MATH 234 Mock Final Exam x+2 . 2 x 8x 5 1. Find the value of lim (a) 1/8 (b) -2/5 (c) 0 (d) + 2. Find the value of lim x2 2x+5 |x+2| . (a) 2 (b) + (c) -2 (d) 3. Find f 0 (x) if f (x) = (a) f 0 (x) = ln x . x2 1 2x2 (b) f 0 (x) = x24 (c) f 0 (x) = 12 ln x x3 (d) f 0 (x) = 1+2 ln x x3 4. Find the derivative of (1 + x4 x1 )5/3 . (a) 5 3 (1 + x4 x1 )2/3 (b) 5 3 3 (4x (c) 5 3 (1 + x4 x1 )2/3 (4x3 + (d) 3 8 (1 + x4 x1 )8/3 1 )(1 x2 + x4 x1 )2/3 1 ) x2 5. Find the equation of the tangent line to the curve y 4 (1 x) + xy = 2 at x = 1. (a) y = 16x 14 (b) y = 14x 12 (c) y 2 = 14(x 1) (d) y 2 = 12(x 1) 6. Given the function f (x) = x3 4x2 + 5x find the interval where f is concave down. \u0001 (a) , 43 \u0001 (b) 1, 35 \u0001 (c) (, 1) 35 , + \u0001 (d) 43 , + 7. For what value(s) of x does the graph of f (x) = 13 x3 x2 + 3 have a horizontal tangent. (a) 0 (b) 2 (c) 0 and 2 (d) None of the above 8. For what value of the constant c is the function f continuous? ( cx2 + 2x, if x < 2 f (x) = 2x + 4, if x 2 (a) 0 (b) 2 (c) 1 (d) 4 9. What is the resulting function if we translate f (x) = x2 + 3x 5 by 3 units to the left and 7 units up. (a) (x 3)2 + 3(x 3) 14 (b) (x 3)2 + 3(x 3) + 2 (c) (x + 3)2 + 3(x + 3) + 2 (d) (x + 3)2 + 3(x + 3) 14 10. Solve for x, the equation 42x = 8x+5 . (a) x = 15 (b) x = 5 (c) x = 15/7 (d) x = 0 11. Write the following expression as a single logarithm: 1 4 (ln 6 3 ln 5) + ln 7. (a) ln (6/4 8) \u0010 \u0011 746 (b) ln 4 125 \u0001 4 (c) ln 7 6 4 125 (d) ln (9/4 + 7) 12. An initial investment of $12,000 is invested for 2 years in an account that earns 4% interest, compounded quarterly. Find the amount of money in the account at the end of the period. (finance and interest rate) (a) $12,994.28 (b) $994.28 (c) $12,979.20 (d) None of the above 13. If $4,000 is invested at 7% compounded annually, how long will it take for it to grow to $6,000, assuming no withdrawals are made? (a) 5 years (b) 2 years (c) 6 years (d) 7 years 14. What will the value of an account (to the nearest cent) be after 8 years if $100 is invested at 6.0% interest compounded continuously? (a) $159.38 (b) $161.61 (c) $175.32 (d) $849.47 15. The demand equation for a certain item is p = 14 q/1000, where q is the demand at price p. The cost equation is C(x) = 7000 + 4x. Find the marginal profit at a production level of 3000. (a) $14 (b) $11 (c) $7 (d) $4 16. Find the absolute extrema, if they exist, for the function f (x) = 12 x 9 x for x > 0. (a) Absolute maximum at x = 3 and absolute minimum at x = 3. (b) Absolute minimum at x = 3 and no absolute maximum. (c) Absolute maximum at x = 3 and absolute minimum at x = 3. (d) Absolute maximum at x = 3 and no absolute minimum. 17. Find the cost function, if you know that the marginal cost function is given by C 0 (x) = x + x12 and that 2 units cost $5.50. (a) C(x) = (x2 /2) (1/x) + 3.5 (b) C(x) = (x2 /2) + (1/x) (c) C(x) = (x2 /2) (1/x) + 4 (d) C(x) = (x2 /2) (1/x) Z 18. Find the integral (x+1)6 6 (a) (x+1)7 7 (b) x(x+1)6 6 +C (c) (x1)x6 6 +C x(x + 1)5 dx. +C (d) (x + 1)6 (x + 1)5 + C Z \u0012 19. Evaluate the integral (a) (ln x)2 3x2 2 3x2 2 \u0013 dx + C. + 7x + C (b) (ln x) 2 (c) (ln x)2 2 3x2 2 + 7x + C (d) (ln x)2 2 3x2 2 +C ln x 3x + 7 x +7+C 20. Find the area between the curves y = 2e2x , y = e2x + 1, x = 2, and x = 1. (a) 2 e2 +e4 2 (b) e2 +e4 2 3 (c) e2 +e4 2 2 (d) 3 e2 +e4 2 21. A worker sketches