calculus

MAC2233 Extra Credit #2

1. Find the derivative of the function. Simplify your answers!

a. 32

583)( xxxh 

b. 204 4

)( 3

 x x

xf

c. 3

2

6 2352)(

83

x xexxf

xx 

d. 3

54 )(

2 

 

x

x xf

e. 10

)37()(  xxg

2. Let f x x x( ) ln( )  2

3 5 . Find the equation of the line tangent to the graph of f at x = 2.

3. The cost (in dollars) of producing q items is given by C q q q q( ) . .  0 01 0 6 13 3 2

a. Find the Marginal Cost function.

b. Find the average cost function.

c. Find the production level that will minimize average cost.

d. What is the minimum average cost?

4. Let C q( ) represent the total cost, in dollars, of producing q items.

If )100(C = 1100 and )100(C = 20,

a. Estimate the additional cost of producing an additional 5 items once 100 items have been produced.

(be careful, we are not asking for )105(C )

b. Estimate )90(C .

5. The demand equation for a quantity q of a product at price p, in dollars, is qp 3200  where 400  q .

Companies producing the product report the cost, C, in dollars, to produce a quantity q is

2 8075)( qqqC  dollars.

a. Express a company’s revenue, in dollars, as a function of q

b. Express a company’s profit, in dollars, as a function of q.

c. Find the production level that will maximize profit.

d. What is the maximum profit?

6. The balance, in dollars, in a bank account t years after a deposit of $5000 is given by t

etA 08.0

5000)(  .

Compute and interpret )5(A . Your interpretation should be in layman’s terms (so the word “derivative”

should not be in the interpretation) .

7. (Bonus Material - Optional) For the equation 2 223  yyxx , compute

dx

dy when 1x and 0y .

8. Suppose $2000 is invested in an account that pays interest at a 7% annual rate.

a. How much is in the account after 10 years if the interest is compounded annually?

b. How much is in the account after 10 years if the interest is compounded continuously?

c. How long will it take for the value of the investment to double if the interest is compounded annually?

d. How long will it take for the value of the investment to double if the interest is compounded

continuously?

9. A radioactive substance has a half-life of 8 years. If 200 grams are present initially,

a. Write a formula that gives the mass of the remains after t years.

b. How much remains at the end of 12 years?

c. After how long will only 1gram remain?

d. At what rate is the mass decreasing after 5 years?

e. How long until only 10% of the original amount remains?

10. For the function 592)( 23  xxxf on -1  x  5

a. Find f 

b. Find f  .

c. Find the critical numbers f.

d. Find the intervals on which f is increasing or decreasing.

e. Find the points where f has local maximum and minimum values.

f. Find the absolute minimum and maximum values.

g. Find the inflection points (be sure to give both the x and y coordinates)

h. Find the intervals of concavity

11. For the function 1)(  xxxf on -1  x  5

a. Find f 

b. Find the critical numbers f.

c. Find the intervals on which f is increasing or decreasing.

d. Find the local maximum and minimum values (be sure to give both the x and y coordinates)

12. The energy expended by a bird per day, E, depends on the time spent foraging for food per day, F hours. Foraging for shorter time requires better territory, which then requires more energy for its defense. Find the

foraging time that minimizes energy expenditure if

E F F

 0 25 1 7

2 .

.

13. A closed box has a fixed surface area of 54 ft 2

and a square base with side x ft.

a. Find a formula for its volume, V, as a function of x.

b. Find the maximum volume of the box.

c. What length of x gives the maximum volume?

14. The demand curve for a product is given by 2

4900 pq  .

a. Find the elasticity of demand when the price is $10

b. Is demand inelastic, elastic, or neither at a price of $10?

c. To collect more revenue, should the price be raised or lowered?

d. What price gives the maximum revenue?

e. What quantity gives the largest revenue?

15. The demand curve for a product is given by qp 18540  .

a. Find the elasticity of demand when the price is $20

b. Is demand inelastic, elastic, or neither at a price of $20?

c. To collect more revenue, should the price be raised or lowered?

d. What price gives the maximum revenue?