Study guide for the Math 115 final Fall 2012

Transcription

1 Study guide for the Math 115 final Fall 2012 This study guide is designed to help you learn the material covered on the Math 115 final. Problems on the final may differ significantly from these problems in their details, but they will not include material not covered by these problems. There are 64 review problems. If you can do these problems, you should do well on the final. Starred problems (*) are for extra credit. 1. Let g(x) = e x4 x. What is g (x)? (a) e x4 x 1 (b) (x 4 x)e x4 x 1 (c) e 4x3 1 (d) (4x 3 1)e 4x3 1 (e) (4x 3 1)e x4 x 2. Let g(x) = ln(x 4 x). What is g (x)? (a) 1 x 4 x (b) (4x 3 1) ln(x 4 x) (c) 4x3 1 x (d) 4x3 1 x 4 x 3. Let g(x) = e x ln x. What s g (x)? (a) e x + e x ln x (b) ex x + ln x (c) ex x + ex ln x (d) ex +ln x e x ln x * 4. Let g(x) = x x2. What s g (x)? [Hint: use logarithmic differentiation.] (a) 2xx x2 (b) x 2 x x2 1 (c) x+2x ln x (d) x 2 ln x(x+2x ln x) (e) x x2 (x+2x ln x) 5. Let x > 0. Which of the following equals ln(3x 2 )? (a) 2 ln 3x (b) 2 ln 3 + 2x (c) 2 ln 3 ln x (d) ln ln x (e) 2 ln 3 + ln x 6. Which of the following equals ln e3 e 4 e x? (a) 12 x x (b) 12 x (c) 7 x (d) 7 x 7. Which of the following does NOT equal 2x? (a) ln e 2x (b) e ln 2x (c) e ln x + ln e x (d) ln((e x ) 2 ) (e) (e ln x ) 2 (f) (e ln 2 )(e ln x ) (g) e ln 3x ln e x 8. e (a+2) ln b = (a) a + b + 2 (b) ab + 2b (c) a b 2 b (d) b a b 2 9. If e 2x = 6, what s x? (a) ln 2 (b) ln 6 (c) ln 2 6 (d) ln 6 2 (e) 6 1/2 (f) 2 1/6 1

2 10. What is e 30x dx? (a) 1 30 e31x + C (b) 1 30 e30x+1 + C (c) 1 30 e30x + C (d) 30e 30x+1 + C (e) 30e 31x + C (f) 30e 30x + C 11. If ln(x 2 a + 1) = 0, where a > 0, what s x? (a) 0 (b) ±1 (c) ±a (d) ± a (e) there is not enough information to decide 12. If e x3 1600x = 1, what s x? (a) 0 (b) 40 (c) -40 (d) all of (a), (b), (c) (e) none of (a), (b), (c) 13. What is 1 dx, where k is a constant, k 0? kx (a) k ln kx + C (b) k ln x + C (c) ln kx + C (d) ln x + C (e) 1 k ln x + C (f) ln x k. Consider the graph of y = 5s s 5 s 13. What is its horizontal asymptote? 5s4 (a) y = 5 (b) y = 5 (c) y = 5 (d) y = (e) y = 0 (f) y = (g) y = 5 (h) y = 5 (j) y = 5 (k) There is no horizontal asymptote + C 15. Consider the graph of y = s13 5s 4 5s. What is its horizontal asymptote? s5 (a) y = 5 (b) y = 5 (c) y = 5 (d) y = (e) y = 0 (f) y = (g) y = 5 (h) y = 5 (j) y = 5 (k) There is no horizontal asymptote 16. Consider the graph of y = s 5s 5 5s. What is its horizontal asymptote? s5 (a) y = 5 (b) y = 5 (c) y = 5 (d) y = (e) y = 0 (f) y = (g) y = 5 (h) y = 5 (j) y = 5 (k) There is no horizontal asymptote 17. What is the slope of the tangent line to the curve y = e 2x ln(x 2 + 1) at the point (0, 1)? (a) -2 (b) -1 (c) 0 (d) 1 (e) What is the equation of the tangent line to the curve y = ln(x + 1) at the point (0,0)? (a) y = x (b) y = 2x (c) y = x (d) y = 2x (e) no such equation the slope is undefined at (0,0). 2

