AP Calculus AB Winter Break Packet Happy Holidays!

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1 AP Calculus AB Winter Break Packet 04 Happy Holidays!

2 Section I NO CALCULATORS MAY BE USED IN THIS PART OF THE EXAMINATION. Directions: Solve each of the following problems. After examining the form of the choices, decide which is the best of the choices given. In this test: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number.. The graph of a piecewise-linear function f, for -3 x 5, as shown. What is the 5 value of f ( x ) dx? 3 (A) 6 (B) 3 (C) 4 (D) (E) -. What is the equation of the line tangent to the graph of f ( x) = 3x 5cos x at x = 0? A) x= 5x 3 B) y= 5x+ 3 C) y= 3x+ 5 D) y= 3x 5 E) y= x 5

3 3. dy x y+ y x The slope field for the differential equation = dx 3x+ y segments when (A) x = 0 or y = 0 (B) y= -x (C) y= -3x (D) y= 5 (E) x = 0, or y = 0, or y = -x will have horizontal 4. The production rate of cola, in thousands of gallons per hour, at the production plant on July is shown in the graph. Of the following, which best approximates the total number of thousands of gallons of cola that were produced that day? (A) 800 (B) 400 (C) 4800 (D) 5000 (E) If f ( x) = ( x ) cos x, then f '(0) = (A) - (B) - (C) 0 (D) (E)

4 6. 3x+ 4 The equation of the tangent line to the curve y= at the point (, 7) is 4x 3 (A) y+ 5x= 3 (B) y 3x= 4 (C) y 7x= 0 (D) y+ 5x= (E) y 5x= 8 7. What are all values of k for which k x 5 dx = 0? (A) - (B) 0 (C) (D) - and (E) -, 0, and 8. A particle moves along the x-axis so that at any time t its position is given by π x( t) = sin t+ cos( t). What is the acceleration of the particle at t=? (A) 0 (B) (C) 3 (D) 5 (E) 7 9. x lim x x x+ 6x (A) -3 is (B) (C) (D) 3 (E) b 0. If g ( x ) dx= 4 a+ b b, then ( ( ) 7) a g x + dx= a (A) 8b a (B) 8b + a (C) 8b - 3a (D) 7b 7a (E) 4a + b +7

5 Questions - refer to the following situation. A spider begins to crawl up a vertical blade of grass at a time t= 0. The velocity v of the spider at time t, 0 t 8, is given by the function whose graph is shown.. At what value of t does the spider change direction? (A) 3 (B) 4 (C) 5 (D) 7 (E) 8. What is the total distance the spider traveled from t = 0 to t = 8? (A) 3 (B) 8 (C) 9 (D) 0 (E) 5 3. Let f be a differentiable function such that f(5) = 3 and f (5) =. If the tangent line to the graph of f at x = 5 is used to find an approximation to a zero of f, that approximation is (A) 6.5 (B) 4.3 (C) 3.5 (D) 0.5 (E) 0.3

6 4. If x = 5 y, what is the value of (A) 5 64 (D) 5 64 (B) (E) 4 3 d y 7 64 dx at the point (3, 4)? (C) 7 64 dy 3 5. If sin x dx =, what is d y dx? (A) 3 3x cos x (B) 3 3x cos x (C) x cos(3 x ) (D) x cos(3 x ) (E) 3 cos x 6. The average rate of change of f ( x) = sin x on the interval [-, 4] is (A) cos 4+ cos 6 (B) sin sin 4 (C) sin 4 sin 6 (D) sin 4+ sin 6 (E) cos 4 cos 6 7. Evaluate ln x x lim x 3 (A) 0 (B) 3 e (C) e (D) 3 (E) The limit does not exist.

7 8. The expression sin sin sin 3... sin approximation for (A) (D) x dx 0 30 (B) sin 30 xdx 0 (E) sin xdx (C) sin xdx is a Riemann sum sin x dx 9. Let ( x+ h) x f ( x) = lim. For what value of x does f(x) = 4? h 0 h (A) - (B) - (C) (D) (E) 4 0. Let this function? I. 3x 5, f ( x) = 6x+, lim f ( x) exists x x. Which of the following are true statements about x > II. III. lim f '( x) exists x f '() exists (A) None (B) II only (C) III only (D) II and III (E) I, II, and III. If the line y= 4x+ 3 is tangent to the curve y= x + c then c is (A) (B) 4 (C) 7 (D) (E) 5

