CALCULUS AB SECTION I, Part A Time 60 minutes Number of questions 30 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM.

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1 AP Calculus AB Portfolio Project Multiple Choice Practice Name: CALCULUS AB SECTION I, Part A Time 60 miutes Number of questios 30 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM. Directios: Solve each of the followig problems, usig the available space for scratch work. After examiig the form of the choices, decided which is the best of the choices give ad fill i the correspodig circle o the aswer sheet. No credit will be give for aythig writte i the exam book. Do ot sped too much time o ay oe problem. I this exam: 1) Uless otherwise specified, the domai of a fuctio f is assumed to be the set of all real umbers x for which f(x) is a real umber. ) The iverse of a trigoometric fuctio f may be idicated usig the iverse fuctio otatio f 1 or with the prefix arc (e.g., si 1 x = arcsi x). 1. 4e 3x + x (A) 4 3 e3x + 1 x + C (B) 4 3 e3x + l x + C (C) 4e 3x C x (D) 4e 3x + l x + C. If f(x) = e 4x (4x 1), the f (1) = (A) 8e 4 (B) 0e 4 (C) 4e 4 (D) 3e 4 3. (8x 3 4x) = 1 (A) -8 (B) 4 (C) 6 (D) 4

2 4. x 3 (x 4 + 1) 8 = (A) 1 4 (x4 + 1) 8 + C (B) 1 4 x4 x + x 8 + C (C) 1 7 (x4 + 1) 9 + C (D) 9 8 (x4 + 1) 9 + C. The values of a cotiuous fuctio f for selected values of x are give i the table. What is the value of the right Riema sum approximatio to 3 4 f(x) usig the subitervals [4, 1], [1, 0], ad [0, 3]? x (A) 76 (B) 113 (C) 180 (D) 07 f(x) Let f be the fuctio defied, where c is a costat. For what value of c, if ay, is f cotiuous at x = 8? f(x) = x3 cos π x 16 for x < 8 x + cx + 8 for x 8 (A) -1 (B) -9 (C) 0 (D) There is o such value of c. 7. The velocity of a particle movig alog the x-axis is give by v(t) = 3t + 1 for time t > 0. What is the average velocity of the particle from time t = 1 to t = 4? (A) 4 3 (B) 3 3 (C) (D) 8. If f(x) = x 3 ad g is a differetiable fuctio of x, what is the derivative of f g(x)? (A) 3 g(x) g (x) (B) 3 g (x) (C) 3x g (x) (D) 3 g(x) 9. lim x x 4 (A) 4 ex is (B) 1 (C) 0 (D) ifiite

3 10. If f (x) = x (x ), the the graph of f is cocave dow for (A) x > (B) x < (C) x < 0 ad x > (E) all real umbers 11. What is the average rate of chage of y = si(4x) o the iterval 0, π 8? (A) 8 π (C) 3 π (B) (D) π 1. d (cos4 (3x )) = (A) si 4 (3x ) (B) 4 si 3 (3x ) (C) 4x si (3x ) (D) 4x cos 3 (3x ) si(3x ) 13. If 4x = y 6 + y, the dy = (A) 0 (C) 8x 6y (B) 8x y (D) 8x 6y The graph of y = f(x) o the closed iterval [0, 4] is show. Which of the followig could be the graph of y = f (x)? (A) (B) (C) (D)

4 1. lim x π cos x+si(x)+1 x π is (A) 1 π (B) 1 π (C) 1 (D) oexistet 16. The graph of the piecewise-defied fuctio f is show i the figure. The graph has a vertical taget lie at x = ad horizotal taget lies at x = 3 ad x = 1. What are all values of x, 4 < x < 3, at which f is cotiuous but ot differetiable? (A) x = 1 (B) x = ad x = 0 (C) x = ad x = 1 (D) x = 0 ad x = A ice sculpture i the form of a sphere melts i such a way that it maitais its spherical shape. The volume of the sphere is decreasig at a costat rate of π cubic meters per hour. At what rate, i square meters per hour, is the surface area of the sphere decreasig at the momet whe the radius is meters? (Note: For a sphere of radius r, the surface area is 4πr ad the volume is 4 3 πr3.) (A) 4π (B) 40π (C) 80π (D) 100π 18. Show is a slope field for which of the followig differetial equatios? (A) dy = xy + x (B) dy = xy + y (C) dy = y + 1 (D) dy = (x + 1)

