Question 1

If a function z = g(x, y)has g(1, 2)= -5, g_x(1, 2)= 4 and g_y(1, 2)= 3, find the equation of the plane tangent to the surface z = g(x, y)at the point where x = 1 and y = 2.

Accepted Answer

The answer of If a function z = g(x, y)has...

Question 2

Let $f ( x , y ) = x ^ { 2 } + 5 y ^ { 2 } - 5 x$ What is the direction of maximum rate of change of f at (1, 1)?

Accepted Answer

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

Question 3

Find the quadratic approximation to the function f(x, y)= cos x cos y valid near the origin.

Accepted Answer

A)  $f ( x , y ) = 1 - \frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 2 }$ 
B)  $f ( x , y ) = 1 + \frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 2 }$ 
C)  $f ( x , y ) = 1 - \frac { x ^ { 2 } } { 2 } - \frac { y ^ { 2 } } { 2 }$ 
D)  $f ( x , y ) = 1 - x ^ { 2 } - y ^ { 2 }$ 
E)  $f ( x , y ) = 1 - \frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 4 }$

Question 4

What is the angle between the unit vector $\vec { u }$ and $\nabla f ( a , b )$ if $f _ { a } ( a , b ) = - 0.5 \| \nabla f ( a , b ) \|$ ?

Accepted Answer

A)  $5 \pi / 6$ 
B)  $2 \pi / 3$ 
C)  $\pi / 3$ 
D)  $\pi / 6$

Question 5

For the function g(x, y)with contour diagram below:  Evaluate g(-1, 1).

Accepted Answer

The answer of For the function g(x, y)with contour diagram...

Question 6

Find the directional derivative of $f ( x , y ) = x ^ { 3 } y + 5 x y ^ { 2 }$ at the point (1, 1)in the direction of $3 \vec { i } + 5 \vec { j }$ .

Accepted Answer

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

Question 7

If f and g have the same partial derivatives at every point in their domains, then $f = g$ .

Accepted Answer

A) True 
 B)False

Question 8

Find the angle between the vector $\vec { c } = 2 \vec { i } + 3 \vec { j } - \vec { k }$ and the positive z-axis.

Accepted Answer

The answer of Find the angle between the vector \(\vec...

Question 9

If $g ( x , y ) = x ^ { 2 } \cos ( x y )$ , then find $\frac { \partial g } { \partial y } \left( 3 , \frac { \pi } { 3 } \right)$ .

Accepted Answer

A) 0 
B)  $- 27$ 
C)  $27$ 
D)  $\pi - 27$

Question 10

Let $f ( x , y , z ) = x ^ { 3 } + 6 x z + 4 y z + z ^ { 2 }$ .Find $\nabla f ( 1 , - 2,6 )$ .

Accepted Answer

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

Question 11

Suppose that the maximum rate of change of f at $( 1 - 1 )$ is 20, and it occurs in the direction of $5 \vec { i } - 5 \vec { j }$ .Find $\nabla f ( 1 , - 1 )$ , $f _ { x } ( 1 , - 1 )$ , and $f _ { y } ( 1 , - 1 )$ .

Accepted Answer

The answer of Suppose that the maximum rate of change...

Question 12

Suppose that the maximum rate of change of f at $( 1 - 1 )$ is 10, and it occurs in the direction of $2 \vec { i } - 2 \vec { j }$ .What is the minimum rate of change of f at $( 1 - 1 )$ , and in which direction does it occur?

Accepted Answer

The answer of Suppose that the maximum rate of change...

Question 13

If $P is invested in a bank account earning r% interest a year, compounded continuously, the balance, $B, at the end of t years is given by $B = f ( P , r , t ) = P e ^ { r t / 100 }$ (a)What are the units of $\partial B / \partial t$ ?
(b)What is the practical interpretation (in terms of money)of $\partial B / \partial t$ ?

Accepted Answer

The answer of If $P is invested in a bank...

Question 14

Estimate $z _ { y } ( 3,2 )$ numerically if $z = ( x + y ) ^ { ( x + y ) } , x , y > 0$

Accepted Answer

The answer of Estimate \(z _ { y } (...

