Deck 14: Differentiating Functions of Several Variables

Question 1

Suppose .
Use a difference quotient to estimate and with h = 0.01.Give your answers to 3 decimal places.

Answer 1

and .

Question 2

Find $f _ { H }$ if $f ( H , T ) = \frac { 5 H + T } { ( 3 - H ) ^ { 3 } }$ .

A) $\frac { 15 + 10 H + 3 T } { ( 3 - H ) ^ { 6 } }$
B) $\frac { 15 - 10 H + 3 T } { ( 3 - H ) ^ { 4 } }$
C) $\frac { 15 + 10 H + 3 T } { ( 3 - H ) ^ { 4 } }$
D) $\frac { 15 - 20 H - 3 T } { ( 3 - H ) ^ { 4 } }$
E) $\frac { 5 + 10 H + 3 T } { ( 3 - H ) ^ { 4 } }$

Answer 2

$\frac { 15 + 10 H + 3 T } { ( 3 - H ) ^ { 4 } }$

Question 3

If $\frac { \partial f } { \partial x } = \frac { \partial f } { \partial y }$ everywhere, then f(x, y)is a constant.

Answer 3

False

Question 4

Find $\frac { \partial } { \partial x } \left( \ln \left( x ^ { 5 } y + 5 \right) \right)$ .

A) $\frac { 5 x ^ { 4 } y } { x ^ { 5 } y + 5 }$
B) $\frac { 5 x ^ { 4 } } { x ^ { 5 } y + 5 }$
C) $\frac { 5 x ^ { 4 } y } { x ^ { 5 } + 5 }$
D) $\frac { x ^ { 4 } y } { x ^ { 5 } y + 5 }$

Question 5

There exists a function f(x, y)with f_x = 2y and f_y = 3x.

Question 6

The ideal gas law states that for a fixed amount of gas, called a mole of gas, where P is the pressure (in atmospheres), V is the volume (in cubic meters), T is the temperature (in degrees Kelvin)and R is a positive constant.
Find

Question 7

Estimate the value of from the given contour diagram of f.

Question 8

The monthly mortgage payment in dollars, P, for a house is a function of three variables P = f(A, r, N), where A is the amount borrowed in dollars, r is the interest rate, and N is the number of years before the mortgage is paid off.It is given that: Estimate the value of

Question 9

The figure below shows the graph of z = f(x, y)and its intersection with various planes.(The x and y-axes have the same scale.) What is the sign of $f _ { y } ( 0,1 )$ ?

A)Negative
B)Positive

Question 10

Find to 2 decimal places if .

Question 11

The level curves of a function z = f(x, y)are shown below.Assume that the scales along the x and y axes are the same. $$f _ { y } ( \mathrm { Q } ) > f _ { y } ( \mathrm { P } )$$

Question 12

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 (a)What are the units of ?
(b)What is the practical interpretation (in terms of money)of ?

Question 13

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

Question 14

Given the contour diagram shown below, state whether $$f _ { x } ( 0.5,1 )$$ is positive, negative or nearly zero.

A)Almost zero
B)Negative
C)Positive
D)Undefined

Question 15

The ideal gas law states that for a fixed amount of gas, called a mole of gas, where P is the pressure (in atmospheres), V is the volume (in cubic meters), T is the temperature (in degrees Kelvin)and R is a positive constant.
A mole of a certain gas is at a temperature of 290° K, a pressure of 1 atmosphere, and a volume of 0.04 m³.
What is for this gas?

Question 16

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 Find

Question 17

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?

A)Negative
B)Positive

Question 18

The monthly mortgage payment in dollars, P, for a house is a function of three variables P = f(A, r, N), where A is the amount borrowed in dollars, r is the interest rate, and N is the number of years before the mortgage is paid off.It is given that: $$\begin{array} { l l l } f ( 100000,7,20 ) = 775.29 , & f ( 100000,8,20 ) = 836.44 , & f ( 100000,7,25 ) = 706.77 \\f ( 120000,7,20 ) = 930.35 , & f ( 120000,8,20 ) = 1003.72 , & f ( 120000,7,25 ) = 848.13\end{array}$$ Estimate the value of $$\left. \frac { \partial f } { \partial A } \right| _ { ( 10000,7,20 ) }$$ and interpret your answer in terms of a mortgage payment.Select all answers that apply.

A)We are currently borrowing $100,000 at 7% interest rate on a 20-year mortgage.
B)The monthly payment will go up by approximately $0.007753 for each extra percentage point charged.
C)The monthly payment will go up by approximately $0.007753 for each extra dollar we borrow.
D)The monthly payment will go up by approximately $0.007753 for each extra year of the mortgage.
E)The monthly payment will go down by approximately $0.007753 for each extra dollar we borrow.

