Functions of several variables

| January 23, 2016

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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