the curves y = x and y = x/2 on a sheet of metal and cut out the region between the curves to form a metal plate. Find the area of the plate. (a) 16 3 4 (b) 1 8 1 4 (c) 1 4 1 8 (d) 4 16 3 22. The rate of reaction to a drug is given by r0 (t) = 2t2 et , where t is the number of hours since the drug was administered. Find the total reaction to the drug from t = 1 to t = 6. (a) 2.866 (b) 3.431 (c) (d) e2 /4 23. Find the absolute maximum and minimum values of f (x) = x3 6x2 + 9x + 13 on the interval [-6, 2]. (a) Maximum value is 17 and minimum value is 13. (b) Maximum value is 17 and minimum value is -365. (c) Maximum value is 15 and minimum value is 13. (d) Maximum value is 15 and minimum value is -365. 24. Use the differential to approximate the quantity 23. (a) 4.800 (b) 4.796 (c) 4.792 (d) 4.804 25. A carpenter is building a rectangular room with a fixed perimeter of 360 ft. What are the dimensions of the largest room that can be built? (a) 90 ft x 90 ft (b) 180 ft x 180 ft (c) 100 ft x 80 ft (d) 70 ft x 110 ft 26. How would you divide a 16 inch line so that the product of the two lengths is a maximum? (a) Divide into 2 inches and 14 inches. (b) Divide into 4 inches and 12 inches. (c) Divide into 6 inches and 10 inches. (d) Divide into 8 inches and 8 inches. 27. Find the vertical asymptotes of the graph of f (x) = x2 4x+4 . x2 4 (a) No vertical asymptotes (b) Vertical asymptote at x = 2 (c) Vertical asymptote at x = 2 (d) Vertical asymptote at x = 2 and x = 2 28. When a management training company prices its seminar on management techniques at $400 per person, 1,000 people will attend the seminar. The company estimates that for each $5 reduction in the price, an additional 20 people will attend the seminar. How much should the company charge for the seminar in order to maximize revenue? Round to the nearest dollar. (a) $400 per person (b) $325 per person (c) $360 per person (d) $385 per person 29. Find f 00 (x) for f (x) = x + 3x5 + 1 x + 12. (a) 1 + 15x4 (1/2x2 ) (b) 60x3 + (1/6x3 ) (c) 60x3 + (2/x3 ) (d) 60x3 + (1/3x3 ) 30. A company estimates that it will sell N (t) large-screen television sets after spending t thousand dollars on advertising, where N (t) = 8t3 0.5t4 + 1000, 0 < t < 15. For which values of t is the rate of change of sales increasing? (a) It increases for 8 < t < 15. (b) It increases for 0 < t < 12. (c) It increases for 0 < t < 0. (d) It increases for 12 < t < 15. 31. The daily cost C(x), if x woks are produced each day, is C(x) = 900 + 60x + 0.16x2 , for 0 < x < 200. Find the minimum average cost. (a) $4725 (b) $1516 (c) $25 (d) $84 32. The cost to produce x units of a product is given by C(x) = 8500 ln(x + 100). Estimate using calculus the cost for producing the 100th unit. (a) 8500 ln(200/199) (b) 8500/201 (c) 8500/200 (d) 8500/199 33. A large pharmacy has an annual need for 320 units of a certain antibiotic. It costs $2 to store one unit for one year. The fixed cost of placing an order amounts to $30 . Find the number of units to order each time, and how many times a year the antibiotic should be ordered. (a) 70 units about every 2.625 months (b) 70 units about every 2.265 months (c) 98 units about every 3.265 months (d) 98 units about every 3.675 months 34. Find the the elasticity of demand when the demand equation is given by p = e0.2q , where q represents the demand for a product priced at p dollars