Question 1

Show that the function is a solution of the wave equation.
-w(x, t) = sin (-10x - 10ct)

Accepted Answer

A) True 
 B)False

Question 2

Find the linearization of the function at the given point.
-$$f ( x , y ) = 8 x ^ { 2 } + 2 y ^ { 2 } + 8 \text { at } ( - 8 , - 3 )$$

Accepted Answer

A)  L(x, y) = 16x + 4y + 538 
B)  L(x, y) = 16x + 4y - 522 
C)  L(x, y) = -128x - 12y - 522 
D)  L(x, y) = -128x - 12y + 538 
A)  L(x, y) = 16x + 4y + 538 
B)  L(x, y) = 16x + 4y - 522 
C)  L(x, y) = -128x - 12y - 522 
D)  L(x, y) = -128x - 12y + 538

Question 3

Answer the question.
-You are hiking on a mountainside, following a trail that slopes downward for a short distance and then begins to climb again. At the bottom of this local "dip", what can be said about the relationship between the trail's
Direction and the contour of the mountainside? [Hint - Think of the trail as a constrained path, g(x, y) = c, on the Mountainside's surface, altitude = f(x, y). Consider only infinitesimal displacements.]

Accepted Answer

A)  At the bottom of the dip, the trail is headed in the direction of the mountain's steepest ascent. 
B)  At the bottom of the dip, the trail is headed along the mountain's contour line which passes through that point. 
C)  The mountainside rises in all directions relative to the dip. 
D)  At the bottom of the dip, the trail is headed perpendicular to the mountain's contour line which passes through that point. 
A)  At the bottom of the dip, the trail is headed in the direction of the mountain's steepest ascent. 
B)  At the bottom of the dip, the trail is headed along the mountain's contour line which passes through that point. 
C)  The mountainside rises in all directions relative to the dip. 
D)  At the bottom of the dip, the trail is headed perpendicular to the mountain's contour line which passes through that point.

Question 4

Estimate the error in the quadratic approximation of the given function at the origin over the given region.
-$$f ( x , y ) = e ^ { 4 x } \cos 2 y , \quad | x | \leq 0.1 , | y | \leq 0.1$$

Accepted Answer

A)  $| \mathrm { E } ( \mathrm { x } , \mathrm { y } ) | \leq 0.0955$ 
B)  $| E ( x , y ) | \leq 0.3819$ 
C)  $| \mathrm { E } ( \mathrm { x } , \mathrm { y } ) | \leq 0.191$ 
D)  $| \mathrm { E } ( \mathrm { x } , \mathrm { y } ) | \leq 0.1273$ 
A)  $| \mathrm { E } ( \mathrm { x } , \mathrm { y } ) | \leq 0.0955$ 
B)  $| E ( x , y ) | \leq 0.3819$ 
C)  $| \mathrm { E } ( \mathrm { x } , \mathrm { y } ) | \leq 0.191$ 
D)  $| \mathrm { E } ( \mathrm { x } , \mathrm { y } ) | \leq 0.1273$

Question 5

Solve the problem.
-If the length, width, and height of a rectangular solid are measured to be 5, 2, and 10 inches respectively and each measurement is accurate to within 0.1 inch, estimate the maximum percentage error in computing the Volume of the solid.

Accepted Answer

A)  9.60% 
B)  8.00% 
C)  7.20% 
D)  6.40% 
A)  9.60% 
B)  8.00% 
C)  7.20% 
D)  6.40%

Question 6

Find the domain and range and describe the level curves for the function f(x,y).
-$$f ( x , y ) = \sqrt { 25 - x ^ { 2 } - y ^ { 2 } }$$

