Final Exam Review (Section 8.3 and Review of Other Sections)

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1 c Kathryn Bollinger, April 29, Final Exam Review (Section 8.3 and Review of Other Sections) Note: This collection of questions is intended to be a brief overview of the material covered throughout the semester (with an emphasis at the beginning on material from Section 8.3, which I have not previously reviewed). This is not intended to represent an actual exam. When studying you should also rework your notes, quizzes and exams, the previous week-in-reviews, and be familiar with your suggested and online homework problems. 1. Find all of the critical points for f(x,y) = 3x 2 +5y 2 8xy +2x+6y Locate any critical points of the following functions, and, if possible, identify each as relative extrema or a saddle point. (a) f(x,y) = 2x 2 +5y 2 +6x 2y +12 (b) f(x,y) = 3x 2 +xy y 2 4x 3y (c) f(x,y) = xy x 3 y 2 3. A firm manufactures and sells two products, X and Y, that sell for $15 and $10 each, respectively. The cost of producing x units of X and y units of Y is C(x,y) = 400+7x+4y +0.01(3x 2 +xy +3y 2 ) Find the values of x and y that maximize the firm s profit. 4. Find the area bounded by f(x) = 8 x 2 and g(x) = x A baker sells onedozen donutswhenadozen of donuts sells for $2.00. For each 2 cent decrease in price, the baker sells an additional dozen of donuts. It costs the baker 25 cents to make a dozen of donuts. Let x be the number of dozen of donuts sold. Find the (a) price-demand function, p(x). (b) cost function, C(x). (c) revenue function, R(x). (d) profit function, P(x). 6. Find dy dx if y = t 3 +4 and t = 3x Find the derivative of f(x) = 2 4x ln 5x 2 +2x 8. What transformation(s) would you apply to the graph of f(x) in order to obtain the graph of g(x) = 2f(x+3) 7? 9. Find the instantaneous rate of change of f(x) = log 3 x 4 at x = 2.

2 c Kathryn Bollinger, April 29, Given f(x) = 4x 3, x < 1 7, x = 1 x+2 6 x, x > 1 (a) find lim x 1 f(x). (b) find lim x f(x). (c) Is f(x) continuous at x = 1? Why or why not? 11. Evaluate b ( ) 2e 3 +π dx. a e Find all asymptotes and holes of f(x) = 13. Find the first derivative of f(x) = (x 6)(x+5) (2x+1)(x 2 3x 18). e (x8) +(ln(x 6 +4)+12) Find the absolute extrema of f(x) = 2x 3 +3x 2 45 on [0,4]. 15. Rewrite f(x) = x2 +2x x+2 as an equivalent piecewise-defined function. 16. Use the given graph to answer the questions which follow. 4 2 (a) If the given graph is f(x), find any absolute extrema of f(x). (b) If the given graph is f(x), where is f (x) < 0? (c) If the given graph is f (x), where is f(x) concave up? (d) If the given graph is f (x), where is f(x) concave down? (e) If the given graph is f (x), where does f(x) have any local extrema? 17. lim x 10e x e x

3 c Kathryn Bollinger, April 29, Find (8x+16)e x2 +4x+5 dx 19. Mr. Barker is adding onto his dog kennel. He needs to fence in 12 equally-sized yard lots (all in a row, connected side-by-side). If Mr. Barker has 400 feet of fencing, what should the dimensions of each dog yard be in order to maximize the total yard area? 4x Evaluate lim x x When using a Riemann sum to approximate the area under f(x) = x 2 +x+5 on the interval [ 3,8], using 10 equally spaced rectangles, what is the area of the last rectangle if you use (a) right endpoints? (b) left endpoints? 22. Suppose water is being pumped out of a well at a rate given by y = 300e 0.3t, where t is the number of years since the pumping began and y is measured in millions of gallons/year. At this rate, how much water will be pumped out during the fourth year? 23. How long will it take for money in an account to quadruple if the account pays 5% annual interest compounded continuously? 24. Evaluate lim x 6 x 2 +x 42 x Find x+ 6 x 9 x dx x Evaluate lim x x 2 2x Sketch a graph of a function with the following properties: f (x) > 0 on (, 3) and (0, ) f (x) < 0 on ( 3, 1) and ( 1,0) f (x) < 0 on (, 1) f (x) > 0 on ( 1, ) VA: x = 1