3 19. The graph of the function g(x) = a2 x ax 3, where a > 0 is a constant. The vertical asymptote a4 is (a) x = a 4 (b) x = 1 a 2 (c) x = 0 (d) x = a (e) x = a + 1 (f) x = a 2 (g) there is no vertical asymptote x How many vertical asymptotes does the graph of y = (x 1)(x 2)(x 3) have? (a) 0 (b) 1 (c) 2 (d) 3 (e) 4 (f) more than If lim x ± f(x) = L then what sort of asymptote(s) must the graph of y = f(x) have? (a) a horizontal asymptote of y = L (b) a vertical asymptote at x = L (c) Can t say, because the information given is not relevant to asymptotes 22. Let f(x) = x If you correctly compute the Riemann sum of f over the interval [0, 1] using two subintervals of equal length and choosing the left endpoint of the interval, what answer do you get? (a) 1 (b) (c) 1.25 (d) 2 (e) (f) If you want to calculate the Riemann sum of x 5 over the interval [2,5] using six subintervals of equal length and choosing the right endpoints, which of the following should you calculate? (a) (2.5) (3.5) (4.5) 5 (b) (2.5) (3.5) (4.5) (c) (2.5) (3.5) (4.5) (d) 1 2 (25 + (2.5) (3.5) (4.5) 5 ) (e) 1 2 (25 + (2.5) (3.5) (4.5) ) (f) 1 2 ((2.5) (3.5) (4.5) ) 24. Let a be a positive constant. The area under the curve y = ax 3 and above the x-axis on the interval [1,2] is (a) 2a (b) 7a (c) 15a (d) 4a (e) 7a (f) 15a Which of the following graphs has a positive first derivative and a negative second derivative in the region shown? (a): (b): (c): (d): 3

4 Questions 26 and 27 concern the graph of the function y = g(x) in the region below; 26. Observe when g is positive, negative, or zero. Then say which of the following statements about g is true: (a) g is never 0 in this region (b) g = 0 at exactly one point in this region (c) g = 0 at exactly two points in this region, and g is negative between those points (d) g = 0 at exactly two points in this region, and g is positive between those points (e) none of the above 27. Observe when g is positive, negative, or zero. Then say which of the following statements about g is true. (a) g (x) > 0 for all x in this region (b) g (x) < 0 for all x in this region (c) g (1) = 0, g (0) > 0, and g (2) < 0 (d) g (1) = 0, g (0) < 0, and g (2) > 0 (e) g (0) = 0, g ( 1) > 0, and g (1) < 0 (f) g (0) = 0, g ( 1) < 0, and g (1) > Consider the graph of the function y = g(x) below: In the region shown, observe when g is positive, negative, or zero. following statements about g is true. (a) if x < 0 then g (x) < 0 and if x > 0 then g (x) > 0 (b) if x 0 then g (x) < 0 (c) if x 0 then g (x) > 0 (d) if x < 0 then g (x) > 0 and if x > 0 then g (x) < 0 Then say which of the 4

5 29. Consider the graph of the following function y = g(x): Where does g fail to have a derivative? (a) at x = 0 only (b) at x = 1 only (c) at x = 1 only (d) at x = 0 and x = 1 only (e) at x = 0 and x = 1 only (f) at x = 1 and x = 1 only (g) at x = 0, x = 1 and x = 1 (h) it has a derivative at every point 30. Consider the following function: Which of the following is its derivative? (a): (b): (c): (d): 31. Which of the following functions is continuous at x = 3? (a) y = x + 3 x 3 { x 3 if x 3 (b) f(x) = x + 3 if x > 3 (c) y = x 3 x + 3 (d) none of the above (e) all of the above Problems 32 through 35 concern the graph of the function y = 10x 9 9x The function has a relative maximum at (a) x = 0 (b) x = 8 9 (c) x = 9 10 (d) x = 1 (e) x = 10 9 (f) x = 9 8 (g) there is no relative maximum 5

