Question 1

Solve the problem.
-What is the distance from the surface $x y - z ^ { 2 } - 6 y + 36 = 0$ to the origin?

Accepted Answer

A)  $6 \sqrt { 2 }$ 
B)  $2 \sqrt { 3 }$ 
C)  $6 \sqrt { 3 }$ 
D)  $2 \sqrt { 6 }$ 
A)  $6 \sqrt { 2 }$ 
B)  $2 \sqrt { 3 }$ 
C)  $6 \sqrt { 3 }$ 
D)  $2 \sqrt { 6 }$

Question 2

Compute the gradient of the function at the given point.
-$$f ( x , y , z ) = 2 x y ^ { 3 } z ^ { 2 } , \quad ( 2,8,4 )$$

Accepted Answer

A)  ${\nabla} f  = 16,384 \mathbf { i } + 8192 \mathbf { j } + 24,576 \mathbf { k }$ 
B)  ${\nabla} f  
= \mathbf { i } + 12,288 \mathbf { j } + 4096 \mathbf { k }$ 
C) ${\nabla} f  = \quad \mathbf { i } + 8192 \mathbf { j } + 6144 \mathbf { k }$ 
D)  ${\nabla} f   = 16,384 i + 12,288 j + 16,384 \mathbf { k }$ 
A)  ${\nabla} f  = 16,384 \mathbf { i } + 8192 \mathbf { j } + 24,576 \mathbf { k }$ 
B)  ${\nabla} f  
= \mathbf { i } + 12,288 \mathbf { j } + 4096 \mathbf { k }$ 
C) ${\nabla} f  = \quad \mathbf { i } + 8192 \mathbf { j } + 6144 \mathbf { k }$ 
D)  ${\nabla} f   = 16,384 i + 12,288 j + 16,384 \mathbf { k }$

Question 3

Give an appropriate answer.
-$f ( x , y ) = x y e ^ { - y }$

Accepted Answer

A)  $\frac { \partial f } { \partial x } = y ^ { - y } ; \frac { \partial f } { \partial y } = - x y e ^ { - y }$ 
B)  $\frac { \partial f } { \partial x } = y e ^ { - y ; } \frac { \partial f } { \partial y } = x e ^ { - y } ( 1 - y )$ 
C)  $\frac { \partial f } { \partial x } = y e ^ { - y } ; \frac { \partial f } { \partial y } = x e ^ { - y }$ 
D)  $\frac { \partial f } { \partial x } = y e ^ { - y } ; \frac { \partial f } { \partial y } = x e ^ { - y } ( y - 1 )$ 
A)  $\frac { \partial f } { \partial x } = y ^ { - y } ; \frac { \partial f } { \partial y } = - x y e ^ { - y }$ 
B)  $\frac { \partial f } { \partial x } = y e ^ { - y ; } \frac { \partial f } { \partial y } = x e ^ { - y } ( 1 - y )$ 
C)  $\frac { \partial f } { \partial x } = y e ^ { - y } ; \frac { \partial f } { \partial y } = x e ^ { - y }$ 
D)  $\frac { \partial f } { \partial x } = y e ^ { - y } ; \frac { \partial f } { \partial y } = x e ^ { - y } ( y - 1 )$

Question 4

Provide an appropriate response.
-For which of the following functions do both $\mathrm { f } _ { \mathrm { X } }$ and $\mathrm { f } _ { \mathrm { y } }$ exist?

Accepted Answer

A)  I. and II. 
B)  II. and IV. 
C)  II., III., and IV. 
D)  IV. only 
A)  I. and II. 
B)  II. and IV. 
C)  II., III., and IV. 
D)  IV. only