Accepted Answer

A)  Domain: all points in the $x y$-plane satisfying $x ^ { 2 } + y ^ { 2 } = 25$; range: real numbers $0 \leq z \leq 5$; level curves: circles with centers at $( 0,0 )$ and radii $r , 0 < r \leq 5$ 
B)  Domain: all points in the $x y$-plane; range: real numbers $0 \leq z \leq 5$; level curves: circles with centers at ( 0 , 0 ) and radii $r , 0 < r \leq 5$ 
C)  Domain: all points in the $x y$-plane; range: all real numbers; level curves: circles with centers at $( 0,0 )$ 
D)  Domain: all points in the $x y$-plane satisfying $x ^ { 2 } + y ^ { 2 } \leq 25$; range: real numbers $0 \leq z \leq 5$; level curves: circles with centers at $( 0,0 )$ and radii $r , 0 < r \leq 5$ 
A)  Domain: all points in the $x y$-plane satisfying $x ^ { 2 } + y ^ { 2 } = 25$; range: real numbers $0 \leq z \leq 5$; level curves: circles with centers at $( 0,0 )$ and radii $r , 0 < r \leq 5$ 
B)  Domain: all points in the $x y$-plane; range: real numbers $0 \leq z \leq 5$; level curves: circles with centers at ( 0 , 0 ) and radii $r , 0 < r \leq 5$ 
C)  Domain: all points in the $x y$-plane; range: all real numbers; level curves: circles with centers at $( 0,0 )$ 
D)  Domain: all points in the $x y$-plane satisfying $x ^ { 2 } + y ^ { 2 } \leq 25$; range: real numbers $0 \leq z \leq 5$; level curves: circles with centers at $( 0,0 )$ and radii $r , 0 < r \leq 5$

Question 7

Solve the problem.
-Find the point on the plane x + 2y - z = 12 that is nearest the origin.

Accepted Answer

A)  (4, 4, 0) 
B)  (2, 4, -2) 
C)  (-2, 8, 2) 
D)  (2, 4, 0) 
A)  (4, 4, 0) 
B)  (2, 4, -2) 
C)  (-2, 8, 2) 
D)  (2, 4, 0)

Question 8

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0).
-$$f ( x , y ) = \frac { x ^ { 2 } } { x ^ { 4 } + y ^ { 2 } }$$

Accepted Answer

Answers will vary. One possibi

Question 9

Find the extreme values of the function subject to the given constraint.
-$$f ( x , y ) = y ^ { 2 } - x ^ { 2 } , \quad x ^ { 2 } + y ^ { 2 } = 16$$

Accepted Answer

A)  Maximum: 32 at $( 0 , \pm 4 \sqrt { 2 } )$; minimum: $- 32$ at $( \pm 4 \sqrt { 2 } , 0 )$ 
B)  Maximum: 16 at $( 0 , \pm 4 )$; minimum: $- 32$ at $( \pm 4 \sqrt { 2 } , 0 )$ 
C)  Maximum: 32 at $( 0 , \pm 4 \sqrt { 2 } )$; minimum: $- 16$ at $( \pm 4,0 )$ 
D)  Maximum: 16 at $( 0 , \pm 4 )$; minimum: $- 16$ at $( \pm 4,0 )$ 
A)  Maximum: 32 at $( 0 , \pm 4 \sqrt { 2 } )$; minimum: $- 32$ at $( \pm 4 \sqrt { 2 } , 0 )$ 
B)  Maximum: 16 at $( 0 , \pm 4 )$; minimum: $- 32$ at $( \pm 4 \sqrt { 2 } , 0 )$ 
C)  Maximum: 32 at $( 0 , \pm 4 \sqrt { 2 } )$; minimum: $- 16$ at $( \pm 4,0 )$ 
D)  Maximum: 16 at $( 0 , \pm 4 )$; minimum: $- 16$ at $( \pm 4,0 )$

Question 10

Find the equation for the level surface of the function through the given point.
-$$f ( x , y , z ) = e ^ { \left( x ^ { 2 } + y ^ { 2 } - z \right) } , ( 3,5,9 )$$