Question 19

Given the contour diagram shown below
(a)Sketch a graph of f(1, y).
(b)Sketch a graph of f(x, 0).

Question 20

Suppose that the price P (in dollars)to purchase a used car is a function of C, its original cost (in dollars), and its age A (in years).So P = f(C,A). What is the sign of $\frac { \partial P } { \partial C } ?$

A)Positive
B)Negative

Question 21

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).

Question 22

Determine the tangent plane to at (x, y)= (2, 1).

Question 23

The equations of the tangent planes to the graph z = f(x, y)at the points (0, -2), (2, 1)are and , respectively.Determine the value of State whether the value you find is exact or an approximation.

Question 24

The volume of a right circular cylinder is to be calculated from measured values of r and h.Suppose r is measured with an error of no more than 2.5% and h with an error of no more than 1%.Using differentials, estimate the percentage error in the calculation of V.
(In general, in measuring a quantity Q, the percentage error is dQ/Q.)

Question 25

Let .Find to 3 decimal places.

Question 26

Let Use the appropriate partial derivative to find the slope of the cross-section at the given point.
(a)The cross-section f(x, 2)at the point (3, 2).
(b)The cross-section f(1, y)at the point (1, -2).

Question 27

If $\vec { v }$ is a unit vector and the level curves of f(x, y)are given below, then at point P we have $f _ { \vec { v } } ( P ) = \| \operatorname { grad } f \| \cos \theta$

Question 28

Estimate numerically if

Question 29

For the function $f ( x , y ) = x ^ { 2 } y ^ { 3 }$ find a unit vector in the direction of the steepest increase at the point (a, b)= (1, 1).

A) $\vec { u } = \frac { 2 } { \sqrt { 13 } } \vec { i } - \frac { 3 } { \sqrt { 13 } } \vec { j }$
B) $\vec { u } = 2 \vec { i } + 3 \vec { j }$
C) $\vec { u } = \frac { 2 } { \sqrt { 13 } } \vec { i } + \frac { 3 } { \sqrt { 13 } } \vec { j }$
D) $\vec { u } = \frac { 2 } { \sqrt { 13 } } \vec { i } + \frac { 6 } { \sqrt { 13 } } \vec { j }$

Question 30

Let Find and to four decimal places.

Question 31

Find an equation for the tangent plane to the graph of at

Question 32

The depth of a pond at the point with coordinates (x, y)is given by .(Assume that x, y, and h are measured in feet.)If a boat at the point (-3, -5)is sailing in the direction of the vector ,
then at what rate is the depth changing?

Question 33

Find the gradient of the function at the point and use the result to obtain a linear approximation for

Question 34

Find $\partial$ z/ $\partial$ x if $z = - 3 \ln x + \sin \left( x y ^ { 5 } \right)$

Question 35

Find the differential of the function at the point (3, 4).
A point is measured to be 3 units from the y-axis with an error of ±0.01 and 4 units from the x-axis with an error of ±0.02.Approximate the error in computing its distance from the origin.

Question 36

The table of some values of f(x, y)is given below.Find a local linearization of f at (1 , 2).

Question 37

If $\vec { u }$ is a unit vector and the level curves of f(x, y)are given below, then at point P we have $f_{u}(P)=\operatorname{grad} f .$

Question 38

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.

Question 39

Calculate the following derivative: .

Question 40

Suppose that $f ( x , y ) = x ^ { 2 } e ^ { y }$ Find an equation for the tangent plane to f at the point (3, 0).

A) $z = - 9 + 6 x + 27 y$
B) $z = 9 + 6 x + 27 y$
C) $z = - 9 + 6 x - 27 y$
D) $z = - 9 - 6 x + 27 y$

Question 41

Let $f ( x , y ) = x ^ { 2 } + 3 y ^ { 2 } - 3 x$ Find the gradient vector of f at the point (-1, 2).

A) $\nabla f = - 5 \vec { i } - 12 \vec { j }$
B) $\nabla f = - 5 \vec { i } + 24 \vec { j }$
C) $\nabla f = 5 \vec { i } + 12 \vec { j }$
D) $\nabla f = - 5 \vec { i } + 12 \vec { j }$

Question 42

Find the equation of the tangent plane to $x ^ { 2 } + 2 x y + 4 y + 6 = z ^ { 2 }$ at the point (-4, 1, 3).