6 33. The function has a relative minimum at (a) x = 0 (b) x = 8 9 (c) x = 9 10 (d) x = 1 (e) x = 10 9 (f) x = 9 8 (g) there is no relative minimum 34. At which of the following values is the graph concave up? (a) x = 1 (b) x = 1 2 (c) x = 1 2 (d) x = The graph has (a) no inflection points (b) exactly one inflection point (c) exactly two inflection points (d) exactly three inflection points 36. Consider the graph of y = 1 20 x x4. Where is the graph concave up? (a) on the interval (, 1000) (b) on the interval (, 0) (c) on the interval ( 1000, 0) (d) on the interval (0, ) (e) on the interval (1000, ) (f) on the interval (0, 1000) 37. Consider the function y = x 170 3x 80 + x 21 5x in the interval [0,3]. Which of the following statements about its absolute maximum or absolute minimum is true? [Hint: don t even think about calculating anything.] (a) it has no absolute maximum or absolute minimum in this interval. (b) it has an absolute maximum but no absolute minimum in this interval (c) it has no absolute maximum but does have an absolute minimum in this interval (d) it has an absolute maximum and an absolute minimum in this interval. 38. Consider the function y = 1000 in the interval [3,10]. Which of the following statements about x 2 its absolute maximum or absolute minimum is true? [Hint: don t even think about calculating anything.] (a) it has no absolute maximum or absolute minimum in this interval. (b) it has an absolute maximum but no absolute minimum in this interval (c) it has no absolute maximum but does have an absolute minimum in this interval (d) it has an absolute maximum and an absolute minimum in this interval. Problems 39 and 40 are about the function g(x) = x 16 16x. 39. What is the absolute minimum value of this function in the interval [-1,2]? (a) (b) -17 (c) -15 (d) -1 (e) 0 (f) 1 (g) 15 (h) 17 (i) (j) there is no minimum value in that interval 40. What is the absolute maximum value of this function in the interval [-1,2]? (a) (b) -17 (c) -15 (d) -1 (e) 0 (f) 1 (g) 15 (h) 17 (i) (j) there is no maximum value in that interval 6

7 Problems 41 through 42 are about the following situation: A toy train is moving back and forth from noon to 8 p.m. on a straight track so that t hours after noon it is 36t t 3 inches north of the toy water tower. 41. At noon it is at the water tower. When does it return? (a) at 2 p.m. (b) at 4 p.m. (c) at 6 p.m. (d) at 8 p.m. (e) it never returns. 42. What is its velocity when it returns? (a) -72 in/hr (b) -36 in/hr (c) -18 in/hr (d) 0 in/hr (e) 18 in/hr (f) 36 in/hr (g) 72 in/hr (h) I already told you, it never returns 43. When does its acceleration equal 0? (a) at noon (b) at 2 p.m. (c) at 4 p.m. (d) at 6 p.m. (e) at 8 p.m. (f) its acceleration is never zero 44. At 6:30 p.m. where is it? (a) north of the water tower (b) south of the water tower (c) at the water tower 45. After noon, when does it reach its maximum distance north of the water tower? a) about 1:30 p.m. (b) about 2:30 p.m. (c) about 3:30 p.m. (d) about 4:30 p.m. (e) never 46. Between noon and 8 p.m. (including noon and 8 p.m.), when does it reach its maximum distance south of the water tower? a) 2 p.m. (b) 4 p.m. (c) 6 p.m. (d) 8 p.m. (e) never Problems 47 and 48 are about the following situation: The weekly revenue function for selling high-end beach umbrellas is R(x) = 0.02x x 47. How many should be sold to maximize the revenue? (a) 1500 (b) 2500 (c) 3000 (d) 4000 (e) How can we know that we ve maximized the revenue? (a) Calculate R at the endpoints and at the critical points. (b) There is only one critical point. (c) Check the second derivative to see if it s negative. (d) Check the second derivative to see if it s positive. (e) There is only one critical point and its second derivative is negative. (f) There is only one critical point and its second derivative is positive. 7