per unit. (a) 5 q (b) 1 2q (c) 1 q (d) 0.2 q 35. The demand for a product is given by q = 400 0.2p2 . Determine whether the demand is elastic, inelastic, or unit elastic when p = $40 and interpret the result. (a) Inelastic, they should increase the price (b) Inelastic, they should decrease the price (c) Elastic, they should decrease the price (d) Unit elastic 36. For a new diet pill, the supply and demand functions are given by p = S(q) = 0.5q + 10 and p = D(q) = 0.5q + 72.50, respectively. Find the equilibrium price and quantity. (a) p = 42.25 and q = 64.5. (b) p = 41.25 and q = 62.5. (c) p = 40.5 and q = 61. (d) p = 42 and q = 64. 37. Determine the local extrema, if any, for the function: f (x) = x3 + 3x2 24x + 6. (a) No local extrema. (b) Local max at x = 2 and local min at x = 4. (c) Local max at x = 5 and x = 3 and local min at x = 0. (d) Local max at x = 4 and local min at x = 2. 38. Find the partial derivative fx (x, y) of f (x, y) = 2x2 y 3 ye2x + ln(3xy). (a) 6x2 y 2 e2x + (1/y) (b) 4xy 3 + 2ye2x + (1/xy) (c) 4xy 3 + 2ye2x + (1/x) (d) 6x2 y 2 e2x + (1/xy) 39. Find the derivative of the function y = (a) (b) (c) (d) xex . ln(x2 1) ln(x2 1)(ex + xex ) xex x22x1 ln(x2 1)2 ln(x2 1)xex xex x22x1 ln(x2 1)2 ln(x2 1)(ex + xex ) xex x2x1 ln(x2 1)2 ln(x2 1)xex (ex + xex ) x22x1 ln(x2 1)2 40. On which intervals is the following function increasing and decreasing: f (x) = x4 + 4x3 + 4x2 + 1? (a) Increasing on (1, 0) (0, ) and decreasing (, 2) (2, 1). (b) Increasing on (2, 1) (, 2) and decreasing (0, ) (1, 0). (c) Increasing on (, 2) (1, 0) and decreasing (2, 1) (0, ). (d) Increasing on (2, 1) (0, ) and decreasing (, 2) (1, 0). 41. Suppose the cost to produce x units of tacos costs C(x) = 5x + 20 dollars and the revenue is given by R(x) = 15x dollars. What is the break even quantity? (a) 2 units (b) 10 units (c) 3 units (d) 4 units 42. Find the relative maximum of the function f (x, y) = 12xy x2 3y 2 subject to the constraint x + y = 16. (a) Relative maximum at (x, y) = (8, 8). (b) Relative maximum at (x, y) = (9, 7). (c) Relative maximum at (x, y) = (5, 11). (d) Relative maximum at (x, y) = (6, 10). 43. Find all points where the function f (x, y) = 3x2 + 7y 3 42xy + 5 has any relative extrema. Also identify any saddle points if they exist. (a) Saddle point at (0, 0), relative minimum at (98, 14). (b) Relative minimum at (0, 0) (c) Saddle point at (0, 0), relative maximum at (98, 14). (d) No relative extrema and no saddle points. Z 44. Compute the following definite integral 0 (a) (1/9) ln 3 (b) (1/6) ln 3 (c) (1/6) ln 2 (d) (1/6) ln 4 1 x2 dx . 2x3 + 1 45. In a midwestern town of 100,000 people, the crime rate is given by C(t) = (1/10)(t 60)2 + 100 where C is the number of crimes per month and t is the average monthly temperature in The average temperature for May was 76 F , and by the end of May the temperature was rising at a rate of 8 F per month. Approximately, how fast is the crime rate rising at the end of May? F . (a) 27 crimes per month (b) 88 crimes per month (c) 25 crimes per month (d) 26 crimes per month Disclaimers: 1. This file was compiled by Paulina Koutsaki and Dileep Menon using problems found on the Internet and the textbook. 2. Solving the above problems should in no way substitute for the entirety of a student's studying regiment. 3. It is possible there are some errors above. If you find any please email us