A) $6 x + 4 y + 6 z = - 4$
B) $- 6 x + 4 y + 6 z = - 2$
C) $6 x - 4 y + 6 z = - 2$
D) $6 x + 4 y + 6 z = 2$
E) $6 x + 4 y + 6 z = - 2$

Question 43

An ant is walking along the surface which is the graph of the function (a)When the ant is at the point (1, 0, 1), what direction should it move in order to be moving on the surface in the direction of greatest ascent?
(b)If the ant moves in this direction at a speed of 6 units per second, what is the rate of change of height of the ant?

Question 44

Given that f(2, 4)= 1.5 and f(2.1, 4.4)= 2.1, estimate the value of , where is the unit vector in the direction of Give your answer to four decimal places.

Question 45

Suppose || $\nabla$ f(a, b, c)||=19.Is it possible to choose a direction from (a, b, c)so that $f _ { \vec { u } }$ in that direction is -19?

Question 46

Let What is the direction of maximum rate of change of f at (1, 1)?

Question 47

Suppose that as you move away from the point (2, 0, 2), the function increases most rapidly in the direction and the rate of increase of f in this direction is 7.At what rate is f increasing as you move away from (2, 0, 2)in the direction of ? Give your answer to 4 decimal places.

Question 48

If $\text { grad } f = \operatorname { grad } g$ , then $f = g$ .

Question 49

Consider the function (a)Describe the level set g = 16.
(b)Find a vector perpendicular to the tangent plane to the level set g = 16 at the point (-1, 2, 2).

Question 50

Let What is the maximum rate of change of f at (2, 1)?

Question 51

Suppose f(x, y)is a function of x and y and define Find given that and

Question 52

Find the equation of the tangent plane to the surface $3 x y z + x ^ { 3 } + y ^ { 3 } + z ^ { 3 } = 15$ at (1, -3, 2).

A) $- 15 x + 33 y + 3 z = - 108$
B) $- 15 x + 33 y + 3 z = 108$
C) $15 x + 33 y + 3 z = - 108$
D) $- 15 x - 33 y + 3 z = - 108$

Question 53

Suppose that as you move away from the point (-1, -1, -1), the function increases most rapidly in the direction and the rate of increase of f in this direction is 4.At what rate is f increasing as you move away from (-1, -1, -1)in the direction of ? Give your answer to 4 decimal places.

Question 54

Let w = 3x cos 4y.If and find at the point t = 3.Give your answer to 2 decimal places.

Question 55

The quantity z can be expressed as a function of x and y as follows: z = f(x, y).Now x and y are themselves functions of r and $\theta$ , as follows: $x = g ( r , \theta )$ and $y = h ( r , \theta )$ Suppose you know that g(1, $\pi$ /2)= -1, and h(1, $\pi$ /2)= 1.In addition, you are told that $$\begin{array} { l } \frac { \partial f } { \partial x } ( - 1,1 ) = 1 , \quad \frac { \partial f } { \partial y } ( - 1,1 ) = 6 , \quad \frac { \partial g } { \partial r } \left( 1 , \frac { \pi } { 2 } \right) = 7 \\\frac { \partial g } { \partial \theta } \left( 1 , \frac { \pi } { 2 } \right) = 7 , \quad \frac { \partial h } { \partial r } \left( 1 , \frac { \pi } { 2 } \right) = 6 , \quad \frac { \partial h } { \partial \theta } \left( 1 , \frac { \pi } { 2 } \right) = 4\end{array}$$ Find $\frac { \partial z } { \partial r } ( 1 , \pi / 2 )$

Question 56

Let w = 3x cos y.If $x = u ^ { 2 } + v ^ { 2 }$ $y=v / \mathcal u_{1}$ find $\partial$ w/ $\partial$ u and $\partial$ w/ $\partial$ v at the point $( u , v ) = ( 2,3 )$ .Give your answers to 2 decimal places.

Question 57

Find the directional derivative of at the point (3, 3, 2), in the direction of the vector

Question 58

Below is a contour diagram for f(x, y)which is defined and continuous everywhere.The z-values have been omitted.Explain why it is true that $f _ { i } ( 1,1 ) = f _ { - 3 } ( 1,1 )$ .

Question 59

Sally is on a day hike at Mt.Baker.From 9 to 11:00 a.m.she zig-zags up z = f(x, y)where x is the number of miles due east of her starting position, y is the number of miles due north of her starting position, and z is her elevation in miles above sea level.Feeling tired, she decides to continue walking, but in such a way that her altitude remains constant from 11 a.m.to noon to settle her stomach for lunch.At 11:30 a.m., she will be passing through (2, -1, 5)where f_x(2, -1)= 3 and f_y(2, -1)= -2.
What is the slope of her "path" in the x, y plane at this instant? (This "path" is among the level curves in the plane.)