Question 5

Give an appropriate answer.
-$$f ( x , y ) = 3 x - 3 y ^ { 2 } - 8$$

Accepted Answer

A)  $\frac { \partial f } { \partial x } = - 5 ; \frac { \partial f } { \partial y } = - 6 y - 8$ 
B)  $\frac { \partial f } { \partial x } = 3 x ; \frac { \partial f } { \partial y } = - 6 y$ 
C)  $\frac { \partial f } { \partial x } = 3 ; \frac { \partial f } { \partial y } = - 6 y$ 
D)  $\frac { \partial f } { \partial x } = - 6 y ; \frac { \partial f } { \partial y } = 3$ 
A)  $\frac { \partial f } { \partial x } = - 5 ; \frac { \partial f } { \partial y } = - 6 y - 8$ 
B)  $\frac { \partial f } { \partial x } = 3 x ; \frac { \partial f } { \partial y } = - 6 y$ 
C)  $\frac { \partial f } { \partial x } = 3 ; \frac { \partial f } { \partial y } = - 6 y$ 
D)  $\frac { \partial f } { \partial x } = - 6 y ; \frac { \partial f } { \partial y } = 3$

Question 6

Determine whether the given function satisfies a Laplace equation.
-$$f ( x , y ) = \frac { x ^ { 2 } } { y }$$

Accepted Answer

A) True 
 B)False

Question 7

Solve the problem.
-The surface area of a hollow cylinder (tube) is given by $\mathrm { S } = 2 \pi \left( \mathrm { R } _ { 1 } + \mathrm { R } _ { 2 } \right) \left( \mathrm { h } + \mathrm { R } _ { 1 } - \mathrm { R } _ { 2 } \right) ,$ where $h$ is the length of the cylinder and $R _ { 1 }$ and $R _ { 2 }$ are the outer and inner radii. If $h _ { \text {, } } R _ { 1 }$, and $R _ { 2 }$ are measured to be 6 inches, 7 inches, and 9 inches respectively, and if these measurements are accurate to within $0.1$ inches, estimate the maximum percentage error in computing $\mathrm { S }$.

Accepted Answer

A)  0.045% 
B)  0.056% 
C)  0.067% 
D)  0.089% 
A)  0.045% 
B)  0.056% 
C)  0.067% 
D)  0.089%

Question 8

Find the specific function value.
-$\text { Find } f ( 4,6 ) \text { when } f ( x , y ) = ( x + y ) ^ { 3 }$

Accepted Answer

A)  100 
B)  1000 
C)  30 
D)  280 
A)  100 
B)  1000 
C)  30 
D)  280

Question 9

Write a chain rule formula for the following derivative.
-$\frac { \partial w } { \partial x }$ for $w = f ( p , q ) ; p = g ( x , y ) , q = h ( x , y )$

Accepted Answer

A)  $\frac { \partial w } { \partial x } = \frac { \partial w } { \partial p } \frac { \partial p } { \partial x } + \frac { \partial w } { \partial q } \frac { \partial q } { \partial x }$ 
B)  $\frac { \partial w } { \partial x } = \frac { \partial w } { \partial p } \frac { \partial p } { \partial x }$ 
C)  $\frac { \partial w } { \partial x } = \frac { \partial w } { \partial x } \frac { \partial x } { \partial p } + \frac { \partial w } { \partial x } \frac { \partial x } { \partial q }$ 
D)  $\frac { \partial w } { \partial x } = \frac { \partial w } { \partial p } + \frac { \partial w } { \partial q }$ 
A)  $\frac { \partial w } { \partial x } = \frac { \partial w } { \partial p } \frac { \partial p } { \partial x } + \frac { \partial w } { \partial q } \frac { \partial q } { \partial x }$ 
B)  $\frac { \partial w } { \partial x } = \frac { \partial w } { \partial p } \frac { \partial p } { \partial x }$ 
C)  $\frac { \partial w } { \partial x } = \frac { \partial w } { \partial x } \frac { \partial x } { \partial p } + \frac { \partial w } { \partial x } \frac { \partial x } { \partial q }$ 
D)  $\frac { \partial w } { \partial x } = \frac { \partial w } { \partial p } + \frac { \partial w } { \partial q }$

