2003 AP Calculus AB Exam Section 2



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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