Question 1

Find all the second order partial derivatives of the given function.
-$$f ( x , y ) = \frac { x } { x + y }$$

Accepted Answer

A)  $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = - \frac { 2 \mathrm { y } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = \frac { 2 \mathrm { x } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \frac { \mathrm { y } - \mathrm { x } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } }$ 
B)  $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = - \frac { \mathrm { y } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = \frac { \mathrm { x } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \frac { \mathrm { x } - \mathrm { y } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } }$ 
C)  $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = \frac { 2 \mathrm { y } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = - \frac { 2 \mathrm { x } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \frac { \mathrm { x } - \mathrm { y } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } }$ 
D)  $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = - \frac { 2 \mathrm { y } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = \frac { 2 \mathrm { x } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \frac { \mathrm { x } - \mathrm { y } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } }$ 
A)  $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = - \frac { 2 \mathrm { y } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = \frac { 2 \mathrm { x } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \frac { \mathrm { y } - \mathrm { x } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } }$ 
B)  $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = - \frac { \mathrm { y } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = \frac { \mathrm { x } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \frac { \mathrm { x } - \mathrm { y } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } }$ 
C)  $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = \frac { 2 \mathrm { y } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = - \frac { 2 \mathrm { x } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \frac { \mathrm { x } - \mathrm { y } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } }$ 
D)  $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = - \frac { 2 \mathrm { y } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = \frac { 2 \mathrm { x } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \frac { \mathrm { x } - \mathrm { y } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } }$

Question 2

Use Taylor's formula to find the requested approximation of f(x, y) near the origin.
-Cubic approximation to f(x, y) = sin(6x + y)

Accepted Answer

A)  $6 x + y - 36 x ^ { 3 } - 18 x ^ { 2 } y - 3 x y ^ { 2 } - \frac { 1 } { 6 } y ^ { 3 }$ 
B)  $6 x + y - 36 x ^ { 3 } - \frac { 1 } { 6 } y ^ { 3 }$ 
C)  $6 x + y - 72 x ^ { 3 } - \frac { 1 } { 3 } y ^ { 3 }$ 
D)  $6 x + y - 72 x ^ { 3 } - 18 x ^ { 2 } y - 3 x y ^ { 2 } - \frac { 1 } { 3 } y ^ { 3 }$ 
A)  $6 x + y - 36 x ^ { 3 } - 18 x ^ { 2 } y - 3 x y ^ { 2 } - \frac { 1 } { 6 } y ^ { 3 }$ 
B)  $6 x + y - 36 x ^ { 3 } - \frac { 1 } { 6 } y ^ { 3 }$ 
C)  $6 x + y - 72 x ^ { 3 } - \frac { 1 } { 3 } y ^ { 3 }$ 
D)  $6 x + y - 72 x ^ { 3 } - 18 x ^ { 2 } y - 3 x y ^ { 2 } - \frac { 1 } { 3 } y ^ { 3 }$

Question 3

Solve the problem.
-$\text { Find the equation for the tangent plane to the surface } z = 7 x ^ { 2 } + 9 y ^ { 2 } \text { at the point } ( 2,1,37 ) \text {. }$

Accepted Answer

A)  28x + 18y - z = 44 
B)  28x + 18y - z = 37 
C)  2x + y + 37z = 1 
D)  2x + y + 37z = 40 
A)  28x + 18y - z = 44 
B)  28x + 18y - z = 37 
C)  2x + y + 37z = 1 
D)  2x + y + 37z = 40

Question 4

Find the limit.
-$$( x , y ) - \left( \frac { \pi } { 4 } , 1 \right) ^ { \frac { y \tan x } { y + 1 } }$$

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 4 \tan 1 } { 5 \pi }$ 
C)  2 
D)  0 
A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 4 \tan 1 } { 5 \pi }$ 
C)  2 
D)  0

Question 5

Find the absolute maximum and minimum values of the function on the given curve.
-$$\text { Function: } f ( x , y ) = x ^ { 2 } + y ^ { 2 } ; \text { curve: } x = 5 t + 1 , y = 5 t - 1,0 \leq t \leq 1 \text {. }$$

