A1. Which of the following functions is continuous at x = 0?

 

A2. Which of the following statements about the graph of  is not true?

(A) The graph is symmetric to the y-axis.
(B) There is no y-intercept.
(C) The graph has one horizontal asymptote.
(D) There is no x-intercept.

A3. 

(A) = –1
(B) = 0
(C) = 1
(D) = 2

A4. The x-coordinate of the point on the curve y = x² – 2x + 3 at which the tangent is perpendicular to the line x + 3y + 3 = 0 is

A5. 

(A) –3
(B) –1
(C) 1
(D) 3

A6. For polynomial function p, p″(2) = –4, p″(4) = 0, and p″(5) = 2. Which must be true?

(A) p has an inflection point at x = 4.
(B) p has a minimum at x = 4.
(C) p has a root at x = 4.
(D) None of (A)–(C) must be true.

A7. 

(A) 6
(B) 8
(C) 10
(D) 12

A8. is

A9. The maximum value of the function f(x) = x⁴ – 4x³ + 6 on [1,4] is

(A) 1
(B) 0
(C) 3
(D) 6

A10. Let , and let f be continuous at x = 5. Then c =

(A) 0
(B) 
(C) 1
(D) 6

A11. 

(A) –1
(B) 
(C) 
(D) 1

A12. If sin x = ln y and 0 < x < π, then, in terms of  equals

(A) e^{sin x} cos x
(B) e^{–sin x} cos x
(C) 
(D) e^{cos x}

A13. If f(x) = x cos x, then  equals
(A) 
(B) 0
(C) –1
(D) 

A14. An equation of the tangent to the curve y = e^x ln x, where x = 1, is

(A) y = ex
(B) y = e(x – 1)
(C) y = ex + 1
(D) y = x – 1

A15. If the displacement from the origin of a particle moving along the x-axis is given by s = 3 + (t – 2)⁴, then the number of times the particle reverses direction is

(A) 0
(B) 1
(C) 2
(D) 3

A16.  equals

(A) 1 – e
(B) e – 1
(C) 
(D) e + 1

A17. If , then  equals

(A) 7
(B) 
(C) 
(D) 9

A18. If the position of a particle on a line at time t is given by s = t³ + 3t, then the speed of the particle is decreasing when

(A) –1 < t < 1
(B) –1 < t < 0
(C) t < 0
(D) t > 0

CHALLENGE A19. A rectangle with one side on the x-axis is inscribed in the triangle formed by the lines y = x, y = 0, and 2x + y = 12. The area of the largest such rectangle is

(A) 6
(B) 3
(C) 
(D) 5

A20. The x-value of the first-quadrant point that is on the curve of x² – y² = 1 and closest to the point (3,0) is

(A) 1
(B) 
(C) 2
(D) 3

A21. If y = ln (4x + 1), then  is

A22. The region bounded by the parabolas y = x² and y = 6x – x² is rotated about the x-axis so that a vertical line segment cut off by the curves generates a ring. The value of x for which the ring of largest area is obtained is

A23.  equals

A24. The volume obtained by rotating the region bounded by x = y² and x = 2 – y² about the y-axis is equal to

A25. The general solution of the differential equation  is a family of

(A) circles
(B) hyperbolas
(C) parabolas
(D) ellipses

A26. Estimate  using a left rectangular sum and two subintervals of equal width.

A27. 

A28. 

A29. 

(A) 0
(B) 
(C) 1
(D) 2

CHALLENGE A30. The number of values of k for which f(x) = e^x and g(x) = k sin x have a common point of tangency is

(A) 0
(B) 1
(C) large but finite
(D) infinite

A31. The curve 2x²y + y² = 2x + 13 passes through (3,1). Use the line tangent to the curve there to find the approximate value of y at x = 2.8.

(A) 0.5
(B) 0.9
(C) 0.95
(D) 1.1

CHALLENGE A32. 

A33. The region bounded by y = tan x, y = 0, and  is rotated about the x-axis. The volume generated equals

 

A34. , for the constant a > 0, equals

(A) 1
(B) a
(C) ln a
(D) a ln a

A35. Solutions of the differential equation whose slope field is shown here are most likely to be

(A) quadratic
(B) cubic
(C) exponential
(D) logarithmic

A36. 