Question 10

Find the requested partial derivative.
-$( \partial \mathrm { w } / \partial \mathrm { y } ) _ { \mathrm { X } } \text { at } ( \mathrm { x } , \mathrm { y } , \mathrm { z } , \mathrm { w } ) = ( 1,1,2,14 ) \text { if } \mathrm { w } = 2 \mathrm { x } + 6 \mathrm { y } + 3 \mathrm { z } \text { and } \mathrm { x } + \mathrm { y } = \mathrm { z }$

Accepted Answer

A)  6 
B)  11 
C)  1 
D)  9 
A)  6 
B)  11 
C)  1 
D)  9

Question 11

Find the specific function value.
-$\text { Find } f ( 3,0,9 ) \text { when } f ( x , y , z ) = 3 x ^ { 2 } + 3 y ^ { 2 } - z ^ { 2 }$

Accepted Answer

A)  108 
B)  -72 
C)  -49 
D)  -54 
A)  108 
B)  -72 
C)  -49 
D)  -54

Question 12

Sketch the surface z = f(x,y).
-$$f ( x , y ) = 2 - x ^ { 2 } - y ^ { 2 }$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 13

Use Taylor's formula to find the requested approximation of f(x, y) near the origin.
-Cubic approximation to f(x, y) = ln(1 + 7x + y)

Accepted Answer

A)  $1 + 7 x + y - \frac { 49 } { 2 } x ^ { 2 } - 7 x y - \frac { 1 } { 2 } y ^ { 2 } + \frac { 343 } { 3 } x ^ { 3 } + \frac { 49 } { 3 } x ^ { 2 } y + \frac { 7 } { 3 } x y ^ { 2 } + \frac { 1 } { 3 } y ^ { 3 }$ 
B)  $7 x + y - \frac { 49 } { 2 } x ^ { 2 } - 7 x y - \frac { 1 } { 2 } y ^ { 2 } + \frac { 343 } { 3 } x ^ { 3 } + \frac { 49 } { 3 } x ^ { 2 } y + \frac { 7 } { 3 } x y ^ { 2 } + \frac { 1 } { 3 } y ^ { 3 }$ 
C)  $1 + 7 x + y - \frac { 49 } { 2 } x ^ { 2 } - 7 x y - \frac { 1 } { 2 } y ^ { 2 } + \frac { 343 } { 3 } x ^ { 3 } + 49 x ^ { 2 } y + 7 x y ^ { 2 } + \frac { 1 } { 3 } y ^ { 3 }$ 
D)  $7 x + y - \frac { 49 } { 2 } x ^ { 2 } - 7 x y - \frac { 1 } { 2 } y ^ { 2 } + \frac { 343 } { 3 } x ^ { 3 } + 49 x ^ { 2 } y + 7 x y ^ { 2 } + \frac { 1 } { 3 } y ^ { 3 }$ 
A)  $1 + 7 x + y - \frac { 49 } { 2 } x ^ { 2 } - 7 x y - \frac { 1 } { 2 } y ^ { 2 } + \frac { 343 } { 3 } x ^ { 3 } + \frac { 49 } { 3 } x ^ { 2 } y + \frac { 7 } { 3 } x y ^ { 2 } + \frac { 1 } { 3 } y ^ { 3 }$ 
B)  $7 x + y - \frac { 49 } { 2 } x ^ { 2 } - 7 x y - \frac { 1 } { 2 } y ^ { 2 } + \frac { 343 } { 3 } x ^ { 3 } + \frac { 49 } { 3 } x ^ { 2 } y + \frac { 7 } { 3 } x y ^ { 2 } + \frac { 1 } { 3 } y ^ { 3 }$ 
C)  $1 + 7 x + y - \frac { 49 } { 2 } x ^ { 2 } - 7 x y - \frac { 1 } { 2 } y ^ { 2 } + \frac { 343 } { 3 } x ^ { 3 } + 49 x ^ { 2 } y + 7 x y ^ { 2 } + \frac { 1 } { 3 } y ^ { 3 }$ 
D)  $7 x + y - \frac { 49 } { 2 } x ^ { 2 } - 7 x y - \frac { 1 } { 2 } y ^ { 2 } + \frac { 343 } { 3 } x ^ { 3 } + 49 x ^ { 2 } y + 7 x y ^ { 2 } + \frac { 1 } { 3 } y ^ { 3 }$

