ࡱ> EZqy GTbjbj  ^{{dK22|l0{.../A8[8 g8zzzzzzz|z9/7A899z./zDDD9./zD9zDDKuhy/0T_7:Nv,z{00{v$F>fFXyy8FSylo808"D88o8o8o8zzBo8o8o80{9999Fo8o8o8o8o8o8o8o8o82 R: 2003 AB Calculus Test Section 1 No Calculator Allowed 1. If EMBED Equation.3 , then  EMBED Equation.3  (A)  EMBED Equation.3  (B)  EMBED Equation.3  (C)  EMBED Equation.3  (D)  EMBED Equation.3  (E)  EMBED Equation.3   EMBED Equation.3  (E) 2.  EMBED Equation.3  (A)  EMBED Equation.3  (B)  EMBED Equation.3  (C)  EMBED Equation.3  (D)  EMBED Equation.3  (E)  EMBED Equation.3   EMBED Equation.3  (D) 3. For EMBED Equation.3 , the horizontal line  EMBED Equation.3  is an asymptote for the graph of the function EMBED Equation.3  Which of the following statements must be true? (A)  EMBED Equation.3  (B)  EMBED Equation.3  for all EMBED Equation.3  (C)  EMBED Equation.3  is undefined (D)  EMBED Equation.3  (E)  EMBED Equation.3   EMBED Equation.3  is a horizontal asymptote for EMBED Equation.3 over  EMBED Equation.3 ; by the definition of a horizontal asymptote  EMBED Equation.3 . (E) 4. If  EMBED Equation.3  then  EMBED Equation.3  (A)  EMBED Equation.3  (B)  EMBED Equation.3  (C)  EMBED Equation.3  (D)  EMBED Equation.3  (E)  EMBED Equation.3   EMBED Equation.3  (D) 5. EMBED Equation.3  (A)  EMBED Equation.3  (B)  EMBED Equation.3  (C)  EMBED Equation.3  (D)  EMBED Equation.3  (E)  EMBED Equation.3   EMBED Equation.3  (D) 6. EMBED Equation.3  (A) 4 (B) 1 (C)  EMBED Equation.3  (D) 0 (E)  EMBED Equation.3   EMBED Equation.3  Or, consider the end behavior of the graph of  EMBED Equation.3  on the right. (C)  7. The graph of  EMBED Equation.3 , the derivative of the function  EMBED Equation.3 , is shown above. Which of the following statements is true about  EMBED Equation.3 ? (A)  EMBED Equation.3  is decreasing for  EMBED Equation.3  (B)  EMBED Equation.3  is increasing for  EMBED Equation.3  (C)  EMBED Equation.3  is increasing for  EMBED Equation.3  (D)  EMBED Equation.3  has a local minimum at  EMBED Equation.3  (E)  EMBED Equation.3  is not differentiable at  EMBED Equation.3  and  EMBED Equation.3  It cannot be (A) since  EMBED Equation.3  for  EMBED Equation.3 ,  EMBED Equation.3  increases It cannot be (C) since  EMBED Equation.3  for  EMBED Equation.3 ,  EMBED Equation.3 decreases It cannot be (D) since  EMBED Equation.3  changes from positive to negative then there is a maximum at x = 0. It cannot be (E) since  EMBED Equation.3  and  EMBED Equation.3  exist. Since the graph of  EMBED Equation.3 , the derivative of  EMBED Equation.3  is positive over EMBED Equation.3 ,  EMBED Equation.3  is increasing over that interval. (B) 8.  EMBED Equation.3  (A)  EMBED Equation.3  (B)  EMBED Equation.3  (C)  EMBED Equation.3  (D) EMBED Equation.3  (E) EMBED Equation.3   EMBED Equation.3  (B) 9. If  EMBED Equation.3 , then  EMBED Equation.3 is (A)  EMBED Equation.3  (B)  EMBED Equation.3  (C) EMBED Equation.3  (D)  EMBED Equation.3  (E) nonexistent  EMBED Equation.3  (A) 10. The function f has the property that  EMBED Equation.3  and  EMBED Equation.3  are negative for all real values x. Which of the following could be the graph of f ?  The only graph that is negative ( EMBED Equation.3  is negative), decreasing ( EMBED Equation.3  is negative), and concave down ( EMBED Equation.3  is negative) over the entire interval is (B.) 11. Using the substitution  EMBED Equation.3  is equivalent to (A)  EMBED Equation.3  (B)  EMBED Equation.3  (C)  EMBED Equation.3  (D)  EMBED Equation.3  (E)  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  (C) 12. The rate of change of the volume,  EMBED Equation.3 , of water in a tank with respect to time,  EMBED Equation.3 , is directly proportional to the square root of the volume. Which of the following is a differential equation that describes this relationship? (A)  EMBED Equation.3  (B)  EMBED Equation.3  (C)  EMBED Equation.3  (D)  EMBED Equation.3  (E)  EMBED Equation.3  The rate of change of the volume is defined by  EMBED Equation.3 . Directly proportional to the square root of the volume indicates the relationship  EMBED Equation.3 , where  EMBED Equation.3  is an arbitrary constant. (E)  13. The graph of a function  EMBED Equation.3  is shown above. At which value of  EMBED Equation.3  is  EMBED Equation.3  continuous, but not differentiable? a (B) b (C) c (D) d (E) e At point a,  EMBED Equation.3 , but the curve is not locally linear at x = a, so it is continuous but not differentiable. (A) 14. If  EMBED Equation.3 , then  EMBED Equation.3  (A)  EMBED Equation.3  (B)  EMBED Equation.3  (C)  EMBED Equation.3  (D)  EMBED Equation.3  (E)  EMBED Equation.3   EMBED Equation.3  (E) 15. Let f be the function with derivative given by  EMBED Equation.3 . On which of the following intervals is f decreasing? (A)  EMBED Equation.3  only (B)  EMBED Equation.3  (C)  EMBED Equation.3 only (D)  EMBED Equation.3  (E)  EMBED Equation.3   f (x) decreases where  EMBED Equation.3  In addition to  EMBED Equation.3 , we must consider x=0 as a critical point as well.  EMBED Equation.3 for x < 0 and  EMBED Equation.3 , for  EMBED Equation.3  (D). 16. If the line tangent to the graph of the function f at the point (1, 7) passes through the point (-2, -2), then  EMBED Equation.3  is (A) -5 (B) 1 (C) 3 (D) 7 (E) undefined Slope of line =  EMBED Equation.3 . Therefore  EMBED Equation.3  (C) 17. Let f be the function given by  EMBED Equation.3 . The graph of f is concave down when (A) x < -2 (B) x > -2 (C) x < -1 (D) x > -1 (E) x < 0  EMBED Equation.3  Never 0 x=-2  EMBED Equation.3  (A) x-4-3-2-101234 EMBED Equation.3 230-3-2-103218. The derivative  EMBED Equation.3  of a function g is continuous and has exactly two zeros. Selected values of  EMBED Equation.3  are given in the table above. If the domain of g is the set of all real numbers, then g is decreasing on which of the following intervals? (A)  EMBED Equation.3  only (B)  EMBED Equation.3  only (C)  EMBED Equation.3  (D)  EMBED Equation.3  only (E)  EMBED Equation.3  or  EMBED Equation.3  g is decreasing when  EMBED Equation.3 . (A) 19. A curve has slope 2x + 3 at each point (x, y) on the curve. Which of the following is an equation for this curve if it passes through the point (1, 2)? (A)  EMBED Equation.3  (B)  EMBED Equation.3  (C)  EMBED Equation.3  (D)  EMBED Equation.3  (E)  EMBED Equation.3   EMBED Equation.3  (D)  EMBED Equation.3  20. Let f be the function given above. Which of the following statements are true about f ? I.  EMBED Equation.3  II. f is continuous at x = 3. III. f is differentiable at x = 3. (A) None (B) I only (C) II only (D) I and II only (E) I, II, and III  EMBED Equation.3   EMBED Equation.3  I and II are true  EMBED Equation.3  from the left is 1 and  EMBED Equation.3  from the right is 4. III is not true. (D)  21. The second derivative of the function f is given by  EMBED Equation.3 . The graph of  EMBED Equation.3 is shown above. For what values of x does the graph of f have a point of inflection? (A) 0 and a only (B) 0 and m only (C) b and j only (D) 0, a, and b (E) b, j, and k At  EMBED Equation.3 changes from positive to negative. At  EMBED Equation.3 changes from negative to positive. At  EMBED Equation.3 there is no change in concavity. (A)  22. The graph of  EMBED Equation.3 , the derivative of f, is the line shown in the figure above. If  EMBED Equation.3 , then f(1) = (A) 0 (B) 3 (C) 6 (D) 8 (E) 11  EMBED Equation.3  (D) 23.  EMBED Equation.3  (A)  EMBED Equation.3  (B)  EMBED Equation.3  (C)  EMBED Equation.3  (D)  EMBED Equation.3  (E)  EMBED Equation.3  By the Fundamental Theorem of Calculus and the chain rule (E) 24. Let  EMBED Equation.3  be the function defined by  EMBED Equation.3 . Which of the following is an equation of the line tangent to the graph of f at the point where x = -1? (A)  EMBED Equation.3  (B)  EMBED Equation.3  (C)  EMBED Equation.3  (D)  EMBED Equation.3  (E)  EMBED Equation.3   EMBED Equation.3  (C) 25. A particle moves along the x-axis so that at time  EMBED Equation.3  its position is given by  EMBED Equation.3 . At what time t is the particle at rest? (A) t = 1 only (B) t = 3 only (C)  EMBED Equation.3  only (D) t = 3 and  EMBED Equation.3  (E) t = 3 and t = 4  EMBED Equation.3  (E) 26. What is the slope of the line tangent to the curve  EMBED Equation.3  at the point (3, 2)? (A)  EMBED Equation.3  (B)  EMBED Equation.3  (C)  EMBED Equation.3  (D)  EMBED Equation.3  (E)  EMBED Equation.3   EMBED Equation.3  (B) 27. Let f be the function defined by  EMBED Equation.3 . If  EMBED Equation.3  and g(2) = 1, what is the value of  EMBED Equation.3 ? (A)  EMBED Equation.3  (B)  EMBED Equation.3  (C)  EMBED Equation.3  (D) 4 (E) 13 To get the inverse function, swap the xs and ys.  