Unformatted text preview: ions still offer
interesting opportunities to investigate concepts and assess student understanding.
Always consult the most recent Course Description on AP Central® for the current
topic outlines for Calculus AB and Calculus BC.
Please note the following:
• The solution to each multiplechoice question suggests one possible way to
solve that question. There are often alternative approaches that produce the
same choice of answer, and for some questions such multiple approaches
are provided. Teachers are also encouraged to investigate how the incorrect
options for each question could be obtained to help students understand
(and avoid) common types of mistakes.
• Scientific (nongraphing) calculators were required on the AP Calculus
Exams in 1993.
• Graphing calculators have been required on the AP Calculus Exams since
1995. In 1997 and 1998, Section I, Part A did not allow the use of a
calculator; Section I, Part B required the use of a graphing calculator.
• Materials included in this resource may not reflect the current AP Course
Description and exam in this subject, and teachers are advised to take this
into account as they use these materials to support their instruction of
students. For uptodate information about this AP course and exam, please
download the official AP Course Description from the AP Central Web site
at apcentral.collegeboard.com. AP Calculus MultipleChoice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. vi 1969 AP Calculus AB: Section I
90 Minutes—No Calculator
Note: In this examination, ln x denotes the natural logarithm of x (that is, logarithm to the base e).
1. Which of the following defines a function f for which f (− x) = − f ( x) ?
(A) (B) f ( x) = sin x (D)
2. f ( x) = x 2 f ( x) = log x (E) f ( x) = e x ln ( x − 2 ) < 0 if and only if (A) x<3 (B) 0< x<3 (D) 3. x>2 (E) x>3 ⎧
2x + 5 − x + 7
, for x ≠ 2,
⎪ f ( x) =
If ⎨
x−2
⎪ f (2) = k
⎩ 8 ∫0 dx
1+ x (A) 1 5. 2< x<3 and if f is continuous at x = 2 , then k = 1
6 (C) 1
3 (D) 1 (E) 7
5 (B) 3
2 (C) 2 (D) 4 (E) 6 (D) 4 (E) not defined = If 3 x 2 + 2 xy + y 2 = 2, then the value of (A) –2 (C) (B) (A) 0 4. f ( x) = cos x (C) (B) 0 dy
at x = 1 is
dx (C) 2 AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 1 1969 AP Calculus AB: Section I
8 6. (A) 0
(E)
7. 1
2 For what value of k will x + (C) 1 (D) The limit does not exist. k
have a relative maximum at x = −2?
x (B) –2 (C) 2 (D) 4 (E) None of these If p ( x) = ( x + 2 )( x + k ) and if the remainder is 12 when p( x) is divided by x − 1, then k =
(A) 2 9. (B) It cannot be determined from the information given. (A) –4
8. 8 ⎛1
⎞
⎛1⎞
8⎜ + h ⎟ − 8⎜ ⎟
2
⎠
⎝2⎠ ?
What is lim ⎝
h →0
h (B) 3 (C) 6 (D) 11 (E) 13 When the area in square units of an expanding circle is increasing twice as fast as its radius in
linear units, the radius is
(A) 1
4π (B) 1
4 (C) 1
π (D) 1 (E) π (E) ln x 10. The set of all points (et , t ) , where t is a real number, is the graph of y = (A) 1
ex (B) 1
ex (C) 1
xex (D) 1
ln x 1⎞
⎛
11. The point on the curve x 2 + 2 y = 0 that is nearest the point ⎜ 0, − ⎟ occurs where y is
2⎠
⎝
1
1
(B) 0
(C) −
(D) −1
(E) none of the above
(A)
2
2 AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 2 1969 AP Calculus AB: Section I
12. If f ( x) = (A) 4
and g ( x) = 2 x, then the solution set of f ( g ( x) ) = g ( f ( x) ) is
x −1 ⎧1 ⎫
⎨⎬
⎩3⎭ (B) {2} (C) {3} (D) {−1, 2} (E) ⎧1 ⎫
⎨ , 2⎬
⎩3 ⎭ 13. The region bounded by the xaxis and the part of the graph of y = cos x between x = − π
and
2 π
π
is separated into two regions by the line x = k . If the area of the region for − ≤ x ≤ k is
2
2
π
three times the area of the region for k ≤ x ≤ , then k =
2
x= ⎛1⎞
(A) arcsin ⎜ ⎟
⎝4⎠
(D) ⎛1⎞
arcsin ⎜ ⎟
⎝3⎠ (E) π
4 (B) π
6 π
3 (C) 14. If the function f is defined by f ( x) = x5 − 1, then f −1 , the inverse function of f , is defined by
f −1 ( x) = (A) (D) 1
5 x +1 5 x −1 (B) (E) 1
5 x +1 5 (C) 5 x −1 x +1 15. If f ′( x) and g ′( x) exist and f ′( x) > g ′( x) for all real x, then the graph of y = f ( x) and the graph
of y = g ( x)
(A) intersect exactly once.
(B) intersect no more than once.
(C) do not intersect.
(D) could intersect more than once.
(E) have a common tangent at each point of intersection. AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 3 1969 AP Calculus AB: Section I
16. If y is a function of x such that y′ > 0 for all x and y′′ < 0 for all x, which of the following could
be part of the graph of y = f ( x) ? 17. The graph of y = 5 x 4 − x5 has a point of inflection at
(A) (0, 0) only...
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