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AP CALCULUS BC BC
Practice Questions for AP Calculus AB Exam: Section I – Version 1
Section I
This section contains 45 multiple-choice questions and contains two parts: Part A and Part B.
Part A has 28 questions that must be solved without a calculator. Part B has 17 questions,
including some questions that require the use of a graphin
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Practice Questions for AP Calculus AB Exam: Section I – Version 1
Section I
This section contains 45 multiple-choice questions and contains two parts: Part A and Part B.
Part A has 28 questions that must be solved without a calculator. Part B has 17 questions,
including some questions that require the use of a graphing calculator.
Part A Multiple-Choice
You may not use a calculator on this portion of the exam.
Directions: Solve each problem in the provided space. Then choose the best option from among
the choices given. Be efficient with the use of your time.
Throughout this exam:
(1) The domain of each function f is the set of all real numbers for which f ( ) x is defined. If
the domain of a particular function differs from this, it will be specified in the problem.
(2) For trigonometric functions, the inverse may be represented with “ -1 ” or “arc”. For example,
the inverse cosine function may be represented as cos-1 x or arccos x .
1. sin 1
4
⎛ ⎞ ⎜ ⎟ x dx =
∫ ⎝ ⎠
(A) cos 1
4
- + ⎛ ⎞ ⎜ ⎟ x C
⎝ ⎠
(B) 4cos 1
4
- + ⎛ ⎞ ⎜ ⎟ x C
⎝ ⎠
(C) 4cos 1
4
⎛ ⎞ ⎜ ⎟ x + C
⎝ ⎠
(D) 1 1 cos
4 4
- + ⎛ ⎞ ⎜ ⎟ x C
⎝ ⎠
(E) 1 1 cos
4 4
⎛ ⎞ ⎜ ⎟ x + C
⎝ ⎠
Practice Questions for AP Calculus AB Exam: Section I
2. Find
2
1
3 2
lim
x 1
x x
→ x
- +
-
.
(A) 2
(B) 1
(C) 0
(D) -1
(E) Does not exist
3. For what value(s) of k, if any, is
2 2
( )
2( )
k x
f x
x k
⎧ -
= ⎨
⎩ +
if 2
if 2
x x
< ≥
continuous on (-∞ ∞ , ) ?
(A) - - 4, 2
(B) 2
(C) 4
(D) -2,4
(E) Does not exist
4. If f ( ) 2 8 1, then '( ) x x x f x = - is
(A) x 8 1 x -
(B) 8
8 1
x
x -
(C) 2 8 1 8 x - + x
(D) 2 8 1 8
8 1
x
x
x
- +
-
(E) 2 8
8 1
x
x
+
-
Practice Questions for AP Calculus AB Exam: Section I – Version 1
5. 8
1 8
dx
x
=
∫ +
(A) 8 ln8 x + + x C
(B) ln 1 8 + + x C
(C) ln 8x + C
(D) 1 ln 1 8
8
+ + x C
(E) 1 ln 1 8
8
- + + x C
6. A particle moves along the x-axis so that at any time t ≥ 0 , its velocity is given by
v t t ( ) cos(4 ) = . If the position of the particle at time is 7
8 4
t x
π
= = , what is the particle’s
position at time t = 0 ?
(A) 2
3
(B) 5
4
(C) 1
2
-
(D) 0
(E) 3
2
Practice Questions for AP Calculus AB Exam: Section I
7. The function f is continuous on the closed interval [0,9] and has the values given in the table
below.
x 0 3 6 9
f(x) 8 15 k 12
The trapezoidal approximation for
9 0
∫ f ( ) x dx found with three subintervals of equal length is
90. What is the value of k?
(A) 2
(B)5
(C) 7
(D) 11
(E) 17
8. dx d ( ) sin3 2 ( ) x =
(A) 6 cos 2 x 2 ( ) x
(B) 6 sin cos x 2 2 2 ( ) ( ) x x
(C) 2 sin ( ) 3 sin (2 ) x 3 2 2 2 x x x +
(D) 6 sin x 2 2 (x )
(E) 3cos 2 2 ( ) x
9. A colony of bacteria starts with 500 and grows at a rate proportional to its size. After 3 hours
there are 8000 bacteria. When will the population reach 25,000?
