A set of AB calculus questions, with answers, similar to the questions in the AP calculus test are presented. The answers to the suggested questions are at the bottom of the page. Detailed solutions and explanations are also included.

=
A) e
B) 1
C) e^{h}
D) e^{4}
E) 4^{e}

The graph of function g defined by
will have vertical asymptotes at
A)x = 1 , 3
B) x = 0
C) x = 1
D) x = 3
E) Function g has no vertical asymptotes

Given that
find
A) 2/3
B) 4/3
C) 1/3
D) 2
E) Does not exist

Function f is defined by
.
Find df(x) / dx.
A) 6x^{2} sin(x)  (1/x^{2})tan(x) + sec(x)
B) 6x^{2} sin(x) + 2x^{3} cos(x)  (1/x^{2})tan(x) + (1/x) sec^{2}(x)) + sec(x) + x sin(x) sec^{2}(x)
C) 2x^{3} cos(x) + 1/x sec^{2}(x)) + x sin(x) sec^{2}(x)
D) 6x^{2} cos(x)  (1/x^{2} sec^{2}(x)) + sec^{2}(x)
E) 6x^{2} sin(x) + 2x^{3} cos(x)  (1/x^{2})tan(x) + (1/x sec^{2}(x)) + sec(x) + x sin(x) sec^{2}(x) + 2

Curve C is described by the equation 0.25x^{2} + y^{2} = 9. Determine the y coordinates of the points on curve C whose tangent lines have slope equal to 1.
A) 3 sqrt(5) / 5 , 3 sqrt(5) / 5
B)  sqrt(35) / 2 , sqrt(35) / 2
C) 3 , 3
D)  sqrt(2) / 2 , sqrt(2) / 2
E) 3 sqrt(2) , 3 sqrt(2)

Find the solution to the differential equation dy/dx = cos(x) / y^{2} , where y(π/2) = 0.
A) y = (3 sin(x)  3)
B) y = sin(x)  1
C) y = (3 sin(x)  3)^{1/3}
D) y = (3 sin(x)  3)^{3}
E) y = (3 sin(x)  3)^{1/3}

A) cos^{5}(x) + C
B) (1/5)sin^{5}(x) + C
C) sin^{5}(x) + C
D) (1/5)cos^{5}(x) + C
E) 5cos^{5}(x) + C

A) 2sin(4x^{2} + 1)
B) 2sin(x^{2} + 1)
C) sin(x^{2} + 1)
D) 2 sin(4x^{2} + 1)  2 sin(3^{2} + 1)
E) 2 sin(4x^{2})

A) 100
B) 108
C) 110
D) 112
E) 114

Evaluate the integral
A) (5 + x^{3/4})^{10}
B) (x^{3/4})^{10}
C) (1/10)(5 + x^{3/4})^{10}
D) (1/10)(5 + x^{3/4})^{10} / x^{1/4}
E) (2/15)(5 + x^{3/4})^{10}

Given that function h is defined by
find h'(x).
A) (3x^{2} / (x^{6} + 2x^{3} + 2) + 2)
B) 4 (arctan(x^{3} + 1) + 2x)^{3} (3x^{2} / (x^{6} + 2x^{3} + 2) )
C) 4 (arctan(x^{3} + 1) + 2x)^{3}
D) 4 (3x^{2} / (x^{6} + 2x^{3} + 2) + 2)
E) (1/4)(arctan(x^{3} + 1) + 2x)^{3}

The graph of function h is shown below. How many zeros does the first derivative h' of h have?
A) 1
B) 2
C) 3
D) 4
E) 5

The graph of a polynomial f is shown below. If f' is the first derivative of f, then the remainder of the division of f'(x) by x  b is more likely to be equal to
A) f(b)
B) 1
C) 0
D) 2
E) 1

The set of all points (ln(t  2) , 3t), where t is a real number greater than 2, is the graph of
A) y = ln(x/3  2)
B) y = 3x
C) x = ln(y  2)
D) y = 3(e^{x} + 2)
E) y = ln(x)

Let P(x) = 2 x^{3} + K x + 1. Find K if the remainder of the division of P(x) by x  2 is equal to 10.
A) 7/2
B) 2/7
C) 7/2
D) 2/7
E) K cannot be determined

Function f is defined by
.
where C is a constant. What must the value of C be equal to for function f to be continuous at x = 0?
A) 0
B) 1/4
C) 1/8
D) 1
E) Any real number

f and g are functions such that f'(x) = g(x) and g'(x) = f(x). The second derivative of (f^{ . }g)(x) is equal to
A) f"(x) g"(x)
B) g'(x) g(x) + f(x) f'(x)
C) 4 g(x) f(x)
D) 2 g(x) f(x)
E) g(x) f(x)

The average rate of change of the function f defined by f(x) = sin(x) + x on the closed interval [0 , pi] is equal to
A) 0
B) 2 pi
C) pi
D) 2
E) 1

The figure shows the graphs of y = sin(x) over half a period and the line y = 1/2. Find area of the shaded region.
.
A) 1
B) 0.5
C) 2 + π/3
D) 2 + π/3  √3
E) 2 + π/3 + √3

Functions f, g and h are defined as follow: g(x) = f(x^{2}), f(x) = h(x^{3} + 1) and h'(x) = 2x + 1. g'(x) =
A) 2 x^{3}
B) 12 x^{9} + 18 x^{3}
C) 2 x^{11} + 3 x^{5}
D) 2 x^{9} + 3 x^{3}
E) 12 x^{11} + 18 x^{5}

