π 2 + h 1 is A) 1 B) 1 C) 0 D) E) none of these

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1 Part A (NO CALCULATORS). lim [] (where [] is the greatest integer in ) is A) B) C) D) E) noneistent lim. h n. = n+ 4 n= sin π + h is A) B) C) D) E) none of these h A) 4 B) C) D) 4 E) diverges 4. The equation of the tangent to the curve y 4 = at the point (,) is A) y = B) y = C) 4y = D) y + = E) 4y + 5 = 5. The nth term of the Taylor series epansion about = of the function f () = A) () n B) n C) n D) ( ) n () n E) ( ) n () n + is 6. When the method of partial fractions is used to decompose + 4, one of the fractions obtained is A) B) C) D) E) 7. A relative maimum value of the function y = ln is A) B) e C) e D) e E) none of these. e 8. When a series is used to approimate d, the value of the integral, to two decimal places, is A). B). C).5 D).8 E).5

2 . A particular solution of the differential equation whose slope field is shown above contains point P. This solution may also contain which other point (A,B,C,D,E)? Answer: D. Let F() = dt t. Which is (are) true? 5 I. The domain of F is ±. II. F() >. III. F is concave upward. A) none B) I only C) II only D) III only E) II and III only. As the tides change, the water-level in a bay varies sinusoidally. At high tide today at 8 A.M., the waterlevel was 5 feet; at low tide, 6 hours later at P.M., it was feet. How fast, in ft. per hr., was the waterlevel dropping at noon today? A) B) π C) D) π E) 6. Let f(t) dt = sin π then f () = A) π B) C) D) E) π u e. du is equal to A) ln ( + e u ) + C B) u + e ln + eu + C C) tan + C D) tan + C E) u u e e tan u e + C 4. Given f() = log and log ().86, which is closest to f ' ()? A).4 B).86 C). D).4 E) 5. If G() = 5 and G ' () =, then an estimate of G(.) using local linearization is approimately A) 5.4 B) 5.5 C) 5.8 D) 8.8 E).8 6. The area bounded by the parabola y = and the lines y = and y = equals A) 8 B) 84 C) 64 4 D) E)

3 7. The first-quadrant region bounded by y =, y =, = q (<q<), and = is rotated about the -ais. The volume obtained as q + equals A) π B) 4π C) π D) 4π E) none of these 8. A curve is given parametrically by the equations = sin t and y = cos t. The length of the arc from t = to t = π is A) π B) π C) + π D) π E) 4π. Suppose the function f is both increasing and concave up on a b. Then, using the same number of subdivision, and with L, R, M, and T denoting respectively Left, Right, Midpoint, and Trapezoid sums, it follows that A) R T M L B) L T M R C) R M T L D) L M T R E) none of these. Which of the following statements about the graph of y = are true? I. The graph has no horizontal asymptote. II. The line = is a vertical asymptote. III. The line y = + is an oblique asymptote. A) I only B) II only C) I and II only D) I and III only E) all three. The only function that does not satisfy the Mean Value Theorem on the interval specified is A) f() = on [,]. B) f() = on [,]. C) f() = + on [,]. D) f() = + on [,]. E) f() = / on,.. e d = A) e B) e C) e D) e E) 4e. A cylindrical tank is partially full of water at time t =, when more water begins flowing in at a constant rate. The tank becomes half full when t = 4, and is completely full when t =. Let h represent the height of the water at time t. During which interval is dh dt increasing? A) never B) < t < 4 C) < t < 8 D) < t < E) 4 < t <