(A) 3.87267 hours
(B) 4.01239 hours
(C) 4.23289 hours
(D) 4.98778 hours
(E) 5.43927 hours
Practice Questions for AP Calculus AB Exam: Section I – Version 1
10. Given
4
2
27
f x ( ) x
- x
= , for what values of x is the graph of f concave downwards?
(A) - < < 3 0 x
(B) 0 3 < < x
(C) 3 < < ∞ x
(D) - < < < < 3 0 and 0 3 x x
(E) -∞ < < - < < ∞ x x 3 and 3
11. Let f be defined by f x x ( ) 6 = - for all real numbers x. For what values of x is the function
increasing?
(A) (-∞ - , 6)
(B) ( ) -∞,6
(C) [-6,0)
(D) ( ) 0,6
(E) ( ) 6,∞
12. Find an equation for the line tangent to the graph of f x x ( ) 7 = - at the point where x =16
(A) y x = - 6 2
(B) 1 1
6 3
y x = -
(C) 1 1
6 3
y x = +
(D) y x = - - 6 2
(E) 1 1
6 3
y x = - +
Practice Questions for AP Calculus AB Exam: Section I
13. d ( ) sin cos 2 x x ( )
dx
=
(A) cos( )cos(2 ) 2sin( )sin(2 ) x x x x -
(B) 2cos( )cos(2 ) sin( )sin(2 ) x x x x -
(C) cos( )cos(2 ) 2sin( )sin(2 ) x x x x +
(D) 2cos( )cos(2 ) 2sin( )sin(2 ) x x x x -
(E) 2cos( )cos(2 ) 2sin( )sin(2 ) x x x x +
14. Find the function y f x = ( ) if dy 2 1 x
dx
= - and f (1) 3 = .
(A) x2 - + x 3
(B) x x 2 - + 5
(C) 2 3 x2 + - x
(D) 1 5 2
2 2
x x - +
(E) x2 - + 4 6 x
15.
2
2
6
1
x
∫ xe dx =
(A)
12 4
8
e e -
(B)
24 6
12
e e -
(C)
24 12
12
e e -
(D)
24 6
12
e e +
(E)
2 12
12
e e -
Practice Questions for AP Calculus AB Exam: Section I – Version 1
16. How many points of inflection does f x x x ( ) 3 2 = + ( )3 .
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
17. Find the area of the region bound by f ( ) x x = 2 and g x x ( ) = 3 .
(A) 1
6
(B) 1
2
(C) 1
8
(D) 1
12
(E) 1
16
18. If
3
10
1
( ) ( ) and ( 2) 5, then '(2)
x
F x f u du f F
-
= - ∫ = =
(A) 60
(B) 30
(C) 10
(D) -5
(E) -10
Practice Questions for AP Calculus AB Exam: Section I
19. What is the equation of the line tangent to the curve y x x = - 3 2 2 at the point ( ) 2,0 ?
(A) y x = - - 4 8
(B) y x = + 4 8
(C) y x = - 4 8
(D) 1 8
4
y x = -
(E) 1 8
4
y x = - +
20. A particle moves along the x-axis so that its velocity at any time t ≥ 0 is given by
v t t t ( ) 4 4 = - 3 . Which of the following expressions could represent the position x( ) t of the
particle at any time t ≥ 0 ?
(A) t t 3 2 - + 2 4
(B) t t 3 3 - 2
(C) t t 4 2 - + 2 4
(D) t t 4 - - 2 3
(E) t t 4 2 - + 4 4
21. What is the slope of the tangent line to x xy y 2 2 + + = 3 that lies in the first quadrant at the
point where y = 1?
(A) -2
(B) -3
(C) 3
(D) -1
(E) 1
Practice Questions for AP Calculus AB Exam: Section I – Version 1
22. If f ( ) ln 2 x x = then what is the derivative of the inverse of f ( ) x at x = 4 ?
(A) 1 2
2
e
(B) 1 4
4
e
(C) 1 2
4
e
(D) 1 4
2
e
(E) Undefined
23. The function
4 3
4 2
3 5 79
( )
7 9 11
x x
f x
x x
- +
=
+ +
has horizontal asymptote(s) at
(A) y = ±3
(B) 7
3
y = ±
(C) 3
7
y =
(D) y = 0
(E) y = 1
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