2003 AP Calculus AB Exam Section 2 - MR. BOGGUS'S COOL …
2003 AB Calculus Test Section 1
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1. If[pic], then [pic]
(A) [pic] (B) [pic] (C) [pic] (D) [pic] (E) [pic]
[pic]
(E)
2. [pic]
(A) [pic] (B) [pic] (C) [pic] (D) [pic] (E) [pic]
[pic]
(D)
3. For[pic], the horizontal line [pic] is an asymptote for the graph of the function[pic] Which of the following statements must be true?
(A) [pic]
(B) [pic] for all[pic]
(C) [pic] is undefined
(D) [pic]
(E) [pic]
[pic] is a horizontal asymptote for[pic]over [pic]; by the definition of a horizontal asymptote [pic]. (E)
4. If [pic] then [pic]
(A) [pic] (B) [pic] (C) [pic] (D) [pic] (E) [pic]
[pic]
(D)
5.[pic]
(A) [pic] (B) [pic] (C) [pic] (D) [pic] (E) [pic]
[pic]
(D)
6.[pic]
(A) 4 (B) 1 (C) [pic] (D) 0 (E) [pic]
[pic]
Or, consider the end behavior of the graph of [pic] on the right. (C)
[pic]
7. The graph of [pic], the derivative of the function [pic], is shown above. Which of the following statements is true about [pic]?
(A) [pic] is decreasing for [pic]
(B) [pic] is increasing for [pic]
(C) [pic] is increasing for [pic]
(D) [pic] has a local minimum at [pic]
(E) [pic] is not differentiable at [pic] and [pic]
It cannot be (A) since [pic] for [pic], [pic] increases
It cannot be (C) since [pic] for [pic], [pic]decreases
It cannot be (D) since [pic] changes from positive to negative then there is a maximum at x = 0.
It cannot be (E) since [pic] and [pic] exist.
Since the graph of [pic], the derivative of [pic] is positive over[pic], [pic] is increasing over that interval.
(B)
8. [pic]
(A) [pic] (B) [pic] (C) [pic]
(D)[pic] (E)[pic]
[pic]
(B)
9. If [pic], then [pic] is
(A) [pic] (B) [pic] (C) [pic] (D) [pic] (E) nonexistent
[pic]
(A)
10. The function f has the property that [pic] and [pic] are negative for all real values x. Which of the following could be the graph of f ?
[pic]
The only graph that is negative ([pic] is negative), decreasing ([pic] is negative), and concave down ([pic] is negative) over the entire interval is (B.)
11. Using the substitution [pic] is equivalent to
(A) [pic] (B) [pic] (C) [pic] (D) [pic] (E) [pic]
[pic] [pic]
(C)
12. The rate of change of the volume, [pic], of water in a tank with respect to time, [pic], is directly proportional to the square root of the volume. Which of the following is a differential equation that describes this relationship?
(A) [pic] (B) [pic] (C) [pic]
(D) [pic] (E) [pic]
The rate of change of the volume is defined by [pic]. “Directly proportional to the square root of the volume” indicates the relationship [pic], where [pic] is an arbitrary constant. (E)
[pic]
13. The graph of a function [pic] is shown above. At which value of [pic] is [pic] continuous, but not differentiable?
A) a (B) b (C) c (D) d (E) e
At point a, [pic], but the curve is not locally linear at x = a, so it is continuous but not differentiable. (A)
14. If [pic], then [pic]
(A) [pic] (B) [pic] (C) [pic]
(D) [pic] (E) [pic]
[pic]
(E)
15. Let f be the function with derivative given by [pic]. On which of the following intervals is f decreasing?
(A) [pic] only (B) [pic] (C) [pic]only (D) [pic] (E) [pic]
f (x) decreases where [pic] In addition to [pic], we must consider x=0 as a critical point as well. [pic]for x < 0 and [pic], for [pic] (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 [pic] is
(A) -5 (B) 1 (C) 3 (D) 7 (E) undefined
Slope of line = [pic]. Therefore [pic] (C)
17. Let f be the function given by [pic]. The graph of f is concave down when
(A) x < -2 (B) x > -2 (C) x < -1 (D) x > -1 (E) x < 0
[pic]
Never 0 x=-2
[pic]
(A)
|x |-4 |
|2 |7 |
|3 |9 |
|4 |12 |
|5 |16 |
|[pic]|[pic] |
|2 |7 |
|3 |11 |
|4 |14 |
|5 |16 |
|[pic]|[pic] |
|2 |16 |
|3 |12 |
|4 |9 |
|5 |7 |
|[pic]|[pic] |
|2 |16 |
|3 |14 |
|4 |11 |
|5 |7 |
|[pic]|[pic] |
|2 |16 |
|3 |13 |
|4 |10 |
|5 |7 |
(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).
82. A particle moves along the x-axis so that at any time t > 0, its acceleration is given by [pic]. 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
[pic]
(E)
83. Let g be the function given by [pic] for [pic]. On which of the following intervals is g decreasing?
(A) [pic] (B) [pic] (C) [pic]
(D) [pic] (E) [pic]
[pic] = [pic]
[pic] is negative for [pic]. (D)
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[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
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