Math 2211 Final Exam Review Name________________________ Just Relax and Think Positively!!!! I Know you CAN Do This!!! 1. Use the given graph to find the indicated quantities: (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) f (-1) (k) f (0) (l) f (1) (m) f (2) (n) Is f(x) continuous at x=2? 2. Find the value of the limit 3. Find the value of the limit 4. Find the value of the limit 5. Find the value of the limit 6. Let f (x) = . Find the following limits. Justify your answers. (a) (b) (c) (d) (e) (f) 7. At what value(s) of x is the function discontinuous? 8. Find the value of the limit 9. Find an equation of the line tangent to f (x) = x2 - 4x at the point (3, - 3). Differentiate the following: 10. f(x) = 30 3 11. F(x) = 4 8 12. h(x) = (x 2)(2x + 3) 13. y = (x - 1) 14. f(x) = 2 3+1 2 15. g(x) = 16. y = 1+ +1 17. y = 3 +2 18. f(x) = sin x 1+sin 19. y = +cos 20. y = tan(x5) 21. f(x) = (4x - x2)100 22. f(x) = (1 + x4)2/3 3 23. f(t) = 1 + tan 24. y = 5 + cos 3x 25. y = e2t cos 4t 26. y = 101 2 27. y = 2x 2 + 1 28. Find an equation of the tangent line to the curve y = x4 + 2x2 x at the point (1, 2). 29. Find an equation of the normal line to the curve y = (1 + 2x)2 at the point (1, 9). 30. The equation of motion of a particle is s = t4 2t3 + t2 - t, where s is in meters and t is in seconds. a. Find the velocity and acceleration as functions of t. b. Find the acceleration after 1 second. 31. Find an equation of the tangent line to the curve y = (1 + 2x)10 at the point (0, 1). Find dy/dx by implicit differentiation: 32. 2 + = 3 33. 2x3 + x2y xy3 = 2 Differentiate the following functions: 34. 35. 36. Find if 37. The population of a bacteria colony after t hours is given by P(t) = rate of the colony when t = 16 hours. 38. A particle moves along a straight line with equation of motion particle at time t = 1. . Find the growth . Find the instantaneous velocity of the 39. Suppose the amount of a drug left in the body t hours after administration is mg. In mg/h, find the rate of decrease of the drug 4 hours after administration. 40. Use logarithmic differentiation to find the derivative of the function: a. y = xx ( 2 +1)4 b. y = (2+1)3 (3 1)5 41. A ladder 10 feet long is leaning against a wall. If the foot of the ladder is being pulled away from the wall at 3 feet per second, how fast is the top of the ladder sliding down the wall when the foot of the ladder is 8 feet away from the wall? 42. If the radius of a sphere is increasing at 1 centimeter per second, find the rate of change of its volume when the radius is 6 centimeters. 43. A child throws a stone into a still millpond causing a circular ripple to spread If the radius of the circle increases at the constant rate of 0.5 meter per second, how fast is the area of the ripple increasing when the radius of the ripple is 20 meters? 44. Find the absolute maximum of the function 45. Consider . (a) Find the intervals on which f is increasing or decreasing. (b) Find the local maximum and minimum values of f. (c) Find the intervals of concavity and the inflection points. 46. Find the value of the limit: 47. Find the value of the limit: on the interval . 48. A farmer has 20 feet of fence, and he wishes to make from it a rectangular pen for his pig Wilbur, using a barn as one of the sides. In square feet, what is the maximum area possible for this pen? 48. A square is to be cut from each corner of a piece of paper which is 8 cm by 10 cm, and the sides are to be folded up to create an open box. What should the side of the square be for maximum volume? (State your answer correct to two decimal places.) 50. A certain company has cost function C(x) = 1000 + 2x + 0.01x2 and Revenue function R(x) = 74x 0.02x2. Find the production level that will maximize the profit. 51. Find the most general antiderivative of the function 52. Find the most general antiderivative of the function . . 53. Find the value of the integral 54. Evaluate the following integrals: (a) (b) 55. Evaluate the following integrals: (a) (b) (c) (d) 56. Find the value of the integral 57. Find the value of the integral 58. Evaluate the following integral 59. Evaluate the following integrals: (a) (b) 60 Suppose we wish to estimate the area under the graph of f (x) = x2 for 0 x 2. What is the value of the estimate using four approximating rectangles and taking sample points to be right-hand endpoints? 10 61. Use the midpoint rule with n = 4 to approximate 2 3 + 1 62. The graph of f is given below. State, with reasons, the number(s) at which (a) f is not differentiable. (b) f is not continuous. Math 2211 Final Exam Review 1. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) f (-1) = 0 (k) f (0) -1.7 (l) f (1) = 1 SOLUTIONS (m) f (2) = -2 (n) No 2. 3 3. 5 4. 1 5. 2 6. (a) 0 7. 1 8. 2 9. 2x y = 9 10. 0 11. 6x7 12. 4x 1 13. 3 12 2 (b) 2 1 2 12 14. 3x2 2x3 1 15. 12 + (2 12 ) 16. 17. 18. 19. (+1)2 2 3 3 2 3 ( 3 + 2)2 cosx + 2 (+)2 20. 5x4 sec2(x5) 21. 100(4x x2)99 (4 2x) 22. 23. 8 3 3 3 1+ 4 2 3 3 (1+)2 (c) DNE (d) 2 (e) 2 (f) 2 24. 3 2 25. 2 2 (24 + 4) 26. 2x(ln10)101 27. 2 2(2 2 +1) 2 +1 28. y = 7x - 5 1 29. y = 12 x + 109 12 30. a. v(t) = 4t3 6t2 +2t 1; a(t) = 12t2 12t + 2 b. 2 m/s2 31. y = 20x + 1 2 6 2 2+ 3 2 3 2 3()2 2+ 2 2 2 +1 32. - 33. 34. 35. 36. 37. 700 38. 0 39. 4 5 mg/h 40. a. (1 + ) b. 1. ( 2 +1)4 8 ( (2+1)3 (3 1)5 2 +1 6 15 3 1) 2+1 41. 42. 43. 44. 45. 4ft/s 144 cm3/s 20m2/s 2 a. Increasing on ( 3, 0) (3, ) 46. 47. 48. 49. 50. and Decreasing on (, 3) (0, 3) b. Local maximum value 0 at x = 0 Local minimum value 9 at x = 3 c. Concave up on (,1) (1, ) Concave down on (1, 1). Inflection points x = 1 0 2 50 1.47 cm 1200 2. 3. 4. 5. 6. 51. 1 4 2 1 3 3+ 5x + C 52. 2+ 2 3+ C 53 7 3 54. a. 33 b. 4ln3 + 2 2 55. a. + ln x+ C b. ex + 15 13 + C c. sec + C d. 25t + C 56. 7 2 57. 0 58. 1 ( 4 +1) 4 + 1 + 2( 2 +6) 1 b. 2 + 6 + 2 59. a. 60. 15 4 61. 124.1644 62. (a) f is not differentiable at x = 6 or at x = 8, because the graph has a corner there; and at x = 11, because there is a discontinuity there. (b) f is not continuous at x = 11 because does not exist