Question 14

Match the surface show below to the graph of its level curves.
-

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 15

Show that the function is a solution of the wave equation.
-w(x, t) = ln cxt

Accepted Answer

A) True 
 B)False

Question 16

Write a chain rule formula for the following derivative.
-$$\frac { \partial \mathrm { u } } { \partial \mathrm { x } } \text { for } \mathrm { u } = \mathrm { f } ( \mathrm { p } , \mathrm { q } ) ; \mathrm { p } = \mathrm { g } ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) , \mathrm { q } = \mathrm { h } ( \mathrm { x } , \mathrm { y } , \mathrm { z } )$$

Accepted Answer

A)  $\frac { \partial \mathrm { u } } { \partial \mathrm { x } } = \frac { \partial \mathrm { u } } { \partial \mathrm { p } } \frac { \partial \mathrm { p } } { \partial \mathrm { x } } + \frac { \partial \mathrm { u } } { \partial \mathrm { q } } \frac { \partial \mathrm { q } } { \partial \mathrm { x } }$ 
B)  $\frac { \partial \mathrm { u } } { \partial \mathrm { x } } = \frac { \partial \mathrm { u } } { \partial \mathrm { p } } \frac { \partial \mathrm { x } } { \partial \mathrm { p } } + \frac { \partial \mathrm { u } } { \partial \mathrm { q } } \frac { \partial \mathrm { x } } { \partial \mathrm { q } }$ 
C)  $\frac { \partial \mathrm { u } } { \partial \mathrm { x } } = \frac { \partial \mathrm { u } } { \partial \mathrm { q } } \frac { \partial \mathrm { q } } { \partial \mathrm { x } }$ 
D)  $\frac { \partial \mathrm { u } } { \partial \mathrm { x } } = \frac { \partial \mathrm { u } } { \partial \mathrm { p } } \frac { \partial \mathrm { p } } { \partial \mathrm { x } } + \frac { \partial \mathrm { u } } { \partial \mathrm { q } } \frac { \partial \mathrm { q } } { \partial \mathrm { y } } + \frac { \partial \mathrm { u } } { \partial \mathrm { q } } \frac { \partial \mathrm { q } } { \partial \mathrm { z } }$ 
A)  $\frac { \partial \mathrm { u } } { \partial \mathrm { x } } = \frac { \partial \mathrm { u } } { \partial \mathrm { p } } \frac { \partial \mathrm { p } } { \partial \mathrm { x } } + \frac { \partial \mathrm { u } } { \partial \mathrm { q } } \frac { \partial \mathrm { q } } { \partial \mathrm { x } }$ 
B)  $\frac { \partial \mathrm { u } } { \partial \mathrm { x } } = \frac { \partial \mathrm { u } } { \partial \mathrm { p } } \frac { \partial \mathrm { x } } { \partial \mathrm { p } } + \frac { \partial \mathrm { u } } { \partial \mathrm { q } } \frac { \partial \mathrm { x } } { \partial \mathrm { q } }$ 
C)  $\frac { \partial \mathrm { u } } { \partial \mathrm { x } } = \frac { \partial \mathrm { u } } { \partial \mathrm { q } } \frac { \partial \mathrm { q } } { \partial \mathrm { x } }$ 
D)  $\frac { \partial \mathrm { u } } { \partial \mathrm { x } } = \frac { \partial \mathrm { u } } { \partial \mathrm { p } } \frac { \partial \mathrm { p } } { \partial \mathrm { x } } + \frac { \partial \mathrm { u } } { \partial \mathrm { q } } \frac { \partial \mathrm { q } } { \partial \mathrm { y } } + \frac { \partial \mathrm { u } } { \partial \mathrm { q } } \frac { \partial \mathrm { q } } { \partial \mathrm { z } }$

