Question 1

Find the directional derivative of the function $f ( x , y ) = y ^ { 2 } \ln x$ at the point $( 1,2 )$ in the direction $( 3,4 )$ .

Accepted Answer

A)  $\frac { 5 } { 16 }$ 
B) 0 
C)  $\frac { 12 } { 5 }$ 
D) 12 
E) 1
F) $\frac { 16 } { 5 }$

G) $\frac { 5 } { 12 }$ 
H) $\frac { 1 } { 12 }$ 
A)  $\frac { 5 } { 16 }$ 
B) 0 
C)  $\frac { 12 } { 5 }$ 
D) 12 
E) 1
F) $\frac { 16 } { 5 }$

G) $\frac { 5 } { 12 }$ 
H) $\frac { 1 } { 12 }$

Question 2

Find the direction of maximum increase of the function $f ( x , y , z ) = x e ^ { - y } + 3 z$ at the point $( 1,0,4 )$ .

Accepted Answer

A)  $\{ 1,1,3 \}$ 
B)  $\{ - 1 , - 1,3 \}$ 
C)  $\{ - 1,1,3 \}$ 
D)  $(1 , - 1,4 )$ 
E)  $\{ 1,3,3 \}$  
F) $\{ 1 , - 3,3 \}$

G) $\{ 1 , - 1,3 \}$

H) $\{ - 1,1,4 \}$ 
A)  $\{ 1,1,3 \}$ 
B)  $\{ - 1 , - 1,3 \}$ 
C)  $\{ - 1,1,3 \}$ 
D)  $(1 , - 1,4 )$ 
E)  $\{ 1,3,3 \}$  
F) $\{ 1 , - 3,3 \}$

G) $\{ 1 , - 1,3 \}$

H) $\{ - 1,1,4 \}$

Question 3

Determine how many critical points the function $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + x ^ { 2 } y + 10$ has.

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 4

If $f ( x , y , z ) = x \ln \left( y z ^ { 2 } \right)$ , find $f _ { x y }$ , $f _ { x x }$ , and $f _ { y z }$ .

Accepted Answer

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

Question 5

Find the directional derivative of $f ( x , y ) = 3 x ^ { 2 } + x y - y ^ { 3 }$ in the direction $\theta = \frac { \pi } { 3 }$ .

Accepted Answer

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

Question 6

Let $f ( x , y ) = \left( x ^ { 2 } + y \right) ^ { 3 }$ . If $x = 1$ , find $f ( x , 2 x )$ .

Accepted Answer

A) 1 
B) 2 
C) 3 
D) 4 
E) 8
F)9
G)16
H)27 
A) 1 
B) 2 
C) 3 
D) 4 
E) 8
F)9
G)16
H)27

Question 7

Let $f ( x , y , z ) = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } - z ^ { 2 } }$ .(a) Evaluate $f ( 1 , - 1,1 )$ .(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 8

Find an equation of the tangent plane to the surface $z = f ( x , y ) = x ^ { 3 } y ^ { 4 }$ at the point $( - 1,2 , - 16 )$ .

Accepted Answer

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

Question 9

Let $f ( x , y , z ) = \sqrt { 2 x + 3 y - z }$ . Find the rate of change of $f$ at the point $( 1,1 , - 4 )$ in the direction $\mathbf { u }$ where $\mathbf { u }$ is a unit vector making an angle $\theta = 150 ^ { \circ }$ with $\nabla f$ .

Accepted Answer

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

Question 10

Find the total differential of $w$ if $w = f ( x , y , z ) = x y ^ { 2 } z ^ { 3 } - e ^ { xz } y$

Accepted Answer

The answer of Find the total differential of $w$ if...

Question 11

The level curves of $f ( x , y )$ are sketched below.   (a) Find $f _ { x } ( 2,1 )$ (b) Find $f _ { y } ( 2,1 )$ (c) Sketch the gradient vector $\nabla f ( 2,1 )$ .

Accepted Answer

(a) About

Question 12

Use implicit differentiation to find $\frac { \delta z } { \delta x }$ on the surface given by $x ^ { 3 } y + y ^ { 2 } z ^ { 2 } + x z ^ { 3 } = 3$ .

Accepted Answer

The answer of Use implicit differentiation to find \(\frac {...

Question 13

Let $f ( x , y , z ) = \sqrt { 2 x + 3 y - z }$ . Find the rate of change of $f$ at the point $( 1,1 , - 4 )$ in the direction $\mathbf { u }$ where $\mathbf { u }$ is a unit vector making an angle $\theta = 60 ^ { \circ }$ with $\nabla f$ .

