Final Exam Review s10

Math 1309 Final Exam Review

1. Find the first derivative. 1 x3 f (x)   x 4 f (x)  (3x  2)(4x  5) 3x  5 f (x)  x 2  3

2. Find the first derivative.

f (x)  3(x 2  2) 4 1 f (x)  4x 2  4x 1 f (x)  3x 2x 2  3

f (x)  ln(x  4)3 f (x)  1 e x  e 2x f (x)  {ln(x 2  3x)}4

2 f (x)  (e x  3)5 e x f (x)  e x 1 f (x)  3x 2 ln(2x)

dy 5. Use implicit differentiation to find dx y 3  x 2 y 2  x 3  2y  3x  6

2 2 e x  y  3ln(xy)  2

6. Find the equation of the tangent line to f(x) at the given point.

a. f (x)  7 11x  6x 2 at x=2 b. f (x)  ln x at x=1 7. Determine intervals where f(x) is increasing and decreasing and find all local extrema.

f (x)  x3  9x 2  7 x  2 f (x)  x  3

8. Determine the intervals where f(x) is concave up and concave down and find all inflection points.

f (x)  x3 18x 2 10x 11 f (x)  1 3x  x3

9. Find the indefinite integral. Use “u” substitution when appropriate.

 6 2   1 dx x   4 3  dx  x x   (5  2x 2 )5 x x  e dx 2 x3 3x2   (x  2x)e dx  4 x    dx x 2 2x3 1dx   x 4   10. Evaluate Use “u” substitution when appropriate.

2 1  (2x 2  3)dx  x 3x 2  2dx 1 0 4 2 x 1 3 xdx dx   2 1 1 2x  4x  4 1 3  (3x 2  x 1 )dx  x x 1dx 3 0

11. Find the area bounded by:

a. y  5  x 2 , y  2  2x,1  x  3

b. y  x 2  x, y  2x,2  x  3

12. a. The cost of producing x guitars is given by C(x)  1000 100x 125x 2 . Find the marginal cost at a production level of 50 guitars and interpret the results. b. The profit from the sale of x grills is given by P(x)  20x  .02x 2  320 . Find the marginal profit at a production level of 40 grills and interpret the results.

13. a. A company estimates that it will sell N(x) speakers after spending $x thousand on advertising, as given by N)x)  3x3  225x 2  3000x 17,000 where 10  x  40. How much should be spent in order to max the rate of change in sales? ( Hint: Find the point of diminishing returns.) b. In a new suburb, it is estimated that the number of registered voters will grow according to N(t)  10  6t 2  t 3 where 0  t  5 and time is t in years and N is in thousands. When will the rate of increase be at a maximum?

14. a. A 300 room hotel in Las Vegas is filled to capacity each night when $80 per room is charged. For each increase of $1 in price, 3 fewer rooms are rented. How much should be charged per room to maximize the revenue? b. A cherry tree grower estimates that if 30 trees are planted per acre, each tree will yield 50 pounds of fruit. For each additional tree planted(up to 20), the yield is reduced by 1 pound. How many trees should be planted to maximize the yield?

15. a. The profit function for the monthly rental of x apartments is given by P(x)  10x 2 1760x  50,000 . Currently they rent 50 units, but have plans to increase that by 5 units per month. Find the rate of change in the profit. b. A division of Ditton Industries manufactures microwave ovens. The daily cost of producing x ovens is given by C(x)  .0002x 3  .06x 2 120x  5000 . Currently they manufacture 100 ovens per day, but plan on increasing that by 25 per day. Find the rate of change in the costs.

16. a. You have $2000 towards the purchase of a computer costing $4000. How long will an investment into an account that earns 12% compounded quarterly take to grow to $4000? b. A woman invests $5000 into an account that earns 9.6% compounded continuously. How long will it take her investment to grow to $8000?

17. a. Parents want to have $120,000 in 15 years for their children’s college education. How much should be deposited quarterly into an account that pays 10% compounded quarterly? b. A bond issue is approved for building a marina in a certain city. After 10 years, the bond obligation will be $5,000,000. The city is required to invest $190,000 semiannually at 6% compounded semiannually. Will they be able to pay the bond?

18. a. A company buys a copy machine for $12,000. It amortizes the cost at 12% monthly over 5 years. How much are the monthly payments? b. At the time they retire, a couple wishes to establish an account that pays 8.4% from which they can withdraw $2000 monthly for 20 years. How much should they invest in this account? 19. a. You deposit $3000 each year into a retirement account that earns 5% compounded continuously. How much will you have at the end of 40 years? b. A shop owner deposits the income from her florist shop at the rate of flow given by f (t)  10,000e.05t . The account earns 10% compounded continuously. How much Will be in tha account after 5 years?

20. Find the Consumers Surplus and the Producers Surplus for the price-demand and supply-demand equations given at a price of $24.

p  D(x)  50  .01x

p  S(x)  11 .05x