Question 17

Write a chain rule formula for the following derivative.
-$\frac { \partial \mathrm { u } } { \partial \mathrm { x } }$ for $\mathrm { u } = \mathrm { f } ( \mathrm { r } , \mathrm { s } , \mathrm { t } ) ; \mathrm { r } = \mathrm { g } ( \mathrm { y } ) , \mathrm { s } = \mathrm { h } ( \mathrm { z } ) , \mathrm { t } = \mathrm { k } ( \mathrm { x } , \mathrm { z } )$

Accepted Answer

A)  $\frac { \partial \mathrm { u } } { \partial \mathrm { x } } = 0$ 
B)  $\frac { \partial \mathrm { u } } { \partial \mathrm { x } } = \frac { \partial \mathrm { u } } { \partial \mathrm { t } }$ 
C)  $\frac { \partial u } { \partial x } = \frac { \partial t } { \partial x }$ 
D)  $\frac { \partial \mathrm { u } } { \partial \mathrm { x } } = \frac { \partial \mathrm { u } } { \partial \mathrm { t } } \frac { \partial \mathrm { t } } { \partial \mathrm { x } }$ 
A)  $\frac { \partial \mathrm { u } } { \partial \mathrm { x } } = 0$ 
B)  $\frac { \partial \mathrm { u } } { \partial \mathrm { x } } = \frac { \partial \mathrm { u } } { \partial \mathrm { t } }$ 
C)  $\frac { \partial u } { \partial x } = \frac { \partial t } { \partial x }$ 
D)  $\frac { \partial \mathrm { u } } { \partial \mathrm { x } } = \frac { \partial \mathrm { u } } { \partial \mathrm { t } } \frac { \partial \mathrm { t } } { \partial \mathrm { x } }$

Question 18

Find the derivative of the function at P0 in the direction of u.
-$$f ( x , y , z ) = \tan ^ { - 1 } \frac { 4 x } { 2 y + 4 z } , \quad P _ { 0 } ( - 9,0,0 ) , \quad \mathbf { u } = 12 \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k }$$

Accepted Answer

A)  $\frac { 2 } { 117 }$ 
B)  $\frac { 1 } { 78 }$ 
C)  $\frac { 1 } { 117 }$ 
D)  $\frac { 5 } { 234 }$ 
A)  $\frac { 2 } { 117 }$ 
B)  $\frac { 1 } { 78 }$ 
C)  $\frac { 1 } { 117 }$ 
D)  $\frac { 5 } { 234 }$

Question 19

At what points is the given function continuous?
-f(x, y, z) = ln(x + y + z - 6)

Accepted Answer

A)  All $( x , y , z )$ such that $x + y + z \geq 6$ 
B)  All $( x , y , z )$ such that $x + y + z > 6$ 
C)  All $( x , y , z )$ in the first octant 
D)  $\operatorname { All } ( x , y , z )$ 
A)  All $( x , y , z )$ such that $x + y + z \geq 6$ 
B)  All $( x , y , z )$ such that $x + y + z > 6$ 
C)  All $( x , y , z )$ in the first octant 
D)  $\operatorname { All } ( x , y , z )$

Question 20

Solve the problem.
-A rectangle with sides parallel to the axes is inscribed in the region bounded by the axes and the line x + 2y = 2. Find the maximum area of this rectangle.

Accepted Answer

A)  $\frac { 1 } { 4 }$ 
B)  $\frac { 1 } { 3 }$ 
C)  $\frac { 2 } { 3 }$ 
D)  $\frac { 1 } { 2 }$ 
A)  $\frac { 1 } { 4 }$ 
B)  $\frac { 1 } { 3 }$ 
C)  $\frac { 2 } { 3 }$ 
D)  $\frac { 1 } { 2 }$

Solve the problem. -What is the distance from the surface $x y - z ^ { 2 } - 6 y + 36 = 0$ to the origin?