Accepted Answer

A)  Absolute maximum: 52 at t = 1; absolute minimum: 2 at t = 0. 
B)  Absolute maximum: 51 at t = 1; absolute minimum: 6 at t = 0. 
C)  Absolute maximum: 51 at t = 1; absolute minimum: 2 at t = 0. 
D)  Absolute maximum: 52 at t = 1; absolute minimum: 6 at t = 0. 
A)  Absolute maximum: 52 at t = 1; absolute minimum: 2 at t = 0. 
B)  Absolute maximum: 51 at t = 1; absolute minimum: 6 at t = 0. 
C)  Absolute maximum: 51 at t = 1; absolute minimum: 2 at t = 0. 
D)  Absolute maximum: 52 at t = 1; absolute minimum: 6 at t = 0.

Question 6

Find the requested partial derivative.
-$\frac { \partial w } { \partial x }$ if $w = x ^ { 3 } + y ^ { 3 } + z ^ { 3 } + 15 x y z$ and $\left( \frac { \partial y } { \partial x } \right) ^ { 2 } + \left( \frac { \partial z } { \partial x } \right) ^ { 2 } = 0$.

Accepted Answer

A)  $15 x ^ { 2 }$ 
B)  $3 x ^ { 2 }$ 
C)  $3 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right)$ 
D)  $15 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right)$ 
A)  $15 x ^ { 2 }$ 
B)  $3 x ^ { 2 }$ 
C)  $3 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right)$ 
D)  $15 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right)$

Question 7

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

Accepted Answer

A)  $\frac { \partial \mathrm { u } } { \partial \mathrm { r } } = \frac { \mathrm { du } } { \mathrm { dx } } \frac { \partial \mathrm { x } } { \partial \mathrm { r } }$ 
B)  $\frac { \partial u } { \partial r } = \frac { d u } { d x }$ 
C)  $\frac { \partial \mathrm { u } } { \partial \mathrm { r } } = \frac { \partial \mathrm { x } } { \partial \mathrm { r } }$ 
D)  $\frac { \partial u } { \partial r } = \frac { d u } { d x } \frac { \partial x } { \partial p } + \frac { d u } { d x } \frac { \partial x } { \partial q } + \frac { d u } { d x } \frac { \partial x } { \partial r }$ 
A)  $\frac { \partial \mathrm { u } } { \partial \mathrm { r } } = \frac { \mathrm { du } } { \mathrm { dx } } \frac { \partial \mathrm { x } } { \partial \mathrm { r } }$ 
B)  $\frac { \partial u } { \partial r } = \frac { d u } { d x }$ 
C)  $\frac { \partial \mathrm { u } } { \partial \mathrm { r } } = \frac { \partial \mathrm { x } } { \partial \mathrm { r } }$ 
D)  $\frac { \partial u } { \partial r } = \frac { d u } { d x } \frac { \partial x } { \partial p } + \frac { d u } { d x } \frac { \partial x } { \partial q } + \frac { d u } { d x } \frac { \partial x } { \partial r }$

Question 8

At what points is the given function continuous?
-$$f ( x , y , z ) = \frac { e ^ { x } } { e ^ { y + z } }$$

Accepted Answer

A)  All $( x , y , z )$ in the first octant 
B)  All $( x , y , z )$ 
C)  All $( x , y , z )$ such that $x \geq 0$ and $y + z \geq 0$ 
D)  All $( x , y , z )$ such that $x - y - z \geq 0$ 
A)  All $( x , y , z )$ in the first octant 
B)  All $( x , y , z )$ 
C)  All $( x , y , z )$ such that $x \geq 0$ and $y + z \geq 0$ 
D)  All $( x , y , z )$ such that $x - y - z \geq 0$