EMBED Equation.3  (B) 28. Let g be a twice-differentiable function with  EMBED Equation.3  and  EMBED Equation.3  for all real numbers x, such that g(4) = 12 and g(5) = 18. Of the following, which is a possible value for g(6)? (A) 15 (B) 18 (C) 21 (D) 24 (E) 27 g is increasing and concave up. The increase from x = 5 to x = 6 is 6. The increase from x = 5 to x = 6 must be > 6. (E) 2003 AP Calculus AB Exam Section 2 Calculator Required A particle moves along the  EMBED Equation.3 -axis so that at any time  EMBED Equation.3 , its velocity is given by  EMBED Equation.3 . What is the acceleration of the particle at time  EMBED Equation.3 ? (A) -2.016 (B) -0.677 (C) 1.633 (D) 1.814 (E) 2.978  EMBED Equation.3  (C)  The regions A, B, and C in the figure above are bounded by the graph of the function  EMBED Equation.3 and the  EMBED Equation.3 -axis. If the area of each region is 2, what is the value of  EMBED Equation.3 ? (A) -2 (B) -1 (C) 4 (D) 7 (E) 12  EMBED Equation.3  (C) The radius of a circle is increasing at a constant rate of 0.2 meters per second. What is the rate of increase in the area of the circle at the instant when the circumference of the circle is  EMBED Equation.3  meters? (A) 0.04 EMBED Equation.3  m EMBED Equation.3 /sec (B) 0.4 EMBED Equation.3  m EMBED Equation.3 /sec (C) 4 EMBED Equation.3  m EMBED Equation.3 /sec (D) 20 EMBED Equation.3  m EMBED Equation.3 /sec (E) 100 EMBED Equation.3  m EMBED Equation.3 /sec   Find: EMBED Equation.3  when C =  EMBED Equation.3  (C) For which of the following does  EMBED Equation.3  exist?  (A) I only (B) II only (C) III only (D) I and II only (E) I and III only Since a limit exists only when  EMBED Equation.3 and  EMBED Equation.3 exist and are equal,  EMBED Equation.3  exists only for graphs I and II. The value of f(4) has nothing to do with this problem. (D) The function  EMBED Equation.3  is continuous for  EMBED Equation.3 and differentiable for -2 <  EMBED Equation.3 < 1. If  EMBED Equation.3  and  EMBED Equation.3 , which of the following statements could be false? (A) There exists c, where -2 < c < 1, such that  EMBED Equation.3 . (B) There exists c, where -2 < c < 1, such that  EMBED Equation.3 . (C) There exists c, where -2 < c < 1, such that  EMBED Equation.3 . (D) There exists c, where -2 < c < 1, such that  EMBED Equation.3 . (E) There exists c, where -2  EMBED Equation.3  c  EMBED Equation.3  1, such that  EMBED Equation.3 for all x on the closed interval  EMBED Equation.3 . A and C must be true because of the Intermediate Value Theorem. D must be true by the Mean Value Theorem. E must be true by the Extreme Value Theorem. B is not necessarily true because there might not be a critical point on the interval  EMBED Equation.3 . (B) Let  EMBED Equation.3  be the function with derivative given by  EMBED Equation.3 . How many relative extrema does  EMBED Equation.3 have on the interval  EMBED Equation.3 ? (A) One (B) Two (C) Three (D) Four (E) Five Over the range [2, 4],  EMBED Equation.3  EMBED Equation.3 crosses the  EMBED Equation.3 -axis 4 times. (D) The rate of change of the altitude of a hot-air balloon is given by  EMBED Equation.3  for  EMBED Equation.3 . Which of the following expressions gives the change in altitude of the balloon during the time the altitude is decreasing? (A)  EMBED Equation.3  (B)  EMBED Equation.3  (C)  EMBED Equation.3  (D)  EMBED Equation.3  (E)  EMBED Equation.3   EMBED Equation.3  is negative over the interval [1.572, 3.514]. The change in altitude is given by r(t), so the