Question 15

The cross-sections of f when x is fixed at x = 1 and when y is fixed at y = 2 are given below.
Determine, if possible, the sign of $$f _ { y } ( 1,3 )$$

Accepted Answer

The answer of The cross-sections of f when x is...

Question 16

The consumption of beef, C (in pounds per week per household)is given by the function C = f(I, p), where I is the household income in thousands of dollars per year, and p is the price of beef in dollars per pound. Do you expect $\partial f / \partial p$ to be positive or negative?

Accepted Answer

A) Negative 
B) Positive

Question 17

Use the differential of $f ( x , y ) = x ^ { 2 } + 2 x \cos ^ { 2 } y$ to find a linear approximation of f at the point (1,  $\pi$/4).

Accepted Answer

The answer of Use the differential of \(f ( x...

Question 18

A gradient is a specific example of a directional derivative.

Accepted Answer

A) True 
 B)False

Question 19

A ball thrown from ground level with initial speed v and at an angle $\alpha$ with the horizontal hits the ground at a distance $s = \frac { v ^ { 2 } \sin ( 2 \alpha ) } { g }$ away from the starting point (where g is the acceleration due to gravity).Calculate the differential ds.

Accepted Answer

A)  $d s = \frac { 2 v \sin ( 2 \alpha ) } { g } d v + \frac { 2 v ^ { 2 } \cos ( 2 \alpha ) } { g } d \alpha$ 
B)  $d s = 2 v \sin ( 2 \alpha ) d v + 2 v ^ { 2 } \cos ( 2 \alpha ) d \alpha$ 
C)  $d s = \frac { 2 v \sin ( 2 \alpha ) } { g } d v + \frac { 2 v ^ { 2 } \sin ( 2 \alpha ) \cos ( 2 \alpha ) } { g } d \alpha$ 
D)  $d s = \frac { 2 v \sin ( 2 \alpha ) } { g } d v - \frac { 2 v ^ { 2 } \cos ( 2 \alpha ) } { g } d \alpha$

Question 20

Find a unit vector perpendicular to both $6 \vec { i } - \vec { j } + \vec { k }$ and $\overrightarrow { 8 i } + \vec { k }$ .

Accepted Answer

A)  $- \frac { 1 } { \sqrt { 69 } } \vec { i } - \frac { 2 } { \sqrt { 69 } } \vec { j } + \frac { 8 } { \sqrt { 69 } } \vec { k }$ 
B)  $- \frac { 1 } { \sqrt { 69 } } \vec { i } + \frac { 2 } { \sqrt { 69 } } \vec { j } + \frac { 8 } { \sqrt { 69 } } \vec { k }$ 
C)  $- \frac { 1 } { \sqrt { 69 } } \vec { i } + \frac { 2 } { \sqrt { 69 } } \vec { j } - \frac { 8 } { \sqrt { 69 } } \vec { k }$ 
D)  $- \vec { i } + 2 \vec { j } + 8 \vec { k }$ 
E)  $\frac { 1 } { \sqrt { 69 } } \vec { i } + \frac { 2 } { \sqrt { 69 } } \vec { j } + \frac { 8 } { \sqrt { 69 } } \vec { k }$

Exam 14: Differentiating Functions of Several Variables

If a function z = g(x, y)has g(1, 2)= -5, g_x(1, 2)= 4 and g_y(1, 2)= 3, find the equation of the plane tangent to the surface z = g(x, y)at the point where x = 1 and y = 2.

Let $f ( x , y ) = x ^ { 2 } + 5 y ^ { 2 } - 5 x$ What is the direction of maximum rate of change of f at (1, 1)?

Find the quadratic approximation to the function f(x, y) = cos x cos y valid near the origin.

What is the angle between the unit vector $\vec { u }$ and $\nabla f ( a , b )$ if $f _ { a } ( a , b ) = - 0.5 \| \nabla f ( a , b ) \|$ ?

For the function g(x, y)with contour diagram below: Evaluate g(-1, 1).

Find the directional derivative of $f ( x , y ) = x ^ { 3 } y + 5 x y ^ { 2 }$ at the point (1, 1)in the direction of $3 \vec { i } + 5 \vec { j }$ .

If f and g have the same partial derivatives at every point in their domains, then $f = g$ .