8 Problems 49 and 50 are about the following situation: Edgar owns a horse and a cow. He is going to fence them in according to the diagram below, where one side of the fenced in area is against the barn (hence doesn t need a fence). y x horse cow barn wall The long side of the fence (y) costs $10 a meter. Each of the short sides (x) costs $5 a meter. The total fenced-in area is 600 square meters. 49. What value of x minimizes the cost? (a) x = 10 (b) x = 15 (c) x = 20 (d) x = 30 (e) x = 45 (f) x = 60 (g) the cost cannot be minimized 50. What is this minimum cost? (a) $1000 (b) $808 (c) $750 (d) $650 (e) $615 (f) $600 (g) I already told you, the cost cannot be minimized 51. A box has a square bottom and no top. It has a volume of 108 cubic inches. If you want to minimize the surface area, what should the dimensions be? y x x (a) x = 1, y = 108 (b) x = 2, y = 27 (c) x = 2, y = 54 (d) x = 3, y = 12 (e) x = 3, y = 36 (f) x = 6, y = 3 (g) x = 6, y = Under the control of an alien power, a cylindrical mechanical room whose height is 8 feet has a radius which is decreasing at a constant rate of 1 2 foot per hour. At what rate is the volume changing when its radius is 4 feet? [Hint: V = πr 2 h, where h is the height and r is the radius.] (a) 128π ft 3 (b) 64π ft 3 (c) 32π ft 3 (d) 16π ft 3 (e) 16π ft 3 (f) 32π ft 3 (g) 64π ft 3 (h) 128π ft 3 Problems 53 and 54 concern a right circular cone whose height equals the radius of its base. The formula for the volume of such a cone is V = 1 3 πr What is dv when r = 10 and dr = 0.2? (a) 1000π 3 (b) 1000π (c) 100π 3 (d) 100π (e) 20π 3 (f) 20π 54. In problem #53, what does dv represent? (a) the volume of the cone when r = 10.2 (b) the volume of the cone when r = 9.8 (c) V (10.2) V (10) (d) V (10) V (9.8) (e) an approximation of V (10.2) V (10) (f) an approximation of V (10) V (9.8) 8

9 55. Two trucks start from the same loading dock. One travels due east at 30 miles an hour; another travels due north at 40 miles an hour. After travelling for 6 minutes, the eastbound truck has gone 3 miles and the northbound truck has gone 4 miles. At that moment, how fast is the distance between them increasing? (a) 25 mph (b) 30 mph (c) 40 mph (d) 50 mph (e) not enough information to answer this question Questions 56 and 57 are about the following situation: A riverbank is eroding exponentially so that every year it loses 10% of its soil. 56. How much of its soil will it have in 20 years? (a) about 20% (b) about 12% (c) about 10% (d) about 6.5% (f) about 5.2% 57. How long will it take for it to lose half its soil? (a) about 20 years (b) about 12 years (c) about 10 years (d) about 6.5 years (e) about 5.2 years 58. A population of bacteria is growing exponentially so that it triples every 20 hours. How long does it take to double? (Pick the closest approximation.) (a) 5.7 hours (b) 6.3 hours (c) 8.9 hours (d) 10 hours (e) 12.6 hours 59. Rain is falling so that t hours after noon the rate at which it is falling is t inches per hour. How much rain fell between 1 and 4 p.m.? (a) 1 3 inches (b) 3 inches (c) 7 3 inches (d) 4 3 inches (e) 20 3 inches (f) there is not enough information to tell 60. A hawk is flying straight down towards a mouse at a velocity of v = 3t feet per second, where t is the number of seconds since it started flying straight down. If it starts flying straight down from 200 feet, what is its height h at t seconds? [Hint: h(0) = 200.] (a) h = t 2 (b) h = t 2 (c) h = 1.5t 2 (d) h = 1.5t 2 (e) h = t (f) h = 200 3t (g) h = 3t (h) h = 3t Problems 61 and 62 concern the following situation: A population of fish has been introduced into a lake. Its population at t years after being introduced into the lake is modeled by P = 150e t2 t fish In the first year, i.e., when 0 t 1, what is the maximum population? 2 (a) 175 (b) 150 (c) 130 (d) 116 (e) 97 1 There s no such thing as a fraction of a fish, but P need not be a whole number. In this model we assume that P may be slightly larger than the real population, but it s never smaller 2 Remember, there s no such thing as a fraction of a fish. 9

10 62. In the first year, i.e., when 0 t 1, what is its minimum population? 3 (a) 175 (b) 150 (c) 130 (d) 116 (e) 97 Problems 63 and 64 are about the following situation: An epidemic follows the logistic growth 5 model of I(t) =, k > 0, where I(t) is the number of people (in thousands) who have been 1+2e kt infected t weeks after the epidemic started. * 63. Long-term, about how many people will have been infected? (a) 0 (b) about 2 thousand (c) 3 thousand (d) about 4 thousand (e) about 5 thousand (f) there s no upper limit * 64. If I(7) = 2, what s k? (a) ln 7 (b) ln 2 (c) ln 4 3 (d) ln 7 4/3 (e) ln(4/3) 7 (f) ln 2 7 (g) ln See previous footnote. 10