# Functions of several variables

1. (1 pt)

Find the limit, if it exists, or type N if it does not exist.

lim 5×2

(x,y)->(0,0) 1X2 + 33,2

Answer(s) submitted:

(incorrect)

2. (1 pt) In this problem we show that the function

4%2 – y2

x‘ i

f( a) x2+yZ

does not have a limit as (x,y) -> (0,0).

(a) Suppose that we consider (x, y) -> (0,0) along the curve

y = 2x. Find the limit in this case:

lim -4x2_y2

(x,2r)->(0,0) x2+¥2

(b) Now consider (x, y) -> (0, 0) along the curve y = 3x. Find

the limit in this case:

lim -4x2_y2

(x,3x)->(0,0) xzfl‘z

(c) Note that the results from (a) and (b) indicate that f has

no limit as (x, y) -> (0,0) (be sure you can explain why!)

To show this more generally, consider (x, y) -> (0.0) along the

curve y = mx, for arbitrary m. Find the limit in this case:

lim 7-74-94 2

(x,mx)->(0,0) x +3,

(Be sure that you can explain how this result also indicates that

f has no limit as (x, y) -> (0,0).

Answer(s) submitted:

(incorrect)

3. (1 pt) For the function f (x,y) below, determine whether

there is a value for 0 making the function continuous every-

where. If so, find it.

4. (1 pt)

The largest set on which the function f (x, y) = l / (11 -x2

yz) is continuous is

A. The interior of the circle x2 + y2 = 11

B. The interior of the circle x2 + yz = 11, plus the circle

C. All of the xy-plane

D. All of the xy-plane except the circle x2 + yz : 11

E. The exterior of the circle x2 + yz : ll

Answer(s) submitted:

(incorrect)

5. (1 pt)

Use thel level curves of the function z = f (x, y) to

determine if each partial derivative at the point P is pos-

itive, negative, or zero.

1- fxx (P)

I2 fix (P)

I3- f (P)

I f (P)

5I f (P)

(x,mx) -> (0,0) x“ +3″

(Be sure that you can explain how this result also indicates that

f has no limit as (x, y) -> (0,0).

Answer(s) submitted:

(incorrect)

3. (1 pt) For the function f (x,y) below, determine whether

there is a value for 6 making the function continuous every-

where. If so, find it.

f( ) C +31 x S 2?.

x’y

5-y x>2

(If there is no value of c that works, enter none, and be sure that

you can explain why there is no such. value.)

Answer(s) submitted:

(incorrect)

6. (2 pts) The temperature adjusted for wind chill is a temper-

ature that tells you how cold it feels, as a result of the combina-

tion of wind speed, w, in miles per hour, and actual temperature,

T, as measured by a thermometer in degrees Fahrenheit. The ta-

ble below shows the temperature adjusted for wind chill, f(w,Ti),

as a function of w and T. The left column of the table gives the

wind speed in miles per hour, the top row gives the actual tem-

perature in degrees Fahrenheit, and the remaining entries give

the temperature adjusted for wind chill in degrees Fahrenheit as

a function of wind speed and actual temperature.

-n

ma

mun

-“

-n

(a) Estimate f“. (15.0) and give a practical interpretation.

f“. (15,0) x

If the wind is blowing _ mph and the actual temperature

is _ degrees Fahrenheit, it will feel _ degrees Fahren-

heit colder for each 1 mph increase in wind speed.

(b) Estimate f7 (10, 10) and give a practical interpretation.

fT(10,10)

If the wind is blowing _ mph and the actual temperature

is _ degrees Fahrenheit, it will feel _ degrees Fahren-

heit warmer for each 1 degree Fahrenheit increase in actual

temperature.

A nswer(s ) submitted:

8. (2 pts)

Compute the partial derivatives:

Z : 8x

3x

3y

Answer(s) submitted:

(incorrect)

9. (1 pt)

Compute the partial derivatives:

z = In (x6 +y4)

3x

a)»

Answer(s) submitted:

(incorrect)

10. (2 pts) Your monthly car payment in dollars is P

(POJ, r), where $Po is the amount you borrowed, t is the num-

ter of months it takes to pay off the loan, and r percent is the

aterest rate.

(a) Is aP/at positive or negative?

Luppose that your bank tells you that the magnitude of 8P/8t is

0.

Vhat are the units of this value?

For this problem, write our your units in full, writing dollars

or $, months for months, percent for %, etc. Note that frac-

ional units generally have a plural numerator and singular de-

ominator.

(b) Is dP/ar positive or negative?

Luppose that your bank tells you that the magnitude of dP/dr is

0.

Vhat are the units of this value?

For this problem, write our your units in full, writing dollars

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**Category**: Essay