Question 60

If you know the directional derivative of f(x, y)in two distinct directions (i.e.not including opposite directions)at a point P then you can find $\frac { \partial f } { \partial x } ( P )$

Question 61

Let Find dz/dt at t = 1 using the chain rule.Give your answer to 4 decimal places.

Question 62

Find the angle between the vector and the positive z-axis.

Question 63

Find the following partial derivative: f_xy if $f ( x , y ) = x ^ { 7 } y ^ { 8 }$ .

A) $f _ { x y } = 8 x ^ { 7 } y ^ { 7 }$
B) $f _ { x y } = 7 x ^ { 6 } y ^ { 8 }$
C) $f _ { x y } = 8 x ^ { 6 } y ^ { 7 }$
D) $f _ { x y } = 7 x ^ { 6 } y ^ { 7 }$
E) $f _ { x y } = 56 x ^ { 6 } y ^ { 7 }$

Question 64

Given that and Suppose that f(1, 1)= 4.Find the quadratic Taylor polynomial of f(x, y)at (1, 1).

Question 65

Using the contour diagram for f(x, y), find the sign of $f _ { y y } ( P )$ given that f_xx(P)< 0.

A)Negative
B)Positive
C)Not possible to decide

Question 66

If and find f_w(1, 1)using the chain rule.Give your answer to 4 decimal places.

Question 67

If x(u, v)= uv and y(u, v)= u + 4v.
If H(u, v)= f(x(u, v), y(u, v)), what is H(0,-1)? Give your answer to 4 decimal places.

Question 68

Suppose that f_x(2, 1)= 2.2, f_x(2.5, 1)= 1, f_x(2, 1.5)= 1.8, f_y(2, 1)= -0.8, f_y(2.5, 1)= -1.2 and f_y(2, 1.5)= -1.4.
If estimate the value of f(1.85, 0.8)using a quadratic Taylor polynomial about (2,1).
Use difference quotients to approximate all second derivatives.

Question 69

If x(u, v)= uv and y(u, v)= u + 3v.
If H(u, v)= f(x(u, v), y(u, v)), what is H_v(0,-2)? Give your answer to 4 decimal places.

Question 70

The table below gives values of a function f(x, y)near x = 1, y = 2. Give the equation of the tangent plane to the graph z = f(x, y)at x = 1, y = 2.

A) $z = 39.5 - 3.5 x - 3 y$
B) $z = 4 + 3.5 x - 3 y$
C) $z = 39.5 + 3.5 x - 3 y$
D) $z = 39.5 + x - y$
E) $z = 27.5 + 3.5 x + 3 y$

Question 71

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

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) $\frac { 1 } { \sqrt { 69 } } \vec { i } + \frac { 2 } { \sqrt { 69 } } \vec { j } + \frac { 8 } { \sqrt { 69 } } \vec { k }$

Question 72

Consider the function $g ( x , y ) = x ^ { \frac { 1 } { 5 } } y ^ { \frac { 1 } { 5 } }$ (a)Find g_x(x, y)and g_y(x, y)for (x, y) $\neq$ (0, 0).
(b)Use the limit definition of partial derivative to show that g_x(0, 0)= 0 and g_y(0, 0)= 0.
(c)Are the functions g_x and g_y continuous at (0, 0)? Explain.
(d)Is g differentiable at (0, 0)? Explain.

Question 73

Find the following partial derivative: H_P(2, 1)if Give your answer to 4 decimal places.

Question 74

The table below gives values of a function f(x, y)near x = 1, y = 2. Estimate .

Question 75

If f_x(0, 0)exists and f_y(0, 0)exists, then f is differentiable at (0, 0).

Question 76

Consider the level curves shown for the function z = f(x, y). Determine the sign of $f _ { y x } ( - 1 , - 5 )$

A)Positive
B)Negative

Question 77

Suppose that , with (a)What is the directional derivative of f at (1, -1)in the direction of ?
(b)What is the smallest value of the directional derivative of f at (1, -1)among all possible directions?

Question 78

Find the directional derivative of at the point (1, 1)in the direction of .

Question 79

Let f be a differentiable function with local linearization L(x, y)= -1 + 4(x - 4)- 2(y - 2)at (4, 2).Evaluate f(4, 2).

Question 80

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

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) $f ( x , y ) = 1 - \frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 4 }$