5 19. Let f be the piecewise-liear fuctio defied above. Which of the followig statemets are true? (A) Noe I. lim x 0 f(3+h) f(3) h II. (C) I ad II oly lim h 0 + f(3+h) f(3) h III. f (3) = x for x < 3 f(x) = x 4 for x 3 (B) II oly = = (D) I, II, ad III 0. If f(x) = (A) (C) 1 x l 1 1+l l t dt for x 1, the f () = (B) (D) 1 1+l 1 1+l 8 1. Which of the followig limits is equal to x 4 3 (A) lim x 3 + k 4 1 k=1 (B) lim x 3 + k 4 k=1 (C) lim x 3 + k 4 1 k=1 (D) lim x 3 + k 4 k=1?. Let y = f(t) be a solutio to the differetial equatio dy = ky, where k is a costat. Values of f for selected values of t are give i the table. Which of the followig is a expressio for f(t)? (A) 4e t l 3 (B) e t l (C) t + 4 (D) 4t Let g be a cotiuous fuctio. Usig the substitutio u = x + 9, the itegral g(x + 9) 0 which of the followig? (A) g(u) du 0 (B) 1 g(u) du 0 (C) (D) 1 g(u) du g(u) du dt is equal to

6 4. The fuctio f is defied by f(x) = 6x 3x +. If g is the iverse fuctio of f ad g(4) = 1, what is the value of g (4)? (A) 1 1 (C) 1 9 (B) 9 (D) x +x 1 = x+4 (A) + 11 l 16 (B) 4 3 (C) 3 + l 4 7 (D) Let y = f(x) be a twice-differetiable fuctio such that f() = 1 ad dy = y4. What is the value of d y at x =? (A) 4 (B) 4 (C) 14 (D) 0 7. If f(x) = si 1 x, the f = (A) π 4 (B) π (C) (D) 1 8. Let f be a differetiable fuctio such that f(3) = 1 ad f (3) =. What is the approximatio for f(3.1) foud by usig the lie taget to the graph of f at x = 3? (A) 0. (B) 1. (C).3 (D) Let g be the fuctio defied by f(x) = x 3 + 3x. How may relative extrema does f have? (A) Zero (B) Oe (C) Two (D) Three 30. Let f ad g be fuctios give by f(x) = e x ad g(x) = 1. Which of the followig gives the area of the x regio eclosed by the graphs of f ad g betwee x = 1 ad x =? (A) e e l (B) l e + e (C) e 1 (D) e e 1

7 CALCULUS AB SECTION I, Part B Time 4 miutes Number of questios 1 A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON THIS PART OF THE EXAM. Directios: Solve each of the followig problems, usig the available space for scratch work. After examiig the form of the choices, decide which is the best of the choices give ad place the letter of your choice i the correspodig box o the aswer sheet. No credit will be give for aythig writte i this exam booklet. Do ot sped too much time o ay oe problem. I this exam: (1) The exact umerical value of the correct aswer does ot always appear amog the choices give. Whe this happes, select from amog the choice the umber that best approximates the exact umerical value. () Uless otherwise specified, the domai of a fuctio f is assumed to be the set of all real umbers x for which f(x) is a real umber. (3) The iverse of a trigoometric fuctio f may be idicated usig the iverse fuctio otatio f 1 or with the prefix arc (e.g., si 1 x = arcsi x). 76. The derivative of a fuctio f is give by f (x) = e si x cos x 1 for 0 < x < 9. O what itervals is f decreasig? (A) 0 < x < ad 4.11 < x < (B) 0 < x < ad.744 < x < 8.30 (C) < x < 4.11 ad < x < 9 (D) < x <.744 ad 8.30 < x < The temperature of a room, i degrees Fahreheit, is modeled by H, a differetiable fuctio of the umber of miutes after the thermostat is adjusted. Of the followig, which is the best iterpretatio of H () =? (A) The temperature of the room is degrees Fahreheit, miutes after the thermostat is adjusted. (B) The temperature of the room icreases by degrees Fahreheit durig the first miutes after the thermostat is adjusted. (C) The temperature of the room is icreasig at a costat rate of degree Fahreheit per miute. (D) The temperature of the room is icreasig at a rate of degrees Fahreheit per miute, miutes after the thermostat is adjusted.