Compute the gradient of the function at the given point. - $f ( x , y , z ) = 2 x y ^ { 3 } z ^ { 2 } , \quad ( 2,8,4 )$

Give an appropriate answer. - $f ( x , y ) = x y e ^ { - y }$

Give an appropriate answer. - $f ( x , y ) = 3 x - 3 y ^ { 2 } - 8$

Determine whether the given function satisfies a Laplace equation. - $f ( x , y ) = \frac { x ^ { 2 } } { y }$

Find the specific function value. - $\text { Find } f ( 4,6 ) \text { when } f ( x , y ) = ( x + y ) ^ { 3 }$

Write a chain rule formula for the following derivative. - $\frac { \partial w } { \partial x }$ for $w = f ( p , q ) ; p = g ( x , y ) , q = h ( x , y )$

Find the specific function value. - $\text { Find } f ( 3,0,9 ) \text { when } f ( x , y , z ) = 3 x ^ { 2 } + 3 y ^ { 2 } - z ^ { 2 }$

Sketch the surface z = f(x,y). - $f ( x , y ) = 2 - x ^ { 2 } - y ^ { 2 }$

Use Taylor's formula to find the requested approximation of f(x, y) near the origin. -Cubic approximation to f(x, y) = ln(1 + 7x + y)

Match the surface show below to the graph of its level curves. -

Show that the function is a solution of the wave equation. -w(x, t) = ln cxt

Find the derivative of the function at P0 in the direction of u. - $f ( x , y , z ) = \tan ^ { - 1 } \frac { 4 x } { 2 y + 4 z } , \quad P _ { 0 } ( - 9,0,0 ) , \quad \mathbf { u } = 12 \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k }$

At what points is the given function continuous? -f(x, y, z) = ln(x + y + z - 6)

Solve the problem. -A rectangle with sides parallel to the axes is inscribed in the region bounded by the axes and the line x + 2y = 2. Find the maximum area of this rectangle.

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Exam 15: Partial Derivatives

Solve the problem. -What is the distance from the surface xy−z2−6y+36=0x y - z ^ { 2 } - 6 y + 36 = 0xy−z2−6y+36=0 to the origin?

Compute the gradient of the function at the given point. - f(x,y,z)=2xy3z2,(2,8,4)f ( x , y , z ) = 2 x y ^ { 3 } z ^ { 2 } , \quad ( 2,8,4 )f(x,y,z)=2xy3z2,(2,8,4)

Give an appropriate answer. - f(x,y)=xye−yf ( x , y ) = x y e ^ { - y }f(x,y)=xye−y

Provide an appropriate response. -For which of the following functions do both fX\mathrm { f } _ { \mathrm { X } }fX​ and fy\mathrm { f } _ { \mathrm { y } }fy​ exist?

Give an appropriate answer. - f(x,y)=3x−3y2−8f ( x , y ) = 3 x - 3 y ^ { 2 } - 8f(x,y)=3x−3y2−8

Determine whether the given function satisfies a Laplace equation. - f(x,y)=x2yf ( x , y ) = \frac { x ^ { 2 } } { y }f(x,y)=yx2​

Find the specific function value. - Find f(4,6) when f(x,y)=(x+y)3\text { Find } f ( 4,6 ) \text { when } f ( x , y ) = ( x + y ) ^ { 3 } Find f(4,6) when f(x,y)=(x+y)3

Write a chain rule formula for the following derivative. - ∂w∂x\frac { \partial w } { \partial x }∂x∂w​ for w=f(p,q);p=g(x,y),q=h(x,y)w = f ( p , q ) ; p = g ( x , y ) , q = h ( x , y )w=f(p,q);p=g(x,y),q=h(x,y)

Find the specific function value. - Find f(3,0,9) when f(x,y,z)=3x2+3y2−z2\text { Find } f ( 3,0,9 ) \text { when } f ( x , y , z ) = 3 x ^ { 2 } + 3 y ^ { 2 } - z ^ { 2 } Find f(3,0,9) when f(x,y,z)=3x2+3y2−z2

Sketch the surface z = f(x,y). - f(x,y)=2−x2−y2f ( x , y ) = 2 - x ^ { 2 } - y ^ { 2 }f(x,y)=2−x2−y2