4 c Kathryn Bollinger, April 29, Given g(x) = x 2 +4 and f(x) = x+20, find the following: (a) (g f)(x) (b) (f g)(x) (c) (g g)(x) 29. Find the first derivative of f(x) = 6 (3x 3 +4x 2 2x+10) Solve for x: x = Evaluate the end behavior of f(x) = 12x 5 +Kx 3 Lx 2 +Bx Given f(x) = 4x 5 x 4, find all values of x where there is a horizontal tangent line to f(x). 33. If we know that f(3) = 4,f (3) = 0 and f (x) is continuous everywhere with f (3) = 5, then what (if anything) can we conclude about f(x) at x = 3? 34. The rate at which the fanbase of a certain band is growing is given by f = 3.7 x + 3 for 1 x 10, where x is the number of years since the band began touring in 1993 and f(x) is measured in tens of fans/year. Evaluate and interpret 1 f(x)dx 35. Given demand d(p) = (225 5p) 1/2, determine the point elasticity of demand at p = 10. At this point is demand inelastic, elastic, or unitary? Should the price be lowered, raised or kept the same to increase revenue? 36. Find the exact value of 0 x x3 +5 dx 37. (a) Which of the following is NOT a function? (b) Which of the functions are one-to-one functions? a. b. c. d. 38. Find the domain of f(x) = ln(3x+8). x

5 c Kathryn Bollinger, April 29, Find lim h 0 f(x+h) f(x) h if f(x) = x+4 4x 2 +3x Evaluate lim x 4 6x 4 +3x Suppose a puppy grows at a rate of y = 0.16e t, where t is the number of years since the beginning of 1999 when the puppy was born and y is measured in inches/year. If the puppy reaches a maximum height of 22 inches at 3 years of age, what is the puppy s height on his first birthday? 42. Solve for x: log 5 (log 3 (log2x)) = Evaluate 2 3 (x 3 +2x 2 +1)dx and determine whether the value represents the total amount of area between a curve and the x-axis or if it represents a net area. 44. Given that 0 2f(x) dx = 20 and 2 f(x) dx = 15, find 0 2 3f(x) dx. 45. Billy s parents want to open an account which pays 6.06% annual interest compounded monthly. They need to have $15,000 for Billy s first year of college tuition. If they open the account exactly 18 years before they plan to withdraw the money, how much should they invest when they open the account to ensure they can pay for Billy s first year of college tuition? 46. Rewrite as a single logarithm: 4 5 (log2x+logx3 (logy 2 +log3y 2 )) 47. Find the first derivative of f(x) = 6x3 e 1/x 48. Krissy s Kosmetics has determined its profit to be given by P(x) = 500x x 2 where 0 x 500 and x measures the number of tubes of lipstick produced and sold. Find P (250) and interpret. 49. Find the domain of f(x) = (x2 1) 3 x+1.

6 c Kathryn Bollinger, April 29, For f(x) = { x, x 2 3+x 2, x < 2, (a) find f( 2),f(2), and f(5). (b) graph f(x). (c) where is f(x) discontinuous? Non-differentiable? Explain your answers. 51. Solve for x: log 7 (2x+1)+log 7 x = Find the equation of the line tangent to the graph of f(x) = (4x 2 +6x)(2x 5x 3 ) at x = Find the exact value of 1 2 x 2 dx 54. Compute the average rate of change of f(x) = x 5 +x 4 x 3 x 2 +5 x 5 x 4 +x 3 over [0,5]. 55. Given f(x) = 1 3 x3 x 2 8x, find (a) all critical values and relative extrema. (b) all inflection points. 56. Where is f(x) = x+12, x 2 x 6 x+1, x > 2 discontinuous? 57. A new moped costs $ in 2002 and its value depreciates at a rate of $ per year. (a) Find an equation for the value of the moped as a function of time. (b) In what year will the moped be worth $ ? 58. What is the effective rate of interest of an account earning interest at a annual rate of 9.25% compounded weekly?

7 c Kathryn Bollinger, April 29, (a) Write the limit that would indicate that the graph of f(x) has a vertical asymptote of x = 7. (b) Write the limit that would indicate that the right-hand side of the graph of f(x) has a horizontal asymptote of y = Find the domain of f(x,y) = 2x+3y 7.