Exam 2 Review (Sections Covered: and )

Transcription

1 Exam 2 Review (Sections Covered: and ) 1. Find the derivative of the following. (a) f(x) = 1 2 x6 3x 4 + 6e x (b) A(s) = s 1/2 ln s ln(13) (c) f(x) = 5e x 8 ln x 2. Given below is the price-demand equation for the production of tennis shoes where x is the number of pairs of tennis shoes that can be sold at a price of p dollars. (Round answers to two decimal places as needed.) p = 40 x (a) Determine the revenue function and the marginal revenue function. (b) Use the marginal revenue function to estimate the revenue from the 150th pair of shoes sold. 3. A company has found that the profit (in dollars) from the sale of x ovens is given by P (x) = 0.01x x x (Round answers to two decimal places as needed.) (a) Determine the exact profit from the sale of the 39th oven. (b) Use the marginal profit to estimate the profit from the 39th oven sold. 4. Differentiate the following functions: (a) f(x) = 2x 5 ln x (b) f(x) = 6x 3 e x (c) f(x) = (x 2 5) x (d) f(x) = (3x 3 + 5)(x 4 4x) (e) f(r) = (r 2 6r)e r (f) f(x) = (x 7 10x 6 + 6) ln(x) 5. Differentiate the following functions: (a) f(x) = 5 ln x (b) f(x) = ex 5x 4 1 (c) f(x) = 8 3e x

2 x4 (d) f(x) = 1 x 3 6. Differentiate the following: (a) F (x) = (x 4 + 3x 2 2) 5 (b) F (x) = ( 6x 3 14) 5 (c) y = 2x + 9x 6 (d) f(x) = 5 ln x + e x (e) g(t) = x ln x (x + 2) 2 7. Find the derivative of the following. (a) g(x) = 7 x (b) g(x) = 4 (x7 +x) (c) y = e (5x5 4x) (d) g(x) = x 4 5 (x7 +2) (e) 5 + 4e 4x 8. Find the derivative of the following. (a) f(x) = log 6 (5x) (b) f(x) = log 15 (7 x 4 ) (c) g(w) = 3 ln(5 + 4w + w 4 ) (d) f(x) = (ln(1 + e x )) 5 (e) f(x) = x 5 log 5 (9x 4 + 5) 9. Given the following price-demand function, find the elasticity of demand, E(p), and determine whether demand is elastic, inelastic, or has unit elasticity for the following values of p. (Round your answers to two decimal places.) x = 225, p 2 (a) E(p)? (b) p = 60 (c) p = 38 (d) p = 65 2 Fall 2016, Maya Johnson

3 10. A product has the following price-demand function where x is the number of items demanded and p is the price in dollars. (Round all answers to two decimal places as needed) x = 168 2p (a) Obtain a formula for the price elasticity of demand function. (b) At what price should the items be sold in order to maximize the revenue? (c) Give the interval where the demand is inelastic. 11. The consumer demand curve for Professor Stefan Schwarzenegger s dumbbells is given by x = (84 3p) 2, 0 < p < 28. Here, p is the price per dumbbell, and x is the demand in weekly sales. Find the price Professor Schwarzenegger should charge for his dumbbells in order to maximize revenue. (Round your answer to the nearest cent.) 12. The Physics Club sells E = mc 2 T-shirts at the local flea market. Unfortunately, the club s previous administration has been losing money for years, so you decide to do an analysis of the sales. A quadratic regression based on old sales data reveals the following demand equation for the T-shirts: x = 4p p, (5 p 8). Here, p is the price the club charges per T-shirt, and x is the number of shirts it can sell each day at the flea market. (a) Obtain a formula for the price elasticity of demand for E = mc 2 T-shirts. (b) How much should the Physics Club charge for the T-shirts in order to obtain the maximum daily revenue? (c) What will the maximum revenue be? 13. For the function below, determine each of the following. f(x) = 6x 3 + 9x 2 360x. (a) Find all critical values of f(x). (b) Find all intervals on which f(x) is increasing and all intervals on which f(x) is decreasing. (c) Find the x-coordinates of all relative extrema on the graph of f(x). 3 Fall 2016, Maya Johnson

4 14. Suppose the function f(x) has a domain of all real numbers except x = 7. The first derivative of f(x) is shown below. f (x) = 5(x 1) (x + 7) 7 (a) Find all intervals on which f(x) is increasing. (b) Find all intervals on which f(x) is decreasing. (c) Find the x-coordinates of all relative extrema on the graph of f(x). 15. Use the graph of f (x) to answer questions about the function f(x). (a) Find all critical values of f(x). (b) Find all intervals on which f(x) is increasing and all intervals on which f(x) is decreasing. (c) Find the x-coordinates of all relative extrema on the graph of f(x). 16. Use the graph of the function f(x) displayed below to answer the following questions. (a) Find the critical values of f(x). (b) Find the x-coordinate(s) of the relative maxima for f(x). 4 Fall 2016, Maya Johnson