Accepted Answer

A)  $\ln \left( x ^ { 2 } + y ^ { 2 } - z \right) = 25$ 
B)  $e ^ { \left( x ^ { 2 } + y ^ { 2 } - z \right) } = \ln ( 25 )$ 
C)  $x ^ { 2 } + y ^ { 2 } - z = \ln ( 25 )$ 
D)  $x ^ { 2 } + y ^ { 2 } - z = e ^ { 25 }$ 
A)  $\ln \left( x ^ { 2 } + y ^ { 2 } - z \right) = 25$ 
B)  $e ^ { \left( x ^ { 2 } + y ^ { 2 } - z \right) } = \ln ( 25 )$ 
C)  $x ^ { 2 } + y ^ { 2 } - z = \ln ( 25 )$ 
D)  $x ^ { 2 } + y ^ { 2 } - z = e ^ { 25 }$

Question 11

Solve the problem.
-Find an equation for the level curve of the function $\mathrm { f } ( \mathrm { x } , \mathrm { y } ) = 16 - \mathrm { x } ^ { 2 } - \mathrm { y } ^ { 2 }$ that passes through the point $( \sqrt { 2 } , \sqrt { 5 } )$.

Accepted Answer

A)  $x ^ { 2 } + y ^ { 2 } = - 7$ 
B)  $x ^ { 2 } - y ^ { 2 } = 7$ 
C)  $x ^ { 2 } + y ^ { 2 } = 23$ 
D)  $x ^ { 2 } + y ^ { 2 } = 7$ 
A)  $x ^ { 2 } + y ^ { 2 } = - 7$ 
B)  $x ^ { 2 } - y ^ { 2 } = 7$ 
C)  $x ^ { 2 } + y ^ { 2 } = 23$ 
D)  $x ^ { 2 } + y ^ { 2 } = 7$

Question 12

Find the limit.
-

Accepted Answer

A)  4 
B)  0 
C)  12 
D)  1 
A)  4 
B)  0 
C)  12 
D)  1

Question 13

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0).
-$$f ( x , y ) = \frac { x ^ { 5 } - y } { x ^ { 5 } + y }$$

Accepted Answer

Answers will vary. O

Question 14

Solve the problem.
-Find an equation for the level surface of the function $f ( x , y , z ) = \int _ { z } ^ { y } e ^ { \theta } d \theta - \int _ { x } ^ { \sqrt { 2 } } t d t$ that passes through the point $( 0 , \ln 2 , \ln 4 )$.

Accepted Answer

A)  $e ^ { y } + e ^ { z } - \frac { x ^ { 2 } } { 2 } = - 2$ 
B)  $e ^ { y } - e ^ { z } + \frac { x ^ { 2 } } { 2 } = 2$ 
C)  $e ^ { y } - e ^ { z } + \frac { x ^ { 2 } } { 2 } = - 2$ 
D)  $e ^ { y } + e ^ { z } - \frac { x ^ { 2 } } { 2 } = 2$ 
A)  $e ^ { y } + e ^ { z } - \frac { x ^ { 2 } } { 2 } = - 2$ 
B)  $e ^ { y } - e ^ { z } + \frac { x ^ { 2 } } { 2 } = 2$ 
C)  $e ^ { y } - e ^ { z } + \frac { x ^ { 2 } } { 2 } = - 2$ 
D)  $e ^ { y } + e ^ { z } - \frac { x ^ { 2 } } { 2 } = 2$

Question 15

Solve the problem.
-Write parametric equations for the tangent line to the curve of intersection of the surfaces $x + y ^ { 2 } + 7 z = 9$ and $x = 1$ at the point $( 1,1,1 )$.

Accepted Answer

A)  x = 1, y = 7t + 1, z = -2t + 1 
B)  x = 1, y = 14t + 1, z = -t + 1 
C)  x = 1, y = 7t + 1, z = -t + 1 
D)  x = 1, y = 14t + 1, z = -2t + 1 
A)  x = 1, y = 7t + 1, z = -2t + 1 
B)  x = 1, y = 14t + 1, z = -t + 1 
C)  x = 1, y = 7t + 1, z = -t + 1 
D)  x = 1, y = 14t + 1, z = -2t + 1

Question 16

Find all the local maxima, local minima, and saddle points of the function.
-$$f ( x , y ) = \left( x ^ { 2 } - 9 \right) ^ { 2 } + \left( y ^ { 2 } - 16 \right) ^ { 2 }$$