Accepted Answer

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

Question 14

Suppose that $z = u ^ { 2 } + u v + v ^ { 3 }$ , and that $u = 2 x ^ { 2 } + 3 x y$ and $v = 2 x - 3 y + 2$ . Find $\frac { \delta z } { \delta x }$ at $( x , y ) = ( 1,2 )$ .

Accepted Answer

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

Question 15

Given that the directional derivative of $f ( x , y )$ at the point $( - 1,2 )$ in the direction of $\{ 3 , - 4 \}$ is $- 10$ and that $| \nabla f ( - 1,2 ) | = 10$ , find $\nabla f ( - 1,2 )$ .

Accepted Answer

The answer of Given that the directional derivative of \(f...

Question 16

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

Accepted Answer

A)  $z = x + y + 1$ 
B)  $z = x + 2 y$ 
C)  $z = x - y + 3$ 
D)  $z = x - 2 y + 4$ 
E)  $z = 2 x + 2 y - 1$ 
F) $z = 2 x + 4 y - 3$

G) $z = 2 x - 2 y + 3$

H) $z = 2 x - 4 y + 5$ 
A)  $z = x + y + 1$ 
B)  $z = x + 2 y$ 
C)  $z = x - y + 3$ 
D)  $z = x - 2 y + 4$ 
E)  $z = 2 x + 2 y - 1$ 
F) $z = 2 x + 4 y - 3$

G) $z = 2 x - 2 y + 3$

H) $z = 2 x - 4 y + 5$

Question 17

Find a normal vector to the surface $x ^ { 2 } + z ^ { 2 } = 5$ at the point $( 1,3 , - 2 )$ .

Accepted Answer

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

H)None of these 
A)  $\{ - 1,0,2 \}$ 
B)  $\{ 1,3 , - 2 \}$ 
C)  $\{ 1 , - 2,0 \}$ 
D)  $\{ - 1,0 , - 3 \}$ 
E)  $\{ 1,0 , - 2 \}$
F) $\{ 0,1 , - 2 \}$ 
 
G) $\{ 1,2,0 \}$

H)None of these

Question 18

Let $f ( x , y , z ) = \frac { x } { y } + \frac { y } { z } + \frac { z } { x }$ . Find the rate of change of $f$ at $P ( 1 , - 1,1 )$ in the direction from $P$ toward $( 2,1,0 )$ .

Accepted Answer

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

Question 19

Suppose you want to give a closed cylindrical tank of radius 20 feet and height 15 feet a cost of paint 0.01 inch thick. Use the differential of the volume of the tank to estimate how many gallons of paint will be required. (1 gallon is approximately 231 cubic inches.)

Accepted Answer

The answer of Suppose you want to give a closed...

Question 20

Let $f ( x , y ) = \frac { x y } { x ^ { 2 } + y ^ { 2 } }$ and let $C _ { m }$ be the curve with the equation $y = m x$ , where $m$ is a constant. The value of the limit of $F ( x , y )$ as $( x , y )$ approaches $( 0,0 )$ along $C _ { m }$ is

Accepted Answer

A) 0 
B)  $\frac { 1 } { 2 }$ 
C) 1 
D)  $\frac { m } { 1 + m ^ { 2 } }$ 
E)  $\frac { m ^ { 2 } } { 1 + m ^ { 2 } }$ 
A) 0 
B)  $\frac { 1 } { 2 }$ 
C) 1 
D)  $\frac { m } { 1 + m ^ { 2 } }$ 
E)  $\frac { m ^ { 2 } } { 1 + m ^ { 2 } }$

Find the directional derivative of the function $f ( x , y ) = y ^ { 2 } \ln x$ at the point $( 1,2 )$ in the direction $( 3,4 )$ .

Find the direction of maximum increase of the function $f ( x , y , z ) = x e ^ { - y } + 3 z$ at the point $( 1,0,4 )$ .

Determine how many critical points the function $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + x ^ { 2 } y + 10$ has.

If $f ( x , y , z ) = x \ln \left( y z ^ { 2 } \right)$ , find $f _ { x y }$ , $f _ { x x }$ , and $f _ { y z }$ .

Find the directional derivative of $f ( x , y ) = 3 x ^ { 2 } + x y - y ^ { 3 }$ in the direction $\theta = \frac { \pi } { 3 }$ .

Let $f ( x , y ) = \left( x ^ { 2 } + y \right) ^ { 3 }$ . If $x = 1$ , find $f ( x , 2 x )$ .

Let $f ( x , y , z ) = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } - z ^ { 2 } }$ .(a) Evaluate $f ( 1 , - 1,1 )$ .(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 $z = f ( x , y ) = x ^ { 3 } y ^ { 4 }$ at the point $( - 1,2 , - 16 )$ .