Find the angle between the vector $\vec { c } = 2 \vec { i } + 3 \vec { j } - \vec { k }$ and the positive z-axis.

If $g ( x , y ) = x ^ { 2 } \cos ( x y )$ , then find $\frac { \partial g } { \partial y } \left( 3 , \frac { \pi } { 3 } \right)$ .

Let $f ( x , y , z ) = x ^ { 3 } + 6 x z + 4 y z + z ^ { 2 }$ .Find $\nabla f ( 1 , - 2,6 )$ .

Suppose that the maximum rate of change of f at $( 1 - 1 )$ is 20, and it occurs in the direction of $5 \vec { i } - 5 \vec { j }$ .Find $\nabla f ( 1 , - 1 )$ , $f _ { x } ( 1 , - 1 )$ , and $f _ { y } ( 1 , - 1 )$ .

Suppose that the maximum rate of change of f at $( 1 - 1 )$ is 10, and it occurs in the direction of $2 \vec { i } - 2 \vec { j }$ .What is the minimum rate of change of f at $( 1 - 1 )$ , and in which direction does it occur?

Estimate $z _ { y } ( 3,2 )$ numerically if $z = ( x + y ) ^ { ( x + y ) } , x , y > 0$

The consumption of beef, C (in pounds per week per household) is given by the function C = f(I, p) , where I is the household income in thousands of dollars per year, and p is the price of beef in dollars per pound. Do you expect $\partial f / \partial p$ to be positive or negative?

Use the differential of $f ( x , y ) = x ^ { 2 } + 2 x \cos ^ { 2 } y$ to find a linear approximation of f at the point (1, $\pi$ /4).

A gradient is a specific example of a directional derivative.

A ball thrown from ground level with initial speed v and at an angle $\alpha$ with the horizontal hits the ground at a distance $s = \frac { v ^ { 2 } \sin ( 2 \alpha ) } { g }$ away from the starting point (where g is the acceleration due to gravity) .Calculate the differential ds.

Find a unit vector perpendicular to both $6 \vec { i } - \vec { j } + \vec { k }$ and $\overrightarrow { 8 i } + \vec { k }$ .

Exam 14: Differentiating Functions of Several Variables

If a function z = g(x, y)has g(1, 2)= -5, gx(1, 2)= 4 and gy(1, 2)= 3, find the equation of the plane tangent to the surface z = g(x, y)at the point where x = 1 and y = 2.

Let f(x,y)=x2+5y2−5xf ( x , y ) = x ^ { 2 } + 5 y ^ { 2 } - 5 xf(x,y)=x2+5y2−5x What is the direction of maximum rate of change of f at (1, 1)?

Find the quadratic approximation to the function f(x, y) = cos x cos y valid near the origin.

What is the angle between the unit vector u⃗\vec { u }u and ∇f(a,b) \nabla f ( a , b ) ∇f(a,b) if fa(a,b) =−0.5∥∇f(a,b) ∥f _ { a } ( a , b ) = - 0.5 \| \nabla f ( a , b ) \|fa​(a,b) =−0.5∥∇f(a,b) ∥ ?

For the function g(x, y)with contour diagram below: Evaluate g(-1, 1).

Find the directional derivative of f(x,y)=x3y+5xy2f ( x , y ) = x ^ { 3 } y + 5 x y ^ { 2 }f(x,y)=x3y+5xy2 at the point (1, 1)in the direction of 3i⃗+5j⃗3 \vec { i } + 5 \vec { j }3i+5j​ .

If f and g have the same partial derivatives at every point in their domains, then f=gf = gf=g .

Find the angle between the vector c⃗=2i⃗+3j⃗−k⃗\vec { c } = 2 \vec { i } + 3 \vec { j } - \vec { k }c=2i+3j​−k and the positive z-axis.

If g(x,y) =x2cos⁡(xy) g ( x , y ) = x ^ { 2 } \cos ( x y ) g(x,y) =x2cos(xy) , then find ∂g∂y(3,π3) \frac { \partial g } { \partial y } \left( 3 , \frac { \pi } { 3 } \right) ∂y∂g​(3,3π​) .

