Extra_Credit_2_MAC2234_withspace-2.pdf

MAC2234 Extra Credit #2

Answer the questions on this pdf.

Then use your answers to fill out the “Quiz” portion of the Extra Credit Assignment in Canvas.

1. For
22

2),( yxyxf  , find )3,2(f .

2. Graph the first-octant portion of the plane 424  zyx

3. For the function
222

523),( yxyxyyxf  , find the following:

a. )2,1(
xx

f

b. )2,1(yyf

c. )2,1(xyf

d. )2,1(yxf

4. A company’s monthly sales, in thousands of dollars, is given by
5.04.0

7),( yxyxS  , where x is the amount spent on

newspaper advertising per month (in thousands of dollars) and y is the amount spent on advertising per month (in

thousands of dollars). Suppose the company currently spends $4000 on newspaper advertising per month and $3000

on radio advertising per month. What would be the approximate effect on sales if the company increases the amount

spent on newspaper advertising to $5000, while the amount spent on radio advertising remains constant?

5. A company has the following production function for a certain product
7.03.0

24),( yxyxf  , where x is the number

of units of labor and y is the number of units of capital. Find the marginal productivity of labor and of capital.

6. Use the discriminant to find all points where the function
33

812),( yxyxyxf  has any relative extrema or

saddle points and identify the type of relative extremum.

7. Use the discriminant to answer the following: Suppose that the labor cost for a building can be approximated by

000,12048012032),(
22

 yxyxyxC where x is the number of days of skilled labor and y is the number

of days of semiskilled labor required. Find the values of x and y that minimize cost and find the minimum labor cost.

8. Use Lagrange multipliers to find the minimum of xyyxyxf 
22

),( , subject to the constraint 10 yx

9. A manufacturer’s production is modeled by the function
4/14/3

100),( yxyxf  . Labor costs are $150 per unit and

capital costs are $250 per unit. If the total cost for labor and capital can not exceed $50,000, how many units of labor

and capital are required to maximize production?

10. Find dz if )25ln( yxz  and 02.0,03.0,12,4  dydxyx

11. Use total differentials to approximate the amount of aluminum needed for a beverage can of radius 2.8 cm and height
16 cm. Assume the walls of the can are 0.1 cm thick.

12. The production function for a certain country is
4.03.0

100 yxz  (note: this is not a Cobb-Douglas function), where x

stands for units of labor and y for units of capital. At present, 29 units of labor and 46 units of capital are utilized. Use

total differentials to estimate the change in production if labor increases to 34 units and capital is increased to 52 units.

13. Evaluate dxe
yx




5

3

4

14. Evaluate   dxdyxyyx 
9

0

2

3

0

65

15. Evaluate   
R

dxdyyyx
4

, where R is the region bounded by 30,20  yx

16. Find the volume under the surface
2

)( yxz  and over the rectangular region bounded by 11,11  yx

17. Evaluate the double integral with variable boundaries   dydxe
y

e

y


99ln

0

18. Evaluate  
R

dydxxy2 , where R is the region bounded by 1,0,,1  xxxyxy

19. Use double integrals to find the average value of the function yxyxf 26),(  over the region

40,60  yx

20. A company’s cost for operating two warehouses is yyxxyyxC 4126),(
22
 dollars, where x is the number of

units in warehouse A and y is the number of units in warehouse B. Find the average cost to store a unit if A has 3 to 5

units and B has 1 to 2 units.

21. Determine whether or not the function
6

1

3

1
)(  xxf defines a probability density function on the interval ]4,3[ .

22. Find a value of k that will make
2

)( xkxf  a probability density function on the interval ]3,0[ .

23. If
10

1

5

1
)(  xxf is a probability density function on ]4,2[ , find the cumulative distribution function for f .

24. If
5/

5

1
)(

x
exf


 is a probability density function on ),0[  , find the following probabilities:

a. )61(  xP

b. )4( xP

c. )2( xP

25. The life (in months) of an automobile battery has a probability density function defined by
4/

4

1
)(

t
etf


 for t in

),0[  . Find the probability that the life of a randomly selected battery is greater than 5 years.