5 (c) Find the x-coordinate(s) of the relative minima for f(x). 17. Find f (x) for the following functions. (a) f(x) = 2x x 5 x 2. (b) y = 9 x 4 ln(x). 2 (c) G(x) = x + 8 x. 18. Use the graph of f (x) to answer questions about the function f(x). (a) Give the intervals where f(x) is concave up. (b) Give the intervals where f(x) is concave down. (c) Find the x-coordinates of the inflection points for f(x). 19. Use the graph of the function f (x) to answer the following questions. (a) Give the intervals where f(x) is concave up. (b) Give the intervals where f(x) is concave down. (c) Find the x-coordinates of the inflection points for f(x). 20. Consider the equation below. f(x) = 4x 3 12x x + 6 (a) Give the intervals where f(x) is concave up. (b) Give the intervals where f(x) is concave down. (c) Find the x-coordinates of the inflection points for f(x). 5 Fall 2016, Maya Johnson

6 21. Suppose the function g(x) has a domain of all real numbers except x = 5. The second derivative of g(x) is shown below. g (x) = (x 5)(x + 4) (x + 5) 3 (a) Give the intervals where f(x) is concave up. (b) Give the intervals where f(x) is concave down. (c) Find the x-coordinates of the inflection points for f(x). 22. Assuming that the function f(x) is continuous on the interval (, ), indicate whether each of the points listed below is a relative maximum, relative minimum, neither or cannot be determined from the information given. (a) if f (1) = 0 and f (1) = 2 (b) if f (0) = 3 and f (0) = 5 (c) if f ( 1) = 0 and f ( 1) = 2 (d) if f (3) = 0 and f (3) = Find the limit, if it exists, of the following. 10x 7 x + 9 (a) lim x 5x 7 + 5x 2 9 6x 3 9 (b) lim x 7x x 5 2x + 7 (c) lim x 6 4x 4 (d) lim x 10x 5 + x + 9 (3x 5 9)x 2 (e) lim x 5x (5x 2 1)(x 2 1) 6 Fall 2016, Maya Johnson

7 24. Find the limits of the following: (a) lim x (x 2 + x 5 ) (b) lim x (x 2 + x 5 ) (c) lim x (x 2 x 5 ) (d) lim x (x 2 x 5 ) 25. Find the limit of the following: (a) lim x e (x2 +10) (b) lim x (c) lim x (d) lim x e x e x 3 + 5e x e x 3 + 5e x 26. Find the horizontal asymptote of the following functions. 5x (a) y = 11x 3 32x 3 6x (b) y = 5x 2 34x 7 (c) y = 10x x 2 62x 9 7 Fall 2016, Maya Johnson

8 27. Which of the graphs of f(x) below has the following properties? ˆ f (x) > 0 on (, a) and (0, g). ˆ f (x) < 0 on (a, 0) and (g, ). ˆ f (x) > 0 on (b, e) ˆ f (x) < 0 on (, b) and (e, ). (a) I (b) II (c) III (d) IV 8 Fall 2016, Maya Johnson

9 28. Sketch the graph of a function that satisfies all of the given conditions. ˆ f (0) = f (2) = f (4) = 0, ˆ f (x) > 0 if x < 0 or 2 < x < 4, ˆ f (x) < 0 if 0 < x < 2 or x > 4, ˆ f (x) > 0 if 1 < x < 3, ˆ f (x) < 0 if x < 1 or x > 3. (a) I (b) II (c) III (d) IV 9 Fall 2016, Maya Johnson

10 29. Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = x 3 6x 2 + 9x + 8 on [1, 8] 30. Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = x + 9 x on [0.2, 12] 31. Find the absolute minimum and absolute maximum values of f on the given interval. f(x) = x ln(2x) on [0.5, 2] 32. Find the absolute maximum and absolute minimum values of f on each interval. f(x) = 6x x (a) (2, 7) (b) (2, 7] 33. Find two positive numbers x and y with xy = 300 such that the sum x + 3y is a minimum. 34. ABC Daycare wants to build a fence to enclose a rectangular playground. The area of the playground is 940 square feet. The fence along three of the sides costs $5 per foot and the fence along the fourth side, which will be made of brick, costs $10 per foot. Find the length of the brick fence that will minimize the cost of enclosing the playground. (Round your answer to one decimal place.) 35. A rancher wants to create two rectangular pens, as shown in the figure, using an existing fence line as one side. If there are 657 feet of fence available, what dimensions should be used to maximize the total area of the pens? 36. If 30, 000cm 2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box. 10 Fall 2016, Maya Johnson