Math 142 Week-in-Review #11 (Final Exam Review: All previous sections as well as sections 6.6 and 6.7) Note: This review is intended to highlight the topics covered on the Final Exam (with emphasis on sections 6.6 and 6.7), but it should not be used as your sole source of practice. When studying, you should also rework your notes, the previous week-in-reviews for this material, as well as your suggested and online homework in order to recall and study the variety of formulas/questions for each topic. 1. Find the area between f (x) = x 2 + 4 and the x-axis on the interval [ 4,4]. Include a sketch with the appropriately shaded area. 2. Find the producers surplus at equilibrium price level for a product whose price-demand equation is given by p = D(x) = 430 0.62x dollars per item and price-supply equation is given by p = S(x) = 0.004x 2 + 25 dollars per item. 1
3. A rectangular box is to have a square base and a volume of 20 cubic feet. If the material for the base costs 30 cents per square foot, the material for the sides costs 10 cents per square foot, and the material for the top costs 20 cents per square foot, determine the dimensions of the box that can be constructed at minimum cost. (source: #8, pg. 365 of Applied Calculus for the Managerial, Life, and Social Sciences, 5th ed., by Tan) 2
4. If we know f (2.5) = 0.25 and f (2.5) = 0, and we also know that f is continuous everywhere and f (2.5) = 2, then what (if anything) can we conclude about the behavior of f at x = 2.5? 5. Find the horizontal asymptotes of the function f (x) = 2e 6x 7e 2x e 2x 3e x, if any exist. 14 6. If y = 3t 4 e t and t = ln(h) + 4 h, find dy/dh. 3
7. The following table shows the number of heartbeats after t minutes of a patient after a surgery. t (min) 36 38 40 42 44 Heartbeats 2256 2232 2264 2352 2496 (a) Find the average heart rate between 38 and 42 minutes. Interpret your answer. (b) The data above can be modeled by the function h(t) = 7t 2 530t + 12,264 heartbeats after t minutes, where 36 t 44. Use this model to find the patient s heart rate after 42 minutes. Interpret your answer. (c) Find the average number of heartbeats between 38 and 42 minutes. Round to the nearest integer. (d) Find the average heart rate between 37 and 41 minutes. 4
8. Compute each of the following by hand. (a) a 1 (7x 1 + x 0.2 )dx where a > 1 (b) 21 9 (t + 1) t 5dt 9. Given 6xe πy + 3 y 2 4x 5 + y = 38, find dy dx. 5
10. Determine where each of the following functions is continuous. (a) f (x) = x + 2 3 x 2 16 (b) f (x) = e x3 x 7 15log 2 (x 4) 11. Find the equation of the tangent line to f (x) = 0.5e 2x at x = 1. 6
12. Find ( 5x 4 + (30x2 + 10e x )ln(x 3 + e x ) + 4) x 3 + e x dx. + 4 13. The cost equation of a company is given by C = 0.3x 2 + 1,000, where C is the total cost when x units are produced. If the number of items manufactured is increasing at a rate of 18 items per week, find the rate of change of C with respect to time when 10 items are manufactured. 7
14. For the function f whose graph is shown below, find each of the following. (Graph courtesty of Heather Ramsey) (a) State the domain of f. (b) Where is f continuous? f(x) 8 (c) For what values of c does lim x c f (x) not exist? 7 7 x 8 (d) Which condition in the definition of continuity fails first at x = 3? (e) For what values of a does f (a) not exist? For each value of a, explain why the derivative does not exist. (f) Find the absolute maximum and minimum values of f on [ 6, 1), if they exist. 8
15. Consider the function f (x) = ex x. (a) Use calculus to find any critical values of f (x), as well as the intervals where the function is increasing/decreasing and the values of any local extrema. (b) Find the absolute minimum of f (x) on the interval [0.5,3], if it exists. (c) Find the absolute minimum of f (x) on the interval [ 1,3], if it exists. (d) Find the absolute maximum of f (x) on the interval [0.5,3), if it exists. 9
16. Find each of the following limits, if they exist. (a) x 2 + 8x + 15 lim x 3 x 2 + x 6 (b) lim x 0 2x 2 7 5x + 14 (c) lim x 4x 2 3x 3 + 2 5 7x + 12x 2 (d) lim x 7x 4 + 5x 10x 3 21x 4 10
17. When finding a right Riemann sum of a function f on the interval [a,b] using n rectangles, write the formula (using sigma notation) for the sum of the areas of the third, fourth, and fifth rectangles. 18. Use calculus to find any holes and horizontal and vertical asymptotes of the function f (x) = If there are vertical asymptotes, use limits to describe the behavior near each asymptote. x 2 2x 8 x 4 x 3 x 2 5x 30. Hint: x 4 x 3 x 2 5x 30 = (x 3)(x 2 + 5)(x + 2). 11