8 78. A fuctio f is cotiuous o the closed iterval [, ] with f() = 17 ad f() = 17. Which of the followig additioal coditios guaratees that there is a umber c i the ope iterval (, ) such that f (c) = 0? (A) No additioal coditios are ecessary. (B) f has a relative extremum o the ope iterval (, ). (C) f is differetiable o the ope iterval (, ). (D) f(x) exists. 79. A rai barrel collects water off the roof of a house durig three hours of heavy raifall. The height of the water i the barrel icreases at the rate of r(t) = 4t 3 e 1.t feet per hour, where t is the time i hours sice the rai bega. At time t = 1 hour, the height of the water is 0.7 foot. What is the height of the water i the barrel at time t = hours? (A) ft (B) 1.00 ft (C) 1.67 ft (D).111 ft 80. A race car is travelig o a straight track at a velocity of 80 meters per secod whe the brakes are applied at time t = 0 secods. From time t = 0 to the momet the race car stops, the acceleratio of the race car is give by a(t) = 6t t meters per secod per secod. Durig this time period, how far does the race car travel? (A) m (B) m (C) m (D) m 81. The graph of f, the derivative of f, is show. Which of the followig statemets is true? (A) f does ot exist at x =. (B) f is decreasig o the iterval (4, 6). (C) The graph of f has a poit of iflectio at x =. (D) f has a local maximum at x = 1.

9 8. A particle moves alog the x-axis so that its positio at time t > 0 is give by x(t) ad = dt 4t + 8t 3 t. The acceleratio of the particle is first zero whe t = (A) (B) 0.41 (C) (D) Let y = f(x) defie a twice-differetiable fuctio ad let y = t(x) be the lie taget to the graph of f at x = 3. If t(x) f(x) for all real x, which of the followig must be true? (A) f(3) 0 (B) f (3) 0 (C) f (3) 0 (D) f (3) The first derivative of the fuctio f is give by f (x) = cos(x ) e x. At which of the followig values of x does f have a local miimum? (A) 0.4 (B) 1.0 (C).061 (D) A vase has the shape obtaied by revolvig the curve y = cos x + from x = 0 to x = 3 about the x-axis, where x ad y are measured i iches. What is the volume, i cubic iches, of the vase? (A) (B) 0.74 (C) (D) The table gives selected values of a fuctio f. The fuctio is twice-differetiable with f (x) > 0. Which of the followig could be the value of f (3)? x 1 3 f(x) (A) 0.7 (B) 1.0 (C) 1.8 (D) Let f ad g be cotiuous fuctios such that f(x) 0 What is the value of ( 1 f(x) g(x))? 0 3 (A) -1 (B) -9 (C) 6 (D) 1 8 = 1, f(x) 0 = 3, ad g(x) = A particle moves alog the x-axis so that its velocity at time t 0 is give by v(t) = cos(x) e x. What is the total distace traveled by the particle from t = 0 to t =? (A).489 (B) 1.3 (C) (D) 7.41

10 89. If f (x) = si (x 3 ) ad f(1) =, the f(3) = (A) 0.3 (B) (C) (D) The rate at which motor oil is leakig from a automobile is modeled by the fuctio L defied by L(t) = 4 + si (t 3 ) for time t 0. L(t) is measured i liter per hour, ad t is measured i hours. How much oil leaks out of the automobile durig the first 4 miutes? (A).001 (B) 6.38 (C) (D).647

Math 105: Review for Final Exam, Part II - SOLUTIONS

Math 105: Review for Final Exam, Part II - SOLUTIONS Math 5: Review for Fial Exam, Part II - SOLUTIONS. Cosider the fuctio f(x) = x 3 lx o the iterval [/e, e ]. (a) Fid the x- ad y-coordiates of ay ad all local extrema ad classify each as a local maximum