BC Practice Test 1

Section I

Part A

TIME: 60 MINUTES

The use of calculators is not permitted for this part of the examination.
There are 30 questions in Part A, for which 60 minutes are allowed. Because there is no deduction for wrong answers, you should answer every question, even if you need to guess.

DIRECTIONS: Choose the best answer for each question.

1. Consider the function f(x), defined above, where b is a constant. What is the value of b for which the function f is continuous at x = 1?
(A) –9
(B) –4
(C) 1
(D) 2

2. Function h is twice differentiable. The table above gives selected values of h. Which of the following must be true?
(A) h has no critical points in the interval –4 < x < 2.
(B) h′(x) = 10 for some value of x in the interval –4 < x < 2.
(C) The graph of h has no points of inflection in the interval – 4 < x < 2.
(D) h′(x) > 0 for all values of x in the interval –4 < x <2.

3. If  and y = sin^–1t, then  equals
(A) 
(B) – t
(C) 2
(D)

4. If  is a geometric series of all-positive terms with b₁ = 90 and b₃ = 10, then 
(A) diverges
(B) = 105
(C) = 135
(D) converges to a sum that cannot be determined

5. The table above shows values of differentiable functions f and g. If h(x) = g(f(x)) then h′(3) =
(A)
(B) 1
(C) 4
(D) 6

6.  is equal to

7. Given the parametric equations. x(t) = 2t² – 1 and. y(t) = 3 + 2t^3/2, which expression gives the length of the curve from. t = 1 to. t = 3?

8. What is the radius of convergence for the series  ?

(A) 
(B) 1
(C) 3
(D) 7

9. Which equation has the slope field shown below?

10. Consider the function h that is continuous on the closed interval [2,6] such that h(4) = 5 and h(6) = 13. Which of the following statements must be true?

(A) h is increasing on the interval [4,6].
(B) h′(x) = 4 has at least one solution in the interval (4,6).
(C) 5 ≤ h(5) ≤ 13.
(D) h(x) = 12 has at least one solution in the interval [4,6].

11. Which of the following series converges conditionally?
I. 
II. 
III. 
(A) I only
(B) II only
(C) I and II only
(D) II and III only

12. If x = 2 sin θ, , then  is equivalent to:

13. 

(A) = 0
(B)
(C) = 1
(D) does not exist

14. You are given two twice-differentiable functions, f(x) and g(x). The table above gives values for f(x) and g(x) and their first and second derivatives at x = 1. Find .

(A) –3
(B) 1
(C) 2
(D) nonexistent

15. What is the volume of the solid generated by rotating the region enclosed by the curve  and the x-axis on the interval [6,∞) about the x-axis?

(A) 
(B) 
(C) 3π
(D) divergent

16. If  is approximated using various sums with the same number of subdivisions, and if L, R, and T denote, respectively, left Riemann Sum, right Riemann Sum, and trapezoidal sum, then it follows that
(A) R ≤ A ≤ T ≤ L
(B) R ≤ T ≤ A ≤ L
(C) L ≤ T ≤ A ≤ R
(D) L ≤ A ≤ T ≤ R

17. If  tan x and y = 3 when x = 0, then, when , y =

18. The parametric equations x(t) = sin(t² + 3) and  give the position of a particle moving in the plane for t ≥ 0. What is the slope of the tangent line to the path of the particle when t = 1?

19. In which of the following series can the convergence or divergence be determined by using the Limit Comparison Test with  ?

20. Find the slope of the curve r = cos 2θ at .

21. A particle moves along a line with velocity, in feet per second, v = t² – t. The total distance, in feet, traveled from t = 0 to t = 2 equals

22. The graph of f′ is shown in the figure above. Of the following statements, which one is true about f at x = 1?
(A) f is not differentiate at x = 1.
(B) f is not continuous at x = 1.
(C) f attains an absolute maximum at x = 1.
(D) There is an inflection point on the graph of f at x = 1.