Let $f ( x , y , z ) = \sqrt { 2 x + 3 y - z }$ . Find the rate of change of $f$ at the point $( 1,1 , - 4 )$ in the direction $\mathbf { u }$ where $\mathbf { u }$ is a unit vector making an angle $\theta = 150 ^ { \circ }$ with $\nabla f$ .

Find the total differential of $w$ if $w = f ( x , y , z ) = x y ^ { 2 } z ^ { 3 } - e ^ { xz } y$

Use implicit differentiation to find $\frac { \delta z } { \delta x }$ on the surface given by $x ^ { 3 } y + y ^ { 2 } z ^ { 2 } + x z ^ { 3 } = 3$ .

Let $f ( x , y , z ) = \sqrt { 2 x + 3 y - z }$ . Find the rate of change of $f$ at the point $( 1,1 , - 4 )$ in the direction $\mathbf { u }$ where $\mathbf { u }$ is a unit vector making an angle $\theta = 60 ^ { \circ }$ with $\nabla f$ .

Suppose that $z = u ^ { 2 } + u v + v ^ { 3 }$ , and that $u = 2 x ^ { 2 } + 3 x y$ and $v = 2 x - 3 y + 2$ . Find $\frac { \delta z } { \delta x }$ at $( x , y ) = ( 1,2 )$ .

Given that the directional derivative of $f ( x , y )$ at the point $( - 1,2 )$ in the direction of $\{ 3 , - 4 \}$ is $- 10$ and that $| \nabla f ( - 1,2 ) | = 10$ , find $\nabla f ( - 1,2 )$ .

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

Find a normal vector to the surface $x ^ { 2 } + z ^ { 2 } = 5$ at the point $( 1,3 , - 2 )$ .

Let $f ( x , y , z ) = \frac { x } { y } + \frac { y } { z } + \frac { z } { x }$ . Find the rate of change of $f$ at $P ( 1 , - 1,1 )$ in the direction from $P$ toward $( 2,1,0 )$ .

Suppose you want to give a closed cylindrical tank of radius 20 feet and height 15 feet a cost of paint 0.01 inch thick. Use the differential of the volume of the tank to estimate how many gallons of paint will be required. (1 gallon is approximately 231 cubic inches.)

Let $f ( x , y ) = \frac { x y } { x ^ { 2 } + y ^ { 2 } }$ and let $C _ { m }$ be the curve with the equation $y = m x$ , where $m$ is a constant. The value of the limit of $F ( x , y )$ as $( x , y )$ approaches $( 0,0 )$ along $C _ { m }$ is

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Exam 11: Partial Derivatives

Find the directional derivative of the function f(x,y)=y2ln⁡xf ( x , y ) = y ^ { 2 } \ln xf(x,y)=y2lnx at the point (1,2)( 1,2 )(1,2) in the direction (3,4)( 3,4 )(3,4) .

Find the direction of maximum increase of the function f(x,y,z)=xe−y+3zf ( x , y , z ) = x e ^ { - y } + 3 zf(x,y,z)=xe−y+3z at the point (1,0,4)( 1,0,4 )(1,0,4) .

Determine how many critical points the function f(x,y)=x2+y2+x2y+10f ( x , y ) = x ^ { 2 } + y ^ { 2 } + x ^ { 2 } y + 10f(x,y)=x2+y2+x2y+10 has.

If f(x,y,z)=xln⁡(yz2)f ( x , y , z ) = x \ln \left( y z ^ { 2 } \right)f(x,y,z)=xln(yz2) , find fxyf _ { x y }fxy​ , fxxf _ { x x }fxx​ , and fyzf _ { y z }fyz​ .

Find the directional derivative of f(x,y)=3x2+xy−y3f ( x , y ) = 3 x ^ { 2 } + x y - y ^ { 3 }f(x,y)=3x2+xy−y3 in the direction θ=π3\theta = \frac { \pi } { 3 }θ=3π​ .

Let f(x,y)=(x2+y)3f ( x , y ) = \left( x ^ { 2 } + y \right) ^ { 3 }f(x,y)=(x2+y)3 . If x=1x = 1x=1 , find f(x,2x)f ( x , 2 x )f(x,2x) .

Let f(x,y,z)=4−x2−y2−z2f ( x , y , z ) = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } - z ^ { 2 } }f(x,y,z)=4−x2−y2−z2​ .(a) Evaluate f(1,−1,1)f ( 1 , - 1,1 )f(1,−1,1) .(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 z=f(x,y)=x3y4z = f ( x , y ) = x ^ { 3 } y ^ { 4 }z=f(x,y)=x3y4 at the point (−1,2,−16)( - 1,2 , - 16 )(−1,2,−16) .