expression EMBED Equation.3  represents the total change in altitude of the balloon during the time the altitude is decreasing (A). The velocity, in ft/sec, of a particle moving along the x-axis is given by the function  EMBED Equation.3 . What is the average velocity of the particle from time  EMBED Equation.3  to time  EMBED Equation.3 ? (A) 20.086 ft/sec (B) 26.447 ft/sec (C) 32.809 ft/sec (D) 40.671 ft/sec (E) 79.342 ft/sec Average Velocity  EMBED Equation.3   (A) A pizza, heated to a temperature of 350 degrees Fahrenheit (F), is taken out of an oven and placed in a 75F room at time  EMBED Equation.3  minutes. The temperature of the pizza is changing at a rate of  EMBED Equation.3  degrees Fahrenheit per minute. To the nearest degree, what is the temperature of the pizza at time t = 5 minutes? (A) 112F (B) 119F (C) 147F (D) 238F (E) 335F Pizza Temperature at time t = 5: T = EMBED Equation.3  (A) If a trapezoidal sum overapproximates  EMBED Equation.3 , and a right Riemann sum underapproximates  EMBED Equation.3 , which of the following could be the graph of  EMBED Equation.3 ?  If a trapezoidal sum overapproximates the area of the function, the function must be concave down. If the right Reimann sums underapproximate the area the function must be decreasing. (A). The base of a solid is the region in the fist quadrant bounded by the y-axis, the graph of  EMBED Equation.3 , the horizontal line  EMBED Equation.3 , and the vertical line  EMBED Equation.3 . For this solid, each cross section perpendicular to the x-axis is a square. What is the volume of the solid? (A) 2.561 (B) 6.612 (C) 8.046 (D) 8.755 (E) 20.773 Area of a square = base2 Volume of slice = (3 - tan -1 x)(thickness)  EMBED Equation.3  = 6.612 (B) The function f has first derivative given by  EMBED Equation.3 . What is the x-coordinate of the inflection point of the graph of f ? (A) 1.008 (B) 0.473 (C) 0 (D) -0.278 (E) The graph of f has no inflection point.  EMBED Equation.3  Or find the zero for  EMBED Equation.3  with graphing calc. (B) On the closed interval [2, 4], which of the following could be the graph of a function f with the property that  EMBED Equation.3 ?  The equation above is the average value of f(x), so by its definition, the average value of x from 2 to 4 must be 1.  EMBED Equation.3  must = 2. The only graph with area under the curve equal to 2 is (C). Let f be a differentiable function with f (2) = 3 and  EMBED Equation.3 , and let g be the function defined by  EMBED Equation.3 . Which of the following is an equation of the line tangent to the graph of g at the point where x = 2? (A)  EMBED Equation.3  (B)  EMBED Equation.3  (C)  EMBED Equation.3  (D)  EMBED Equation.3  (E)  EMBED Equation.3   Point (2,6) Slope = -7 (D) For all x in the closed interval [2, 5], the function f has a positive first derivative and a negative second derivative. Which of the following could be a table of values for f ?  EMBED Equation.3  EMBED Equation.3 2739412516  EMBED Equation.3  EMBED Equation.3 27311414516  EMBED Equation.3  EMBED Equation.3 2163124957  EMBED Equation.3  EMBED Equation.3 21631441157  EMBED Equation.3  EMBED Equation.3 21631341057(A) (B) (C) (D) (E) Since the first derivative is positive, f must be increasing and the answer must be A or B. If the second derivative is negative, the first derivative is decreasing, meaning that the distance between the y values is decreasing. The answer is (B). A particle moves along the x-axis so that at any time t > 0, its acceleration is given by  EMBED Equation.3 . If the velocity of the particle is 2 at time t = 1, then the velocity of the particle at time t = 2 is (A) 0.462 (B) 1.609 (C) 2.555 (D) 2.886 (E) 3.346  EMBED Equation.3  (E) Let g be the function given by  EMBED Equation.3  for  EMBED Equation.3 . On which of the following intervals is g decreasing? (A)  EMBED Equation.3  (B)  EMBED Equation.3  (C)  EMBED Equation.3  (D)  EMBED Equation.3  (E)  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  is negative for  EMBED Equation.3 . 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