Accepted Answer

A)  f(0, 0) = 337, local maximum; f(-3, -4) = 0, local minimum 
B)  f(0, 0) = 337, local maximum; f(0, 4) = 81, saddle point; f(3, 0) = 256, saddle point; f(3, 4) = 0, local minimum; f(-3, -4) = 0, local minimum 
C)  f(0, 0) = 337, local maximum; f(0, 4) = 81, saddle point; f(0, -4) = 81, saddle point; f(3, 0) = 337, saddle point; f(3, 4) = 0, local minimum; f(3, -4) = 0, local minimum;
F(-3, 0) = 256, saddle point; f(-3, 4) = 0, local minimum; f(-3, -4) = 0, local minimum 
D)  f(0, 0) = 337, local maximum; f(3, 4) = 0, local minimum; f(3, -4) = 0, local minimum; f(-3, 4) = 0, local minimum; f(-3, -4) = 0, local minimum 
A)  f(0, 0) = 337, local maximum; f(-3, -4) = 0, local minimum 
B)  f(0, 0) = 337, local maximum; f(0, 4) = 81, saddle point; f(3, 0) = 256, saddle point; f(3, 4) = 0, local minimum; f(-3, -4) = 0, local minimum 
C)  f(0, 0) = 337, local maximum; f(0, 4) = 81, saddle point; f(0, -4) = 81, saddle point; f(3, 0) = 337, saddle point; f(3, 4) = 0, local minimum; f(3, -4) = 0, local minimum;
F(-3, 0) = 256, saddle point; f(-3, 4) = 0, local minimum; f(-3, -4) = 0, local minimum 
D)  f(0, 0) = 337, local maximum; f(3, 4) = 0, local minimum; f(3, -4) = 0, local minimum; f(-3, 4) = 0, local minimum; f(-3, -4) = 0, local minimum

Question 17

Find all the local maxima, local minima, and saddle points of the function.
-$$f ( x , y ) = \left( x ^ { 2 } - 81 \right) ^ { 2 } - \left( y ^ { 2 } - 64 \right) ^ { 2 }$$

Accepted Answer

A)  f(0, 0) = 2465, saddle point; f(0, 8) = 6561, local maximum; f(0, -8) = 6561, local maximum; f(9, 0) = -4096, local minimum; f(9, 8) = 0, saddle point; f(9, -8) = 0, saddle point;
F(-9, 0) = -4096, local minimum; f(-9, 8) = 0, saddle point; f(-9, -8) = 0, saddle point 
B)  f(0, 0) = 2465, saddle point; f(9, 8) = 0, saddle point; f(9, -8) = 0, saddle point; f(-9, 8) = 0, saddle point; f(-9, -8) = 0, saddle point 
C)  f(0, 0) = 2465, saddle point; f(0, 8) = 6561, local maximum; f(9, 0) = -4096, local minimum 
D)  f(0, 8) = 6561, local maximum; f(0, -8) = 6561, local maximum; f(9, 0) = -4096, local minimum; f(-9, 0) = -4096, local minimum 
A)  f(0, 0) = 2465, saddle point; f(0, 8) = 6561, local maximum; f(0, -8) = 6561, local maximum; f(9, 0) = -4096, local minimum; f(9, 8) = 0, saddle point; f(9, -8) = 0, saddle point;
F(-9, 0) = -4096, local minimum; f(-9, 8) = 0, saddle point; f(-9, -8) = 0, saddle point 
B)  f(0, 0) = 2465, saddle point; f(9, 8) = 0, saddle point; f(9, -8) = 0, saddle point; f(-9, 8) = 0, saddle point; f(-9, -8) = 0, saddle point 
C)  f(0, 0) = 2465, saddle point; f(0, 8) = 6561, local maximum; f(9, 0) = -4096, local minimum 
D)  f(0, 8) = 6561, local maximum; f(0, -8) = 6561, local maximum; f(9, 0) = -4096, local minimum; f(-9, 0) = -4096, local minimum

Question 18

Provide an appropriate answer.
-Suppose that $w = x ^ { 2 } + y ^ { 2 } + 20 z + t$ and $x + 4 z + t = 8$. Assuming that the independent variables are $x$, $y$, and $z$, find $\frac { \partial w } { \partial x }$.