8 . If C( x ) gives the cost in dollars of producing x items of a certain product, which of the following statements are true about dc, the derivative of ( ) dx C x? I. The units of dc dx II. additional III. The value of dc dx item. dc dx are dollars per item. at any value of x is the cost of producing one is the rate at which items are produced. (A) I only (B) II only (C) III only (D) I and II only (E) I, II, and III 4 3. The function f defined by f ( x) = x x has a relative minimum at x = (A) (B) (C) (D) (E) 0 4. Find the slope of the line normal to the curve y = x+ 4 at the point where x = 0. (A) -4 (D) 4 (B) (E) 4 4 (C) 8 5. A particle moves along the x-axis so that its acceleration at any time t is a( t) = t 3. If the initial velocity of the particle is -4, at what time t in the time interval 0 t 5 is the particle farthest left? (A) 0 (B) 3 (C) 3 (D) 4 (E) 5

9 6. 0 x dx= (A) 0 (B) (C) (D) (E) 3 7. The function f is continuous at the point (c, f(c) ). Which of the following statements could be false? (A) lim f ( x ) exists (B) lim f ( x ) = f ( c ) x c x c (C) lim f ( x) = lim f ( x) (D) f ( c ) is defined (E) f '( c ) exists x c x c+ 8. If xy+ x = 6, then the value of dy dx at x= is (A) -7 (B) - (C) 0 (D) (E) 3

10 Section II A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS IN THIS PART OF THE EXAMINATION. Directions: Solve each of the following problems. After examining the form of the choices, decide which is the best of the choices given. In this test:. The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value.. Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. 9. A particle moves along the x-axis so that its position at any time t 0 is given by x( t) = t t. The particle is at rest when t = + 4 (A) 0 (B) 4 (C) (D) (E) 4 30.

11 3. Let f ( x) = x. If the rate of change of f at x = c is four times its rate of change at x =, then c = (A) 6 (B) (D) (E) 3 (C) 3. At time t 0, the acceleration of a particle that is moving along the x-axis is a( t) = t+ sin t. At t = 0, the velocity of the particle is -4. For what value of t will the velocity of the particle be zero? (A) 0 (B).0 (C).78 (D).3 (E) 3.87 x 33. Let f ( x) = h( t) dt, where h has the graph shown below. Which of the following could be the graph of f? a

12 34. Which of the following are antiderivatives of f ( x) = 4sin x cos x? I. F( x) = cos x II. F( x) = sin x III. F( x) = cos x (A) I only (B) II only (C) III only (D) I and II (E) I, II, and III 35. Let f be a function such that f "( x ) < 0 for all x in the closed interval [3, 4], with selected values shown in the table. Which of the following must be true about f '(3.3)? x f(x) (A) f '(3.3) < 0 (B) 0 < f '(3.3) <.6 (C).6 < f '(3.3) <.8 (D).8 < f '(3.3) <.0 (E) f '(3.3) >.0 36.

13 37. The position of an object oscillating on the x-axis is given by x( t) = sin 4t cos 4t, where t is the time in seconds. In the first 5 seconds, how many times is the velocity of the object equal to 0? (A) 0 (B) 4 (C) 5 (D) 6 (E) Let f be a continuous function on the closed interval [-, 5]. If f ( ) = 3 and f (5) = 7, then the Intermediate Value Theorem guarantees that (A) 7 f ( x) 3 for all x between - and 5 0 (B) f '( c ) = for at least one c between - and 5 7 (C) f ( c ) = 3 for at least one c between - and 5 (D) f ( c ) = 0 for at least one c between -7 and 3 (E) f ( x ) is continually decreasing between - and 5 39.

14 Find all values c that satisfy the Mean Value Theorem for the function f ( x) = ( + x ) on the interval [-, ]. (A) (B) (C) 0.0 and.80 (D) and.449 (E) None exist in the interval 4. If the function f is differentiable at the point ( a, f ( a )), then which of the following are true? f ( a+ h) f ( a) I. f '( a) = lim h 0 h II. f '( a) = f a f a h lim h 0 ( ) ( ) h III. f '( a) = lim h 0 f ( a+ h) f ( a h) h (A) I only (B) I and II only (C) I and III only (D) II and III only (E) I, II, III

15 43. In the right triangle shown, θ is increasing at a constant rate of radians per minute. In units per minute, at what rate is x increasing when x =? x 3 θ (A) (B) 4 (C) 5 (D) 0 (E) Boyle s Law states that if the temperature of a gas remains constant, then the pressure P and the volume V of the gas satisfy the equation PV = c where c is a constant. If the volume is decreasing at the rate of 0 in 3 per second, how fast is the pressure increasing when the pressure is 00 lb/in and the volume is 0 in 3? lb / in lb / in lb / in (A) 5 (B) 0 (C) 50 sec sec sec (D) 00 lb / in sec (E) 500 lb / in sec 45. Suppose the continuous function f is defined on the closed interval [0, 3] such that x its derivative f ' is defined by f '( x) = e sin( x ). Which of the following are true about the graph of f? I. f has exactly one relative maximum point II. f has two relative minimum points III. f has two inflection points. (A) I only (B) II only (C) III only (D) I and II only (E) I, II, III