y + y z + x zf(x,y,z)=xy+yz+xz at (1,1,1)( 1,1,1 )(1,1,1) ?

If w=x2z+xy2−yz2w = x ^ { 2 } z + x y ^ { 2 } - y z ^ { 2 }w=x2z+xy2−yz2 , find δwδx,δwδy, and δwδz\frac { \delta w } { \delta x } , \frac { \delta w } { \delta y } \text {, and } \frac { \delta w } { \delta z }δxδw​,δyδw​, and δzδw​ .

Find the local maximum and minimum values and saddle points of the function f(x,y)=4xy−x4−y4+116f ( x , y ) = 4 x y - x ^ { 4 } - y ^ { 4 } + \frac { 1 } { 16 }f(x,y)=4xy−x4−y4+161​ .

Find the differential of z=1x+y2z = \frac { 1 } { x + y ^ { 2 } }z=x+y21​ at the point (x,y)=(1,1)( x , y ) = ( 1,1 )(x,y)=(1,1) .

Find the directional derivative of the function f(x,y,z)=xexy/zf ( x , y , z ) = x e ^ { x y / z }f(x,y,z)=xexy/z at the point PPP (3,0,1)( 3,0,1 )(3,0,1) in the direction from PPP toward the point (2,2,3)( 2,2,3 )(2,2,3) ..

Find the minimum value of the function f(x,y,z)=2x+y−2zf ( x , y , z ) = 2 x + y - 2 zf(x,y,z)=2x+y−2z subject to the constraint that x2+y2+z2=4x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4x2+y2+z2=4 .

Compare the minimum value of ZZZ and sketch a portion of the graph of z=3x2+6x+2y2−8yz = 3 x ^ { 2 } + 6 x + 2 y ^ { 2 } - 8 yz=3x2+6x+2y2−8y near its lowest point.

Let f(x,y)=∫0xysin⁡(t2)dtf ( x , y ) = \int _ { 0 } ^ { x y } \sin \left( t ^ { 2 } \right) d tf(x,y)=∫0xy​sin(t2)dt . Find fx(1,π2)f _ { x } \left( 1 , \sqrt { \frac { \pi } { 2 } } \right)fx​(1,2π​​) .

Let f(x,y,z)=3x2+4y2+z2f ( x , y , z ) = 3 \sqrt { x ^ { 2 } + 4 y ^ { 2 } + z ^ { 2 } }f(x,y,z)=3x2+4y2+z2​ . Find fz(2,1,1)f _ { z } ( 2,1,1 )fz​(2,1,1) .

Consider f(x,y,z)=x3y2z+1x2+y2−z2f ( x , y , z ) = \frac { x ^ { 3 } y ^ { 2 } z + 1 } { x ^ { 2 } + y ^ { 2 } - z ^ { 2 } }f(x,y,z)=x2+y2−z2x3y2z+1​ . Where fff is continuous?

Find the second directional derivative of f(x,y)=x2eyf ( x , y ) = x ^ { 2 } e ^ { y }f(x,y)=x2ey at the point (2,0)( 2,0 )(2,0) in the direction (3,−4)(3 , - 4 )(3,−4) .

Let f(x,y,z)=4−x2−y2f ( x , y , z ) = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } }f(x,y,z)=4−x2−y2​ .(a) Evaluate f(1,−1,2)f ( 1 , - 1,2 )f(1,−1,2) .(b) Sketch the domain of fff .(c) What is the range of the function fff ?

Find the domain of the function f(x,y)=ln⁡(x−y2)f ( x , y ) = \ln \left( x - y ^ { 2 } \right)f(x,y)=ln(x−y2) .

Let f(x,y)=xey/xf ( x , y ) = x e ^ { y / x }f(x,y)=xey/x . Find the value of the partial derivative fx(2,4)f _ { x } ( 2,4 )fx​(2,4) .

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Multiple Integrals

Vector Calculus

Filters

Find the range of the function $f ( x , y ) = \sqrt { x - y ^ { 2 } }$ .

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

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

What is the direction of maximal decrease for $f ( x , y , z ) = x y + y z + x z$ at $( 1,1,1 )$ ?

If $w = x ^ { 2 } z + x y ^ { 2 } - y z ^ { 2 }$ , find $\frac { \delta w } { \delta x } , \frac { \delta w } { \delta y } \text {, and } \frac { \delta w } { \delta z }$ .

Find the local maximum and minimum values and saddle points of the function $f ( x , y ) = 4 x y - x ^ { 4 } - y ^ { 4 } + \frac { 1 } { 16 }$ .

Find the differential of $z = \frac { 1 } { x + y ^ { 2 } }$ at the point $( x , y ) = ( 1,1 )$ .

Find the directional derivative of the function $f ( x , y , z ) = x e ^ { x y / z }$ at the point $P$ $( 3,0,1 )$ in the direction from $P$ toward the point $( 2,2,3 )$ ..

Find the minimum value of the function $f ( x , y , z ) = 2 x + y - 2 z$ subject to the constraint that $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ .

Compare the minimum value of $Z$ and sketch a portion of the graph of $z = 3 x ^ { 2 } + 6 x + 2 y ^ { 2 } - 8 y$ near its lowest point. $Compare the minimum value of Z and sketch a portion of the graph of z = 3 x ^ { 2 } + 6 x + 2 y ^ { 2 } - 8 y near its lowest point.$

Let $f ( x , y ) = \int _ { 0 } ^ { x y } \sin \left( t ^ { 2 } \right) d t$ . Find $f _ { x } \left( 1 , \sqrt { \frac { \pi } { 2 } } \right)$ .

Let $f ( x , y , z ) = 3 \sqrt { x ^ { 2 } + 4 y ^ { 2 } + z ^ { 2 } }$ . Find $f _ { z } ( 2,1,1 )$ .

Consider $f ( x , y , z ) = \frac { x ^ { 3 } y ^ { 2 } z + 1 } { x ^ { 2 } + y ^ { 2 } - z ^ { 2 } }$ . Where $f$ is continuous?

Find the second directional derivative of $f ( x , y ) = x ^ { 2 } e ^ { y }$ at the point $( 2,0 )$ in the direction $(3 , - 4 )$ .

Let $f ( x , y , z ) = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } }$ .(a) Evaluate $f ( 1 , - 1,2 )$ .(b) Sketch the domain of $f$ .(c) What is the range of the function $f$ ?

Find the domain of the function $f ( x , y ) = \ln \left( x - y ^ { 2 } \right)$ .

Let $f ( x , y ) = x e ^ { y / x }$ . Find the value of the partial derivative $f _ { x } ( 2,4 )$ .