Accepted Answer

A)  $\frac { \partial w } { \partial x } = 2 x - 5$ 
B)  $\frac { \partial w } { \partial x } = 2 x + 5$ 
C)  $\frac { \partial w } { \partial x } = 2 x + 1$ 
D)  $\frac { \partial w } { \partial x } = 2 x - 1$ 
A)  $\frac { \partial w } { \partial x } = 2 x - 5$ 
B)  $\frac { \partial w } { \partial x } = 2 x + 5$ 
C)  $\frac { \partial w } { \partial x } = 2 x + 1$ 
D)  $\frac { \partial w } { \partial x } = 2 x - 1$

Question 19

Provide an appropriate response.
-Find $F ^ { \prime } ( x )$ if $F ( x ) = \int _ { x } ^ { 1 } \sqrt { t ^ { 4 } + x } d t$.

Accepted Answer

A)  $\int _ { x } ^ { 1 } \frac { 1 } { \sqrt { t ^ { 4 } + x } } d t - \sqrt { x ^ { 4 } + x }$ 
B)  $\int _ { x } ^ { 1 } \frac { 1 } { \sqrt { t ^ { 4 } + x } } d t - x \sqrt { x ^ { 4 } + x }$ 
C)  $\int _ { x } ^ { 1 } \frac { 1 } { 2 \sqrt { t ^ { 4 } + x } } d t - \sqrt { x ^ { 4 } + x }$ 
D)  $\sqrt { x ^ { 4 } + x } - \int _ { x } ^ { 1 } \frac { 1 } { 2 \sqrt { t ^ { 4 } + x } } d t$ 
A)  $\int _ { x } ^ { 1 } \frac { 1 } { \sqrt { t ^ { 4 } + x } } d t - \sqrt { x ^ { 4 } + x }$ 
B)  $\int _ { x } ^ { 1 } \frac { 1 } { \sqrt { t ^ { 4 } + x } } d t - x \sqrt { x ^ { 4 } + x }$ 
C)  $\int _ { x } ^ { 1 } \frac { 1 } { 2 \sqrt { t ^ { 4 } + x } } d t - \sqrt { x ^ { 4 } + x }$ 
D)  $\sqrt { x ^ { 4 } + x } - \int _ { x } ^ { 1 } \frac { 1 } { 2 \sqrt { t ^ { 4 } + x } } d t$

Question 20

$\text { Find } \mathrm { f } _ { \mathbf { x } } , \mathrm { f } _ { \mathbf { y } } \text {, and } \mathrm { f } _ { \mathbf { Z } }$
-$f ( x , y , z ) = x ^ { 2 } y + y ^ { 2 } z + x z ^ { 2 }$

Accepted Answer

A)  $f _ { x } = 2 x y + z ^ { 2 } ; f _ { y } = x ^ { 2 } + 2 y z ; f _ { z } = y ^ { 2 } + 2 x z$ 
B)  $f _ { X } = 2 y + z ^ { 2 } ; f _ { y } = x ^ { 2 } + 2 z ; f _ { z } = y ^ { 2 } + 2 x$ 
C)  $f _ { x } = 2 x y ; f _ { y } = x ^ { 2 } + 2 y z ; f _ { z } = y ^ { 2 } + 2 x z$ 
D)  $f _ { x } = 2 x y + z ^ { 2 } ; f _ { y } = x ^ { 2 } + y z ; f _ { z } = y ^ { 2 } + x z$ 
A)  $f _ { x } = 2 x y + z ^ { 2 } ; f _ { y } = x ^ { 2 } + 2 y z ; f _ { z } = y ^ { 2 } + 2 x z$ 
B)  $f _ { X } = 2 y + z ^ { 2 } ; f _ { y } = x ^ { 2 } + 2 z ; f _ { z } = y ^ { 2 } + 2 x$ 
C)  $f _ { x } = 2 x y ; f _ { y } = x ^ { 2 } + 2 y z ; f _ { z } = y ^ { 2 } + 2 x z$ 
D)  $f _ { x } = 2 x y + z ^ { 2 } ; f _ { y } = x ^ { 2 } + y z ; f _ { z } = y ^ { 2 } + x z$