4 4. The graph of function f consists of quarter-circles. Which of the following is equivalent to f() d? I. f() d II. f() d III. 4 f() d 4 A) I only B) II only C) III only D) I and II only E) all of these 5. The base of a solid is the first-quadrant region bounded by y = 4 4, and each cross-section perpendicular to the -ais is a semicircle with a diameter in the y-plane. The volume of the solid is A) π 4 d B) π 8 4 d C) π 8 4 d D) π 4 (4 y 4 ) dy 4 4 (4 y 4 ) dy E) π 8 6. The average value of f() = + on the interval [,4] is A) B) C) 4 D) 5 E) 6 7. The area inside the circle r = sin θ and outside the cardioid r = + sin θ is given by π/ π/ A) [ sin θ ( + sin θ) ] dθ B) ( sin θ ) dθ C) 5 (8 sin θ ) dθ D) π 4 5 ( + sin θ) dθ E) none of these 6, 6 8. Let f() = 6, = 6 Which of the following statements is (are) true? I. f is defined at = 6. II. lim 6 f () eists. III. f is continuous at = 6. A) I only B) II only C) I and II only D) I, II, and III E) none of the statements

5 PART B (CALCULATORS ARE ALLOWED). Two objects in motion from t = to t = seconds have positions, (t) = cos (t + ) and (t) = et t respectively. How many times during the three seconds do the objects have the same velocity? A) B) C) D) E) 4 f ''() The table above shows values of f '' () for various values of. The function f could be A) a linear function B) a quadratic function C) a cubic function D) a fourth-degree function E) an eponential function. Where, in the first quadrant, does the rose r = sin θ have a vertical tangent? A) nowhere B) θ =. C) θ =.47 D) θ =.5 E) θ =.6. A cup of coffee placed on a table cools at a rate of dh dt =.5(H 7) degrees per minute, where H represents its temperature and t is time in minutes. If the coffee was at F initially, what will its temperature be minutes later? A) 7 F B) 5 F C) F D) 8 F E) 4 F. An investment of $4 grows at the rate of e.8t dollars per year after t years. Its value after years is approimately A) $4 B) $8 C) 7 D) $, E) none of these 4. The sketch shows the graphs of f () = 4 5 and the line = k. The regions labeled A and B have equal areas if k = A) 5 B) C) 7.8 D) 8 E)

6 The graph above is for questions 5 and 6. It shows the velocity of an object during the interval t. 5. The object attains its greatest speed at t = A) B) C) 5 D) 6 E) 8 6. The object was at the origin at t =. It returned to the origin A) at t = 5 B) at t = 6 C) during 6 < t < 7 D) at t = 7 E) during 7< t < 8 7. An object in motion in the plane has acceleration vector a (t) = (sint, e t ) for t 5. It is at rest when t =. What is the maimum speed it attains? A). B).44 C).7 D).58 E).6 8. If we replace by u, then 6 d is equivalent to A) u du u + B) u du u C) u u + du D) u du u + E) u u + du n. The set of all for which the power series converges is n n= ( n + ) i A) {-,} B) < C) > D) < E) < 4. A particle moves along a line with acceleration a = 6t. If, when t =, v =, then the total distance traveled between t = and t = equals A) B) 8 C) 7 D) 6 E) none of these 4. The definite integral point (,), then an equation describing the curve is A) y = ln + B) y = + ln + C) y = D) + d represents the length of an arc. If one end of the arc is at the y = + E) 4. Suppose f () =, f ' () = 5, and f '' () =. Then d d (f ()) at = is equal to A) B) C) D) 8 E) 4 y = + + 8

7 4. Which statement is true? A) If f () is continuous at = c, then f ' (c) eists. B) If f ' (c) =, then f has a local maimum or minimum at (c,f (c)). C) If f '' (c) =, then f has an inflection point at (c,f (c)). D) If f is differentiable at = c, then f is continuous at = c. E) If f is continuous on (a,b), then f maintains a maimum value on (a,b). 44. The graph of f ' is shown above. Which statements about f must be true for a < < b? I. f is increasing II. f is continuous III. f is differentiable A) I only B) II only C) I and II only D) I and III only E) all three 45. After a bomb eplodes, pieces can be found scattered around the center of the blast. The density of bomb fragments lying meters from ground zero is given by N () = + / fragments per square meter. How many fragments will be found within meters of the point where the bomb eploded? A) B) 78 C) 556 D) 7 E) 48