Question 9

Use the limit definition of the partial derivative to compute the indicated partial derivative of the function at the specified point.
-$\text { Find } \frac { \partial f } { \partial x } \text { at the point } ( 10 , - 5 ) : f ( x , y ) = 5 x ^ { 2 } + 4 x y + 3 y ^ { 2 }$

Accepted Answer

None

Question 10

Find the limit.
-$$\lim _ { ( x , y ) \rightarrow ( 1,1 ) } \ln \left| \frac { x + y } { x y } \right|$$

Accepted Answer

A)  -ln 2 
B)  ln 2 
C)  0 
D)  No limit 
A)  -ln 2 
B)  ln 2 
C)  0 
D)  No limit

Question 11

Compute the gradient of the function at the given point.
-$$f ( x , y , z ) = \ln \left( x ^ { 2 } + 5 y ^ { 2 } - 7 z ^ { 2 } \right) , \quad ( 5,5,5 )$$

Accepted Answer

A) ${\nabla} f = - i - 2 j + \frac { 10 } { 7 } \mathbf { k }$ 
B) ${\nabla} f  = - \frac { 2 } { 5 } \mathbf { i } - 2 \mathbf { j } + \frac { 14 } { 5 } \mathbf { k }$ 
C) ${\nabla} f = - \frac { 2 } { 5 } \mathbf { i } - 2 \mathbf { j } + \frac { 10 } { 7 } \mathbf { k }$ 
D) ${\nabla} f  = - \mathrm { i } - 2 \mathrm { j } + \frac { 14 } { 5 } \mathbf { k }$ 
A) ${\nabla} f = - i - 2 j + \frac { 10 } { 7 } \mathbf { k }$ 
B) ${\nabla} f  = - \frac { 2 } { 5 } \mathbf { i } - 2 \mathbf { j } + \frac { 14 } { 5 } \mathbf { k }$ 
C) ${\nabla} f = - \frac { 2 } { 5 } \mathbf { i } - 2 \mathbf { j } + \frac { 10 } { 7 } \mathbf { k }$ 
D) ${\nabla} f  = - \mathrm { i } - 2 \mathrm { j } + \frac { 14 } { 5 } \mathbf { k }$

Question 12

Use the limit definition of the partial derivative to compute the indicated partial derivative of the function at the specified point.
-$\text { Find } \frac { \partial f } { \partial y } \text { at the point } ( - 10,9 ) \text { : } f ( x , y ) = 5 x ^ { 2 } + 7 x y + 7 y ^ { 2 }$

Accepted Answer

None

Question 13

Provide an appropriate response.
-Which order of differentiation will calculate $f _ { x y }$ faster, $x$ first or $y$ first?
$$f ( x , y ) = y \ln ( x ) - 3 \sin ( x )$$

Accepted Answer

A)  x first 
B)  y first 
A)  x first 
B)  y first

Question 14

Find all the second order partial derivatives of the given function.
-$$\mathrm { f } ( \mathrm { x } , \mathrm { y } ) = x \mathrm { y } ^ { 2 } + y \mathrm { e } ^ { \mathrm { x } ^ { 2 } + 5 }$$