19. Consider the graph of a function shown below. (a) If the graph is f (x), find the partition numbers of f (x) and the critical values of f (x). y (b) If the graph is f (x), find where f (x) < 0. 9 7 5 3 2 3 4 6 8 x (c) If the graph is f (x), find where f (x) is increasing. x=3 (d) If the graph is f (x), and f is continuous on its domain of (, 9) ( 9,3) (3, ), find the partition numbers of f (x) and the critical values of f (x). (e) If the graph is f (x), and f is continuous on its domain of (, 9) ( 9,3) (3, ), find where f (x) is concave down. (f) If the graph is f (x), and f is continuous on its domain of (, 9) ( 9,3) (3, ), find where f (x) has local extrema and classify the type of extrema. (g) If the graph is f (x), and f is continuous on its domain of (, 9) ( 9,3) (3, ) and f is continuous on its domain of (, 9) ( 9,3) (3, ), find where f (x) has any inflection points. (h) If the graph is f (x), and f is continuous on its domain of (, 9) ( 9,3) (3, ) and f is continuous on its domain of (, 9) ( 9,3) (3, ), find where f (x) is decreasing. 12
x 2 + 4x 21 20. Find lim. x 7 + x + 7 21. Consider the function g(x) = 2 3x. (a) Find an expression for the slope of the secant line through (x,g(x)) and (x + h,g(x + h)). (b) Find an expression for the slope of the tangent line at the point (x,g(x)). 13
22. Find lim x ( ax3 + bx c), where a > 0, b < 0, and c > 0. 23. Find the area bounded by the curves f (x) = x 2 and g(x) = x + 4. 24. Use the definition of the derivative to find f (x) for f (x) = x 4x 2. 14
25. Acme Music Company has determined the price-demand function for its home audio system to be p = 3080 10x dollars per system. Acme Music has fixed costs that amount to $56,250 and variable costs of $580 per system. (a) Find the profit and marginal profit from the production and sale of 175 home audio systems, and interpret your answers. (b) Approximate the profit from the production and sale of 176 home audio systems. (c) Find the exact profit from the production and sale of 176 home audio systems. 15
26. Find the derivative of each of the following functions. (a) f (x) = 3x 2 5 4 x + e x + 7 2x e 2 (b) f (x) = lnx + 4x7 2x 9 (c) f (x) = e 4.5ln(x2 +1) (d) f (x) = 6 (10x 3 4x + π 2 ) 5 [ ln ( 4x 3 + 9 x)] 16
(e) f (x) = log ( 8 x 2 4x + 5 ) e x + 6x 3 ( (f) f (x) = 3 x 2 + 1 5 ) 12 x (g) f (x) = 17e x + 25e 3x + 9x 5 8e x 17
27. Suppose x 2 2xy y 2 = 7, where x and y are functions of t. If dx dt = 3 when x = 1 and y = 2, find dy dt. 28. Given 5 2 f (x)dx = 10, 5 10 f (x)dx = 3, and 10 2 10 2 g(x)dx = 4, find [3 f (x) 4g(x)]dx 18
29. Given the function f (x) below, (a) determine where f (x) is continuous. f (x) = 8 x < 0 13 x 5 x 3 3e x 0 x < 9 ln(x 7) x 2 7x 30 x > 9 (b) find f ( 4), if it exists. (c) find lim f (x), if it exists. x 10 (d) find lim x 0 f (x), if it exists. 19
30. Find the intervals where f (x) = 3e 2x 8e x is concave upward and concave downward, and find the coordinates of any inflection points. 31. Sketch the graph of the derivative of f (x) shown below. y x 20
32. A company s marginal profit can be modeled by m(x) = 18 x + 5 dollars per item, where x is the number of items produced and sold. This company s profit is $11,475 when 95 items are produced and sold. (a) Find the profit when 120 items are produced and sold. (b) Find the average marginal profit when the number of items sold is between 100 and 120. (c) Find the change in profit when the number of items sold increases from 100 to 120. (d) Find the average profit when the number of items sold is between 100 and 120. (e) Estimate the profit from the sale of the 90 th item. 21
33. Find bounds to estimate the value of 1 6 (x 2 + 8x + 9) dx. 34. Ship A is observed to be 4 miles due north of port and traveling due north. At the same time, ship B is observed to be 3 miles due west of port and traveling due east on its way back to port at 4 miles per hour. If the distance between the ships is increasing at a rate of 2 miles per hour, how fast is ship A traveling at this time? Round your answer to one decimal place if necessary. 22
35. Use the graph of g (x) below to answer the following: Note: Each grid mark represents one unit on the graph below. (a) Estimate 7 1 three rectangles. g (x) dx using a left Riemann sum with y 6 g (x) 3 (b) Find 8 5 g (x) dx. 6 3 3 6 x 3 6 (c) Find 1 3 g (x) dx. (d) Find g(3) if g(0) = 25. 23
36. Compute each of the following. (a) (4e x 7x 3 + 4x 1 e 2) dx (b) 4t 3 t 4 7dt (c) x 6 4x 7 5 dx (d) (4x + 2)e x2 +x dx 24
(e) 2 x 5 + lnx dx 37. Find the area bounded by the curves y = 0.6 x + 3 and y = x 2 3x + 1 on the interval [ 3,2]. 25
38. Reid is selling lemonade. When she charges $5 per glass, she can sell 36 glasses of lemonade. For every $0.10 that she lowers the price, an additional 4 glasses of lemonade will be sold. (a) Use optimization techniques from calculus to determine what price Reid should charge per glass to maximize her revenue. (b) Find the exact revenue from selling the 45 th glass of lemonade. 39. The price-demand equation for a certain item is given by p = D(x) = 0.002(x + 100) 2 + 7000 dollars per item, where x is the number of items that can be sold at a price of $p. If the current price per item is $4,580, find the consumers surplus. Shade the region on an appropriately labeled graph, and explain what your answer represents. 26
40. Sketch the graph of a function that satisfies all of the following conditions: f ( 3) = f ( 5) = 0 f (x) < 0 if 5 < x < 3 f (x) > 0 if x < 5, 3 < x < 1, or x > 1 f (x) < 0 if x < 4 or x > 1 f (x) > 0 if 4 < x < 1 lim f (x) = x 1 + lim f (x) = x 1 lim f (x) = 2 x 41. Find π (3 e t )e t dt. 27