Let f(x,y,z)=x3+6xz+4yz+z2f ( x , y , z ) = x ^ { 3 } + 6 x z + 4 y z + z ^ { 2 }f(x,y,z)=x3+6xz+4yz+z2 .Find ∇f(1,−2,6)\nabla f ( 1 , - 2,6 )∇f(1,−2,6) .

Suppose that the maximum rate of change of f at (1−1)( 1 - 1 )(1−1) is 10, and it occurs in the direction of 2i⃗−2j⃗2 \vec { i } - 2 \vec { j }2i−2j​ .What is the minimum rate of change of f at (1−1)( 1 - 1 )(1−1) , and in which direction does it occur?

Estimate zy(3,2)z _ { y } ( 3,2 )zy​(3,2) numerically if z=(x+y)(x+y),x,y>0z = ( x + y ) ^ { ( x + y ) } , x , y > 0z=(x+y)(x+y),x,y>0

The cross-sections of f when x is fixed at x = 1 and when y is fixed at y = 2 are given below. Determine, if possible, the sign of fy(1,3)f _ { y } ( 1,3 )fy​(1,3)

The consumption of beef, C (in pounds per week per household) is given by the function C = f(I, p) , where I is the household income in thousands of dollars per year, and p is the price of beef in dollars per pound. Do you expect ∂f/∂p\partial f / \partial p∂f/∂p to be positive or negative?

Use the differential of f(x,y)=x2+2xcos⁡2yf ( x , y ) = x ^ { 2 } + 2 x \cos ^ { 2 } yf(x,y)=x2+2xcos2y to find a linear approximation of f at the point (1, π\piπ /4).

A gradient is a specific example of a directional derivative.

Find a unit vector perpendicular to both 6i⃗−j⃗+k⃗6 \vec { i } - \vec { j } + \vec { k }6i−j​+k and 8i→+k⃗\overrightarrow { 8 i } + \vec { k }8i+k .

If a function z = g(x, y)has g(1, 2)= -5, g_x(1, 2)= 4 and g_y(1, 2)= 3, find the equation of the plane tangent to the surface z = g(x, y)at the point where x = 1 and y = 2.

Let $f ( x , y ) = x ^ { 2 } + 5 y ^ { 2 } - 5 x$ What is the direction of maximum rate of change of f at (1, 1)?

What is the angle between the unit vector $\vec { u }$ and $\nabla f ( a , b )$ if $f _ { a } ( a , b ) = - 0.5 \| \nabla f ( a , b ) \|$ ?

Find the directional derivative of $f ( x , y ) = x ^ { 3 } y + 5 x y ^ { 2 }$ at the point (1, 1)in the direction of $3 \vec { i } + 5 \vec { j }$ .

If f and g have the same partial derivatives at every point in their domains, then $f = g$ .

Find the angle between the vector $\vec { c } = 2 \vec { i } + 3 \vec { j } - \vec { k }$ and the positive z-axis.

If $g ( x , y ) = x ^ { 2 } \cos ( x y )$ , then find $\frac { \partial g } { \partial y } \left( 3 , \frac { \pi } { 3 } \right)$ .

Let $f ( x , y , z ) = x ^ { 3 } + 6 x z + 4 y z + z ^ { 2 }$ .Find $\nabla f ( 1 , - 2,6 )$ .

Suppose that the maximum rate of change of f at $( 1 - 1 )$ is 10, and it occurs in the direction of $2 \vec { i } - 2 \vec { j }$ .What is the minimum rate of change of f at $( 1 - 1 )$ , and in which direction does it occur?

Estimate $z _ { y } ( 3,2 )$ numerically if $z = ( x + y ) ^ { ( x + y ) } , x , y > 0$

The consumption of beef, C (in pounds per week per household) is given by the function C = f(I, p) , where I is the household income in thousands of dollars per year, and p is the price of beef in dollars per pound. Do you expect $\partial f / \partial p$ to be positive or negative?

Use the differential of $f ( x , y ) = x ^ { 2 } + 2 x \cos ^ { 2 } y$ to find a linear approximation of f at the point (1, $\pi$ /4).

Find a unit vector perpendicular to both $6 \vec { i } - \vec { j } + \vec { k }$ and $\overrightarrow { 8 i } + \vec { k }$ .