Accepted Answer

A)  $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = 2 y \mathrm { e } ^ { \mathrm { x } ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = 2 \mathrm { x } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = 2 \mathrm { y } + 2 x e ^ { \mathrm { x } ^ { 2 } }$ 
B)  $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = 2 y \mathrm { e } ^ { x ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial y ^ { 2 } } = 2 \mathrm { x } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = 2 x \mathrm { e } ^ { \mathrm { x } ^ { 2 } }$ 
C)  $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = y \mathrm { e } ^ { \mathrm { x } ^ { 2 } } \left( 1 + 2 \mathrm { x } ^ { 2 } \right) ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = \mathrm { x } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \mathrm { y } + x \mathrm { x } ^ { 2 }$ 
D)  $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = 2 \mathrm { ye } ^ { \mathrm { x } ^ { 2 } } \left( 1 + 2 \mathrm { x } ^ { 2 } \right) ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = 2 \mathrm { x } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = 2 \mathrm { y } + 2 \mathrm { xe } ^ { \mathrm { x } ^ { 2 } }$ 
A)  $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = 2 y \mathrm { e } ^ { \mathrm { x } ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = 2 \mathrm { x } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = 2 \mathrm { y } + 2 x e ^ { \mathrm { x } ^ { 2 } }$ 
B)  $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = 2 y \mathrm { e } ^ { x ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial y ^ { 2 } } = 2 \mathrm { x } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = 2 x \mathrm { e } ^ { \mathrm { x } ^ { 2 } }$ 
C)  $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = y \mathrm { e } ^ { \mathrm { x } ^ { 2 } } \left( 1 + 2 \mathrm { x } ^ { 2 } \right) ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = \mathrm { x } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \mathrm { y } + x \mathrm { x } ^ { 2 }$ 
D)  $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = 2 \mathrm { ye } ^ { \mathrm { x } ^ { 2 } } \left( 1 + 2 \mathrm { x } ^ { 2 } \right) ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = 2 \mathrm { x } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = 2 \mathrm { y } + 2 \mathrm { xe } ^ { \mathrm { x } ^ { 2 } }$

Question 15

Find an upper bound for the magnitude |E| of the error in the approximation f(x, y) ≈ L(x, y) at the given point over the
given region R.
-$f ( x , y ) = \ln ( 8 x + 7 y )$ at $( 1,1 ) ; R : | x - 1 | \leq 0.1 , | y - 1 | \leq 0.1$

Accepted Answer

A)  $| \mathrm { E } | \leq 0.007$ 
B)  $| \mathrm { E } | \leq 0.0057$ 
C)  $| E | \leq 0.0089$ 
D)  $| \mathrm { E } | \leq 0.0047$ 
A)  $| \mathrm { E } | \leq 0.007$ 
B)  $| \mathrm { E } | \leq 0.0057$ 
C)  $| E | \leq 0.0089$ 
D)  $| \mathrm { E } | \leq 0.0047$

Question 16

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0).
-$\text { Define } f ( 0,0 ) \text { in a way that extends } f ( x , y ) = \frac { 6 x ^ { 2 } - x ^ { 2 } y + 6 y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } } \text { to be continuous at the origin. }$

Accepted Answer

A)  f(0, 0) = 0 
B)  f(0, 0) = 6 
C)  f(0, 0) = 2 
D)  f(0, 0) = 12 
A)  f(0, 0) = 0 
B)  f(0, 0) = 6 
C)  f(0, 0) = 2 
D)  f(0, 0) = 12

Question 17

Solve the problem.
-Find the point on the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ that is closest to the point $( 3,1 , - 1 )$.

Accepted Answer

A)  $\left( \frac { 6 } { \sqrt { 11 } } , \frac { 2 } { \sqrt { 11 } } , \frac { 2 } { \sqrt { 11 } } \right)$ 
B)  $\left( \frac { 6 } { \sqrt { 11 } } , \frac { 2 } { \sqrt { 11 } } , - \frac { 2 } { \sqrt { 11 } } \right)$ 
C)  $\left( - \frac { 6 } { \sqrt { 11 } } , \frac { 2 } { \sqrt { 11 } } , \frac { 2 } { \sqrt { 11 } } \right)$ 
D)  $\left( - \frac { 6 } { \sqrt { 11 } } , \frac { 2 } { \sqrt { 11 } } , - \frac { 2 } { \sqrt { 11 } } \right)$ 
A)  $\left( \frac { 6 } { \sqrt { 11 } } , \frac { 2 } { \sqrt { 11 } } , \frac { 2 } { \sqrt { 11 } } \right)$ 
B)  $\left( \frac { 6 } { \sqrt { 11 } } , \frac { 2 } { \sqrt { 11 } } , - \frac { 2 } { \sqrt { 11 } } \right)$ 
C)  $\left( - \frac { 6 } { \sqrt { 11 } } , \frac { 2 } { \sqrt { 11 } } , \frac { 2 } { \sqrt { 11 } } \right)$ 
D)  $\left( - \frac { 6 } { \sqrt { 11 } } , \frac { 2 } { \sqrt { 11 } } , - \frac { 2 } { \sqrt { 11 } } \right)$