Show that the function is a solution of the wave equation. -w(x, t) = sin (-10x - 10ct)

Find the linearization of the function at the given point. - $f ( x , y ) = 8 x ^ { 2 } + 2 y ^ { 2 } + 8 \text { at } ( - 8 , - 3 )$

Estimate the error in the quadratic approximation of the given function at the origin over the given region. - $f ( x , y ) = e ^ { 4 x } \cos 2 y , \quad | x | \leq 0.1 , | y | \leq 0.1$

Solve the problem. -If the length, width, and height of a rectangular solid are measured to be 5, 2, and 10 inches respectively and each measurement is accurate to within 0.1 inch, estimate the maximum percentage error in computing the Volume of the solid.

Find the domain and range and describe the level curves for the function f(x,y). - $f ( x , y ) = \sqrt { 25 - x ^ { 2 } - y ^ { 2 } }$

Solve the problem. -Find the point on the plane x + 2y - z = 12 that is nearest the origin.

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0). - $f ( x , y ) = \frac { x ^ { 2 } } { x ^ { 4 } + y ^ { 2 } }$

Find the extreme values of the function subject to the given constraint. - $f ( x , y ) = y ^ { 2 } - x ^ { 2 } , \quad x ^ { 2 } + y ^ { 2 } = 16$

Find the equation for the level surface of the function through the given point. - $f ( x , y , z ) = e ^ { \left( x ^ { 2 } + y ^ { 2 } - z \right) } , ( 3,5,9 )$

Solve the problem. -Find an equation for the level curve of the function $\mathrm { f } ( \mathrm { x } , \mathrm { y } ) = 16 - \mathrm { x } ^ { 2 } - \mathrm { y } ^ { 2 }$ that passes through the point $( \sqrt { 2 } , \sqrt { 5 } )$ .

Find the limit. -

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0). - $f ( x , y ) = \frac { x ^ { 5 } - y } { x ^ { 5 } + y }$

Solve the problem. -Find an equation for the level surface of the function $f ( x , y , z ) = \int _ { z } ^ { y } e ^ { \theta } d \theta - \int _ { x } ^ { \sqrt { 2 } } t d t$ that passes through the point $( 0 , \ln 2 , \ln 4 )$ .

Solve the problem. -Write parametric equations for the tangent line to the curve of intersection of the surfaces $x + y ^ { 2 } + 7 z = 9$ and $x = 1$ at the point $( 1,1,1 )$ .

Find all the local maxima, local minima, and saddle points of the function. - $f ( x , y ) = \left( x ^ { 2 } - 9 \right) ^ { 2 } + \left( y ^ { 2 } - 16 \right) ^ { 2 }$

Find all the local maxima, local minima, and saddle points of the function. - $f ( x , y ) = \left( x ^ { 2 } - 81 \right) ^ { 2 } - \left( y ^ { 2 } - 64 \right) ^ { 2 }$

Provide an appropriate answer. -Suppose that $w = x ^ { 2 } + y ^ { 2 } + 20 z + t$ and $x + 4 z + t = 8$ . Assuming that the independent variables are $x$ , $y$ , and $z$ , find $\frac { \partial w } { \partial x }$ .

Provide an appropriate response. -Find $F ^ { \prime } ( x )$ if $F ( x ) = \int _ { x } ^ { 1 } \sqrt { t ^ { 4 } + x } d t$ .

$\text { Find } \mathrm { f } _ { \mathbf { x } } , \mathrm { f } _ { \mathbf { y } } \text {, and } \mathrm { f } _ { \mathbf { Z } }$ - $f ( x , y , z ) = x ^ { 2 } y + y ^ { 2 } z + x z ^ { 2 }$

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Exam 15: Partial Derivatives