(A) 0
(B) 1
(C) 
(D) 

A37. The graph of g, shown below, consists of the arcs of two quarter-circles and two straight-line segments. The value of  is

A38. Which of these could be a particular solution of the differential equation whose slope field is shown here?

(A) 
(B) y = ln x
(C) y = e^x
(D) y = e^–x

A39. What is the domain of the particular solution, y = f(x), for  containing the point (–1, ln 3)?

(A) x < 0
(B) x > –2
(C) –2 < x < 2
(D) x ≠ ±2

A40. The slope field shown here is for the differential equation

(A) 
(B) y′ = ln x
(C) y′ = e^x
(D) y′ = y²

A41. If we substitute x = tan θ, which of the following is equivalent to 

CHALLENGE *A42. If x = 2 sin u and y = cos 2u, then a single equation in x and y is

(A) x² + 4y² = 4
(B) x² + 2y = 2
(C) x² + y² = 4
(D) x² – 2y = 2

*A43. The area bounded by the lemniscate with polar equation r² = 2 cos 2θ is equal to

(A) 4
(B) 1
(C) 
(D) 2

*A44. 

(A) = 0
(B) 
(C) = π
(D) = 2π

*A45. The first four nonzero terms of the Maclaurin series (the Taylor series about x = 0) for  are

(A) 1 + 2x + 4x² + 8x³
(B) 1 – 2x + 4x² – 8x³
(C) –1 – 2x – 4x² – 8x³
(D) 1 – x + x² – x³

*A46. 

(A) 
(B) –x²e^–x + 2xe^–x + C
(C) –x²e^–x – 2xe^–x – 2e^–x + C
(D) –x²e^–x+ 2xe^–x – 2e^–x + C

CHALLENGE *A47.  is equal to

*A48. If x = a cot θ and y = a sin²θ, then , when , is equal to

(A) 
(B) –2
(C) 2
(D) 

*A49. Which of the following improper integrals diverges?

*A50. equals

CHALLENGE *A51.  is

(A) 0
(B) 1
(C) ∞
(D) nonexistent

*A52. A particle moves along the parabola x = 3y – y² so that  at all time t. The speed of the particle when it is at position (2,1) is equal to

(A) 0
(B) 3
(C) 
(D) 

*A53. 

(A) –∞
(B) –1
(C) 0
(D) 1 

*A54. When rewritten as partial fractions,  includes which of the following?

(A) none
(B) I only
(C) II only
(D) I and III only

*A55. Using three terms of an appropriate Maclaurin series, estimate 

*A56. The slope of the spiral r = θ at  is

A57.  equals

A58. A particle moves along a line with acceleration a = 6t. If, when t = 0, v = 1, then the total distance traveled between t = 0 and t = 3 equals

(A) 30
(B) 28
(C) 27
(D) 26

A59. Air is escaping from a balloon at a rate of  cubic feet per minute, where t is measured in minutes. How much air, in cubic feet, escapes during the first minute?

(A) 15π
(B) 30
(C) 45
(D) 30 ln 2

B1. The graph of function h is shown here. Which of these statements is (are) true?

I. The first derivative is never negative.
II. The second derivative is constant.
III. The first and second derivatives equal 0 at the same point.

(A) I only
(B) III only
(C) I and II only
(D) I and III only

B2. Graphs of functions f(x), g(x), and h(x) are shown below.

Consider the following statements:

I. g(x) = f′(x)
II. f(x) = g′(x)
III. h(x) = g″(x)

Which of these statements is (are) true?

(A) I only
(B) II only
(C) II and III only
(D) I, II, and III

B3. If , then 

B4. If , then 

(A) –6
(B) –5
(C) 5
(D) 6

B5. At what point in the interval [1,1.5] is the rate of change of f(x) = sin x equal to its average rate of change on the interval?

(A) 1.058
(B) 1.239
(C) 1.253
(D) 1.399

B6. Suppose f′(x) = x² (x – 1). Then f″(x) = x (3x – 2). Over which interval(s) is the graph of f both increasing and concave up?

I. x < 0
II. 
III. 
IV. x > 1

(A) I only
(B) II and IV only
(C) I and III only
(D) IV only

B7. Suppose f′(x) = x² (x – 1). Then f″(x) = x (3x – 2). Which of the following statements is true about the graph of f(x)?