Question 18

Find all the local maxima, local minima, and saddle points of the function.
-$f ( x , y ) = 2 x y - 5 x + 3 y$

Accepted Answer

A)  $\left\{ \left( - \frac { 3 } { 2 } , \frac { 5 } { 2 } \right) = \frac { 15 } { 2 } \right.$, saddle point 
B)  $\left\{ - \frac { 3 } { 2 } , - \frac { 5 } { 2 } \right) = \frac { 15 } { 2 }$, saddle point; $\left\{ \left( \frac { 3 } { 2 } , \frac { 5 } { 2 } \right) = \frac { 15 } { 2 } \right.$, saddle point 
C)  $\left\{ - \frac { 3 } { 2 } , - \frac { 5 } { 2 } \right) = \frac { 15 } { 2 }$, local minimum; $\left( \frac { 3 } { 2 } , \frac { 5 } { 2 } \right) = \frac { 15 } { 2 }$, local 
D)  $\left\{ \left( \frac { 3 } { 2 } , - \frac { 5 } { 2 } \right) = - \frac { 45 } { 2 } \right.$, local maximum 
A)  $\left\{ \left( - \frac { 3 } { 2 } , \frac { 5 } { 2 } \right) = \frac { 15 } { 2 } \right.$, saddle point 
B)  $\left\{ - \frac { 3 } { 2 } , - \frac { 5 } { 2 } \right) = \frac { 15 } { 2 }$, saddle point; $\left\{ \left( \frac { 3 } { 2 } , \frac { 5 } { 2 } \right) = \frac { 15 } { 2 } \right.$, saddle point 
C)  $\left\{ - \frac { 3 } { 2 } , - \frac { 5 } { 2 } \right) = \frac { 15 } { 2 }$, local minimum; $\left( \frac { 3 } { 2 } , \frac { 5 } { 2 } \right) = \frac { 15 } { 2 }$, local 
D)  $\left\{ \left( \frac { 3 } { 2 } , - \frac { 5 } { 2 } \right) = - \frac { 45 } { 2 } \right.$, local maximum

Question 19

Solve the problem.
-Find the derivative of the function $f ( x , y , z ) = \frac { x } { y } + \frac { y } { z } + \frac { z } { x }$ at the point $( - 7,7 , - 7 )$ in the direction in which the function increases most rapidly.

Accepted Answer

A)  $\frac { 2 } { 7 } \sqrt { 2 }$ 
B)  $\frac { 2 } { 7 } \sqrt { 3 }$ 
C)  $\frac { 3 } { 7 } \sqrt { 2 }$ 
D)  $\frac { 3 } { 7 } \sqrt { 3 }$ 
A)  $\frac { 2 } { 7 } \sqrt { 2 }$ 
B)  $\frac { 2 } { 7 } \sqrt { 3 }$ 
C)  $\frac { 3 } { 7 } \sqrt { 2 }$ 
D)  $\frac { 3 } { 7 } \sqrt { 3 }$

Question 20

Solve the problem.
-$$\text { Find parametric equations for the normal line to the surface } \mathrm { z } = \mathrm { e } ^ { 8 \mathrm { x } ^ { 2 } + 10 \mathrm { y } ^ { 2 } } \text { at the point } ( 0,0,1 ) \text {. }$$

Accepted Answer

A)  x = t, y = t, z = t - 1 
B)  x = 0, y = 0, z = t + 1 
C)  x = t, y = t, z = -t - 1 
D)  x = 0, y = 0, z = t - 1 
A)  x = t, y = t, z = t - 1 
B)  x = 0, y = 0, z = t + 1 
C)  x = t, y = t, z = -t - 1 
D)  x = 0, y = 0, z = t - 1

Find all the second order partial derivatives of the given function. - $f ( x , y ) = \frac { x } { x + y }$

