Exam Four - Olson PDF

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Brief Calculus Exam 4 Practice Assignment 1.

Given the function f (x) = 2x 3 −15x 2 − 84x : a. Find the intervals on which f (x) is increasing. b. Find the intervals on which f (x) is decreasing. c. Find the local extrema.

2.

Given the function f (x) = x 4 − 72x 2 : a. Find the intervals on which f (x) is increasing. b. Find the intervals on which f (x) is decreasing. c. Graph the function and add horizontal tangent lines.

3.

Find the x and y coordinates of all inflections points for f (x) = x 3 + 21x 2 .

4.

Given the function f (x) = −x 6 +12 x 5 −12x + 3 : a. For what intervals of x is the graph concave upward? b. For what intervals of x is the graph concave downward? c. Determine the x coordinates of any inflection points of the graph.

5.

A candy box is made from a piece of cardboard that measures 11 by 7 inches. Squares of equal size will be cut out of each corner. The sides will then be folded up to form a rectangular box. What size square should be cut from each corner to obtain maximum volume?

6.

A company manufactures and sells x television sets per month. The monthly cost and pricex demand equations are C(x) = 72,000 + 80x and p(x) = 250 − , 0 ≤ x ≤5000. 20 a. Find the maximum revenue b. Find the maximum profit, the production level that will realize the maximum profit, and the price the company should charge for each television set.

Evaluate the following integrals. 8.

∫x

∫ x dx

10.

∫ x −6 dx

11.

∫ ( ex + 3) dx

12.

13.

e x − 5x ∫ 8 dx

7.

∫ 11dx

9.

5

10

∫

dx

9+u du u

Answers 1. a. (−∞,−2) , (7, ∞) , b. (−2, 7) , c. The function has a local maximum at x = –2 and a local minimum at x = 7.

2. a. (−6, 0) , (6, ∞) , b. (−∞,−6) , (0, 6) , c. 4. a. (0,8) , b. (−∞, 0) , (8, ∞) , c. x = 0, 8

3. (–7, 686) 5. 1.39 inches on a side.

6. a. $312,500.00 maximum revenue, b. $72,500.00 maximum profit at a production level of 1700 sets at a price $165.00.

x11 +C 11

7. 11x + C

8.

9. 5 ln x + C

10. −

11. e x + 3x + C

12. 9 ln u + u + C

13.

1 ⎛ x 5x 2 e − 2 8 ⎜⎝

⎞ 1 x 5 2 ⎟ + C = 8 e − 16 x + C ⎠

x −5 +C 5...