(A) The graph has one relative extremum and one inflection point.
(B) The graph has one relative extremum and two inflection points.
(C) The graph has two relative extrema and one inflection point.
(D) The graph has two relative extrema and two inflection points.

B8. The nth derivative of ln (x + 1) at x = 2 equals

B9. If f(x) is continuous at the point where x = a, which of the following statements may be false?

(A) 
(B) f′(a) exists
(C) f(a) is defined
(D) 

B10. Suppose , where k is a constant. Then  equals

(A) 3
(B) 4 – k
(C) 4
(D) 4 + k

CHALLENGE B11. The volume, in cubic feet, of an “inner tube” with inner diameter 4 feet and outer diameter 8 feet is

(A) 6π²
(B) 12π²
(C) 24π²
(D) 48π²

B12. If f(u) = tan^–1u² and g(u) = e^u, then the derivative of f(g(u)) is

B13. If sin(xy) = y, then  equals

(A) y cos(xy) – 1
(B) 
(C) 
(D) cos(xy)

B14. Let x > 0. Suppose  and ; then 

(A) f(x²)
(B) 2xg(x²)
(C) 
(D) 2g(x²) + 4x²f(x)

B15. The region bounded by y = e^x, y = 1, and x = 2 is rotated about the x-axis. The volume of the solid generated is given by the integral

B16. Suppose the function f is continuous on 1 ≤ x ≤ 2, that f′(x) exists on 1 < x < 2, that f(1) = 3, and that f(2) = 0. Which of the following statements is not necessarily true?

(A)  exists.
(B) There exists a number c in the open interval (1,2) such that f′(c) = 0.
(C) If k is any number between 0 and 3, there is a number c between 1 and 2 such that f(c) = k.
(D) If c is any number such that 1 < c < 2, then  exists.

B17. The region S in the figure is bounded by y = sec x, the y-axis, and y = 4. What is the volume of the solid formed when S is rotated about the y-axis?

 

(A) 2.279
(B) 5.692
(C) 11.385
(D) 17.217

B18. If 40 grams of a radioactive substance decomposes to 20 grams in 2 years, then, to the nearest gram, the amount left after 3 years is

(A) 10
(B) 12
(C) 14
(D) 16

B19. An object in motion along a line has acceleration  and is at rest when t = 1. Its average velocity from t = 0 to t = 2 is

(A) 0.362
(B) 0.274
(C) 3.504
(D) 4.249

B20. Find the area bounded by y = tan x and x + y = 2, and above the x-axis on the interval [0,2].

(A) 0.919
(B) 1.013
(C) 1.077
(D) 1.494

CHALLENGE B21. An ellipse has major axis 20 and minor axis 10. Rounded off to the nearest integer, the maximum area of an inscribed rectangle is

(A) 50
(B) 79
(C) 82
(D) 100

B22. The average value of y = x ln x on the interval 1 ≤ x ≤ e is

(A) 0.772
(B) 1.221
(C) 1.359
(D) 2.097

B23. Let  for 0 ≤ x ≤ 2π. On which interval is f increasing?

(A) 0 < x < π
(B) 0.654 < x < 5.629
(C) 0.654 < x < 2π
(D) π < x < 2π

B24. The table shows the speed of an object (in ft/sec) at certain times during a 6-second period. Estimate its acceleration (in ft/sec²) at t = 2 seconds.

 

(A) –10
(B) –6
(C) –5
(D) 

B25. A maple syrup storage tank 16 feet high hangs on a wall. The back is in the shape of the parabola y = x² and all cross sections parallel to the floor are squares. If syrup is pouring in at the rate of 12 ft³/hr, how fast (in ft/hr) is the syrup level rising when it is 9 feet deep?

B26. In a protected area (no predators, no hunters), the deer population increases at a rate of , where P(t) represents the population of deer at t years. If 300 deer were originally placed in the area and a census showed the population had grown to 500 in 5 years, how many deer will there be after 10 years?

(A) 608
(B) 643
(C) 700
(D) 833

B27. Shown is the graph of .

Let . The local linearization of H at x = 1 is

(A) y = 2x
(B) y = –2x – 4
(C) y = 2x + π – 2
(D) y = –2x + π + 2

B28. A smokestack 100 feet tall is used to treat industrial emissions. The diameters, measured at selected heights, are shown in the table. Using a left Riemann Sum with the 4 subintervals indicated in the table, estimate the volume of the smokestack to the nearest cubic foot.