Use Taylor's formula to find the requested approximation of f(x, y) near the origin. -Cubic approximation to f(x, y) = sin(6x + y)

Solve the problem. - $\text { Find the equation for the tangent plane to the surface } z = 7 x ^ { 2 } + 9 y ^ { 2 } \text { at the point } ( 2,1,37 ) \text {. }$

Find the limit. - $( x , y ) - \left( \frac { \pi } { 4 } , 1 \right) ^ { \frac { y \tan x } { y + 1 } }$

Find the absolute maximum and minimum values of the function on the given curve. - $\text { Function: } f ( x , y ) = x ^ { 2 } + y ^ { 2 } ; \text { curve: } x = 5 t + 1 , y = 5 t - 1,0 \leq t \leq 1 \text {. }$

Find the requested partial derivative. - $\frac { \partial w } { \partial x }$ if $w = x ^ { 3 } + y ^ { 3 } + z ^ { 3 } + 15 x y z$ and $\left( \frac { \partial y } { \partial x } \right) ^ { 2 } + \left( \frac { \partial z } { \partial x } \right) ^ { 2 } = 0$ .

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

At what points is the given function continuous? - $f ( x , y , z ) = \frac { e ^ { x } } { e ^ { y + z } }$

Use the limit definition of the partial derivative to compute the indicated partial derivative of the function at the specified point. - $\text { Find } \frac { \partial f } { \partial x } \text { at the point } ( 10 , - 5 ) : f ( x , y ) = 5 x ^ { 2 } + 4 x y + 3 y ^ { 2 }$

Find the limit. - $\lim _ { ( x , y ) \rightarrow ( 1,1 ) } \ln \left| \frac { x + y } { x y } \right|$

Compute the gradient of the function at the given point. - $f ( x , y , z ) = \ln \left( x ^ { 2 } + 5 y ^ { 2 } - 7 z ^ { 2 } \right) , \quad ( 5,5,5 )$

Use the limit definition of the partial derivative to compute the indicated partial derivative of the function at the specified point. - $\text { Find } \frac { \partial f } { \partial y } \text { at the point } ( - 10,9 ) \text { : } f ( x , y ) = 5 x ^ { 2 } + 7 x y + 7 y ^ { 2 }$

Provide an appropriate response. -Which order of differentiation will calculate $f _ { x y }$ faster, $x$ first or $y$ first? $f ( x , y ) = y \ln ( x ) - 3 \sin ( x )$

Find all the second order partial derivatives of the given function. - $\mathrm { f } ( \mathrm { x } , \mathrm { y } ) = x \mathrm { y } ^ { 2 } + y \mathrm { e } ^ { \mathrm { x } ^ { 2 } + 5 }$

Find an upper bound for the magnitude |E| of the error in the approximation f(x, y) ≈ L(x, y) at the given point over the given region R. - $f ( x , y ) = \ln ( 8 x + 7 y )$ at $( 1,1 ) ; R : | x - 1 | \leq 0.1 , | y - 1 | \leq 0.1$

Solve the problem. -Find the point on the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ that is closest to the point $( 3,1 , - 1 )$ .

Find all the local maxima, local minima, and saddle points of the function. - $f ( x , y ) = 2 x y - 5 x + 3 y$

Solve the problem. -Find the derivative of the function $f ( x , y , z ) = \frac { x } { y } + \frac { y } { z } + \frac { z } { x }$ at the point $( - 7,7 , - 7 )$ in the direction in which the function increases most rapidly.

Solve the problem. - $\text { Find parametric equations for the normal line to the surface } \mathrm { z } = \mathrm { e } ^ { 8 \mathrm { x } ^ { 2 } + 10 \mathrm { y } ^ { 2 } } \text { at the point } ( 0,0,1 ) \text {. }$

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Exam 15: Partial Derivatives

Find all the second order partial derivatives of the given function. - f(x,y)=xx+yf ( x , y ) = \frac { x } { x + y }f(x,y)=x+yx​