23. The Maclaurin series for a function h is given by  and converges to h on the interval –3 ≤ x ≤ 3. If  is approximated with the sixth-degree Maclaurin polynomial, what is the Alternating Series Error Bound for this approximation?

24. ∫x cos x dx =
(A) x sin x + cos x + C
(B) x sin x – cos x + C
(C) 
(D) 

25. Which one of the following series converges?

26. The coefficient of the (x – 8)² term in the Taylor polynomial for y = x^2/3 centered at x = 8 is

27. If f′(x) = h(x) and g(x) = x³, then 

(A) h(x³)
(B) 3x²h(x)
(C) 3x²h(x³)
(D) h(3x²)

28. 

(A) — 2
(B) 1
(C) 2
(D) ∞

29. The graph of f(x) consists of two line segments as shown above. If g(x) = f^–1(x), the inverse function of f(x), find g′(4).

30. Choose the integral that is the limit of the Riemann Sum: 

 

Part B

TIME: 45 MINUTES

Some questions in this part of the examination require the use of a graphing calculator. There are 15 questions in Part B, for which 45 minutes are allowed. Because there is no deduction for wrong answers, you should answer every question, even if you need to guess.

DIRECTIONS: Choose the best answer for each question. If the exact numerical value of the correct answer is not listed as a choice, select the choice that is closest to the exact numerical answer.

31. Consider h′(x) = 3e^x–1 – 5 sin(x³), the first derivative of the function h. The graph of h has a local maximum at which of the following values of x?

(A) 0.318
(B) 0.801
(C) 1.116
(D) 1.300

32. The graph of f′, which consists of a quarter-circle and two line segments, is shown above. At x = 2, which of the following statements is true?
(A) f is not continuous.
(B) f is continuous but not differentiate.
(C) f has a local maximum.
(D) The graph of f has a point of inflection.

33. Let , where f is the function whose graph appears below.

The local linearization of H(x) near x = 3 is H(x) ≈
(A) –2x + 8
(B) 2x – 4
(C) –2x + 4
(D) 2x – 8

34. Consider the velocity vector 〈ln(t⁴ + 1),cos(t³)) for a particle moving in the plane. The position vector of the particle at t = 2 is (4,–2). What is the x-coordinate of the position vector of the particle at t = 0.5?

(A) 0.061
(B) 1.941
(C) 2.059
(D) 5.941

35. The function g is defined on the closed interval [0,4] such that g(0) = g(2) = g(4). The function g is continuous and strictly decreasing on the open interval (0,4). Which of the following statements is true?

(A) g has neither an absolute minimum nor an absolute maximum on [0,4].
(B) g has an absolute minimum but not an absolute maximum on [0,4].
(C) g has an absolute maximum but not an absolute minimum on [0,4].
(D) g has both an absolute minimum and an absolute maximum on [0,4].

36. Find the volume of the solid generated when the region bounded by the y-axis, y = e^x, and y = 2 is rotated around the y-axis.
(A) 0.592
(B) 1.214
(C) 2.427
(D) 3.998

37. The path of a satellite is given by the parametric equations

The upward velocity at t = 1 equals
(A) 3.073
(B) 3.999
(C) 12.287
(D) 12.666

38. If g is a differentiable function with g(1) = 4 and g′(1) = 3, which of the following statements could be false?

39. A particle is moving in the xy-plane. The position of the particle is given by x(t) = t² + ln (t² + 1) and y(t) = 2t + 5cos(t²). What is the speed of the particle when t = 3.1?
(A) 3.641
(B) 3.807
(C) 10.269
(D) 12.04

40. Which definite integral represents the length of the first-quadrant arc of the curve defined by x(t) = e^t, y(t) = 1 – t²?

41. For which function is  the Taylor series about 0?
(A) e^–x
(B) sin x
(C) cos x
(D) ln (1 + x)

42. The graph of the function f is shown in the figure above. Find the value of .

(A) 3
(B) 4
(C) 5
(D) nonexistent

43. At how many points on the interval [0,π] does f(x) = 2 sin x + sin 4x satisfy the Mean Value Theorem?

(A) 1
(B) 2
(C) 3
(D) 4

44. As a cup of hot chocolate cools, its temperature after t minutes is given by H(t) = 70 + ke^–0.4t. If its initial temperature was 120°F, what was its average temperature (in °F) during the first 10 minutes?