(A) 1160
(B) 5671
(C) 8718
(D) 11765

For Questions B29–B33, the table shows the values of differentiable functions f and g.

B29. If , then P′(3) =

(A) –2
(B) 
(C) 
(D) 2

B30. If H(x) = f(g(x)), then H′(3) =

(A) 1
(B) 2
(C) 3
(D) 6

B31. If M(x) = f(x) · g(x), then M′(3) =

(A) 2
(B) 6
(C) 8
(D) 16

B32. If K(x) = g^–1(x), then K′(3) =

B33. If , then R′(3) =

B34. Water is poured into a spherical tank at a constant rate. If W(t) is the rate of increase of the depth of the water, then W is

(A) linear and increasing
(B) linear and decreasing
(C) concave up
(D) concave down

B35. The graph of f′ is shown below. If f(7) = 3 then f(1) =

(A) -10
(B) –4
(C) 10
(D) 16

B36. At an outdoor concert, the crowd stands in front of the stage filling a semicircular disk of radius 100 yards. The approximate density of the crowd x yards from the stage is given by

people per square yard. About how many people are at the concert?

(A) 200
(B) 19500
(C) 21000
(D) 165000

B37. The Centers for Disease Control announced that, although more AIDS cases were reported this year, the rate of increase is slowing down. If we graph the number of AIDS cases as a function of time, the curve is currently

(A) increasing and concave down
(B) increasing and concave up
(C) decreasing and concave down
(D) decreasing and concave up

The graph below is for Questions B38–B40. It shows the velocity, in feet per second, for 0 < t < 8, of an object moving along a straight line.

B38. The object’s average speed (in ft/sec) for this 8-second interval was

B39. When did the object return to the position it occupied at t = 2?

(A) t = 6
(B) t = 7
(C) t = 8
(D) never

B40. The object’s average acceleration (in ft/sec²) for this 8-second interval was

B41. If a block of ice melts at the rate of , how much ice melts during the first 3 minutes?

(A) 8 cm³
(B) 16 cm³
(C) 21 cm³
(D) 40 cm³

*B42. A particle moves counterclockwise on the circle x² + y² = 25 with a constant speed of 2 ft/sec. Its velocity vector, v, when the particle is at (3,4), equals

*B43. Let R = 〈a cos kt,a sin kt〉 be the (position) vector from the origin to a moving point P(x,y) at time t, where a and k are positive constants. The acceleration vector, a, equals

(A) –k²R
(B) a²k²R
(C) –a^R
(D) –ak² 〈a cos kt,a sin kt〉

B44. The length of the curve y = 2^x between (0,1) and (2,4) is

(A) 3.141
(B) 3.664
(C) 4.823
(D) 7.199

*B45. The position of a moving object is given by P(t) = (3t,e^t). Its acceleration is

(A) constant in both magnitude and direction
(B) constant in magnitude only
(C) constant in direction only
(D) constant in neither magnitude nor direction

*B46. Suppose we plot a particular solution of  from initial point (0,1) using Euler’s method. After one step of size Δx = 0.1, how big is the error?

(A) 0.09
(B) 1.09
(C) 1.49
(D) 1.90

*B47. We use the first three terms to estimate . Which of the following statements is (are) true?

I. The estimate is 0.7.
II. The estimate is too low.
III. The estimate is off by less than 0.1.

(A) I only
(B) III only
(C) I and III only
(D) I, II, and III

*B48. Which of these diverges?

*B49. Find the radius of convergence of 

(A) 0
(B) 
(C) 1
(D) e

*B50. When we use  to estimate , the Lagrange remainder is no greater than

(A) 0.021
(B) 0.035
(C) 0.057
(D) 0.063

*B51. An object in motion along a curve has position P(t) = (tan t,cos 2t) for 0 ≤ t ≤ 1. How far does it travel?

(A) 0.952
(B) 1.726
(C) 1.910
(D) 2.140

B52. The region S in the figure shown above is bounded by y = sec x and y = 4. What is the volume of the solid formed when S is rotated about the x-axis?

(A) 11.385
(B) 23.781
(C) 53.126
(D) 108.177