Find the total differential of www if w=f(x,y,z)=xy2z3−exzyw = f ( x , y , z ) = x y ^ { 2 } z ^ { 3 } - e ^ { xz } yw=f(x,y,z)=xy2z3−exzy

The level curves of f(x,y)f ( x , y )f(x,y) are sketched below. (a) Find fx(2,1)f _ { x } ( 2,1 )fx​(2,1) (b) Find fy(2,1)f _ { y } ( 2,1 )fy​(2,1) (c) Sketch the gradient vector ∇f(2,1)\nabla f ( 2,1 )∇f(2,1) .

Use implicit differentiation to find δzδx\frac { \delta z } { \delta x }δxδz​ on the surface given by x3y+y2z2+xz3=3x ^ { 3 } y + y ^ { 2 } z ^ { 2 } + x z ^ { 3 } = 3x3y+y2z2+xz3=3 .

Suppose that z=u2+uv+v3z = u ^ { 2 } + u v + v ^ { 3 }z=u2+uv+v3 , and that u=2x2+3xyu = 2 x ^ { 2 } + 3 x yu=2x2+3xy and v=2x−3y+2v = 2 x - 3 y + 2v=2x−3y+2 . Find δzδx\frac { \delta z } { \delta x }δxδz​ at (x,y)=(1,2)( x , y ) = ( 1,2 )(x,y)=(1,2) .

Find an equation of the tangent plant to the surface z=x2+2y2z = x ^ { 2 } + 2 y ^ { 2 }z=x2+2y2 at the point (1,1,3)( 1,1,3 )(1,1,3) .

Find a normal vector to the surface x2+z2=5x ^ { 2 } + z ^ { 2 } = 5x2+z2=5 at the point (1,3,−2)( 1,3 , - 2 )(1,3,−2) .

Let f(x,y,z)=xy+yz+zxf ( x , y , z ) = \frac { x } { y } + \frac { y } { z } + \frac { z } { x }f(x,y,z)=yx​+zy​+xz​ . Find the rate of change of fff at P(1,−1,1)P ( 1 , - 1,1 )P(1,−1,1) in the direction from PPP toward (2,1,0)( 2,1,0 )(2,1,0) .

Suppose you want to give a closed cylindrical tank of radius 20 feet and height 15 feet a cost of paint 0.01 inch thick. Use the differential of the volume of the tank to estimate how many gallons of paint will be required. (1 gallon is approximately 231 cubic inches.)

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Find the directional derivative of the function $f ( x , y ) = y ^ { 2 } \ln x$ at the point $( 1,2 )$ in the direction $( 3,4 )$ .

Find the direction of maximum increase of the function $f ( x , y , z ) = x e ^ { - y } + 3 z$ at the point $( 1,0,4 )$ .

Determine how many critical points the function $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + x ^ { 2 } y + 10$ has.

If $f ( x , y , z ) = x \ln \left( y z ^ { 2 } \right)$ , find $f _ { x y }$ , $f _ { x x }$ , and $f _ { y z }$ .

Find the directional derivative of $f ( x , y ) = 3 x ^ { 2 } + x y - y ^ { 3 }$ in the direction $\theta = \frac { \pi } { 3 }$ .

Let $f ( x , y ) = \left( x ^ { 2 } + y \right) ^ { 3 }$ . If $x = 1$ , find $f ( x , 2 x )$ .

Let $f ( x , y , z ) = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } - z ^ { 2 } }$ .(a) Evaluate $f ( 1 , - 1,1 )$ .(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 $z = f ( x , y ) = x ^ { 3 } y ^ { 4 }$ at the point $( - 1,2 , - 16 )$ .

Find the total differential of $w$ if $w = f ( x , y , z ) = x y ^ { 2 } z ^ { 3 } - e ^ { xz } y$

Use implicit differentiation to find $\frac { \delta z } { \delta x }$ on the surface given by $x ^ { 3 } y + y ^ { 2 } z ^ { 2 } + x z ^ { 3 } = 3$ .

Suppose that $z = u ^ { 2 } + u v + v ^ { 3 }$ , and that $u = 2 x ^ { 2 } + 3 x y$ and $v = 2 x - 3 y + 2$ . Find $\frac { \delta z } { \delta x }$ at $( x , y ) = ( 1,2 )$ .

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

Find a normal vector to the surface $x ^ { 2 } + z ^ { 2 } = 5$ at the point $( 1,3 , - 2 )$ .

Let $f ( x , y , z ) = \frac { x } { y } + \frac { y } { z } + \frac { z } { x }$ . Find the rate of change of $f$ at $P ( 1 , - 1,1 )$ in the direction from $P$ toward $( 2,1,0 )$ .