(A) 79.1
(B) 82.3
(C) 95.5
(D) 99.5

45. The rate at which a purification process can remove contaminants from a tank of water is proportional to the amount of contaminant remaining. If 20% of the contaminant can be removed during the first minute of the process and 98% must be removed to make the water safe, approximately how long will the decontamination process take?
(A) 2 minutes
(B) 5 minutes
(C) 7 minutes
(D) 18 minutes

 

Section II

Part A

TIME: 30 MINUTES
2 PROBLEMS

A graphing calculator is required for some of these problems. See instructions on page 8.

1. The velocity vector of an object in motion in the plane for 0 ≤ t ≤ 6 is given by its components  and . At time t = 5, the position of the object is (3,–2).

(a) Find the speed of the object at time t = 4, and find the acceleration vector of the object at time t = 4.
(b) Find the position of the object at time t = 4.
(c) Find the total distance traveled by the object from t = 0 to t = 4.

 

2. The rate of sales of a new software product is given by the differentiate and increasing function S(t), where S is measured in units per month and t is measured in months from the initial release date. The software company recorded these sales data:

(a) Use the data in the table to estimate S′(7). Show the work that leads to your answer. Indicate units of measure.
(b) Use a left Riemann Sum with the four subintervals indicated in the table to estimate the total number of units of software sold during the first 9 months of sales. Is this an underestimate or an overestimate of the total number of software units sold? Give a reason for your answer.
(c) For 1 ≤ t ≤ 9, must there be a time t when the rate of sales is increasing at  per month? Justify your answer.
(d) For 9 ≤ t ≤ 12, the rate of sales is modeled by the function . Given that the actual sales in the first 9 months is 3636 software units, use this model to find the number of units of the software sold during the first 12 months of sales. Round your answer to the nearest whole unit.

 

Part B

TIME: 60 MINUTES
4 PROBLEMS

No calculator is allowed for any of these problems.

If you finish Part B before time has expired, you may return to work on Part A, but you may not use a calculator.

3. The graph of function y = f(x) passes through the point (1,1) and satisfies the differential equation .

(a) Sketch the slope field for the differential equation at the 12 indicated points on the axes provided.

(b) Find an equation of the line tangent to f(x) at the point (1,1) and use the linear equation to estimate f(1.2).
(c) Solve the differential equation, and find the particular solution for y = f(x) that passes through the point (1,1).

 

4. Shown above are the graphs of the polar curves r = 3 and r = 3 – 2 cos(3θ) for 0 ≤ θ ≤ π. Region R is the shaded region inside the graph of r = 3 – 2 cos(3θ) and outside the graph of r = 3. The two curves intersect at , , and .

(a) Write, but do not evaluate, an integral expression for the area of region R.
(b) For the graph of r = 3 – 2 cos(3θ), write expressions for in terms of θ.
(c) Write an equation, in terms of x and y, for the tangent line to the curve r = 3 – 2 cos(3θ) when . Show the computations that lead to your answer.

 

5. The graph of the function f is given above. f is twice differentiate and is defined on the interval –8 ≤ x ≤ 6. The function g is also twice differentiate and is defined as .

(a) Write an equation for the line tangent to the graph of g(x) at x = – 8. Show the work that leads to your answer.
(b) Using your tangent line from part (a), approximate g(–7). Is your approximation greater than or less than g(–7)? Give a reason for your answer.
(c) Evaluate: . Show the work that leads to your answer.
(d) Find the absolute minimum value of g on the interval – 8 ≤ x ≤ 6. Justify your answer.

 

6. 

(a) Write the first four nonzero terms and the general term of the Maclaurin series for f(x) = ln (e + x).
(b) What is the radius of convergence?
(c) Use the first three terms of that series to write an expression that estimates the value of .