AP Calculus AB Problems with Solutions (Part - 1)

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Problem 1 :

Evaluate the limit :

Solution :

Problem 2 :

Find equations for the lines that are tangent and normal to the graph of y = sin x + 3 at x = 𝜋.

Solution :

Problem 3 :

Show that the graph of y = sec x has a horizontal tangent at x = 0.

Solution :

Problem 4 :

Find equations for the lines that are tangent and normal to the curve y = √2 cos x at the point (𝜋/4, 1).

Solution :

Problem 5 :

Find equations for the lines that are tangent and normal to the curve y = √2 cos x at the point (𝜋/4,1).

Solution :

Problem 6 :

Using the product rule, determine 𝑓′'(x), if 𝑓(x)=sin²x.

Solution :

Problem 7 :

If f(x) = 3x³ - x + 4, then (f^-1)'(6) =

A) 1/8

B) 1/6

C) 1/3

D) 3

Solution :

Problem 8 :

The graph of the function f shown above consists of three line segments. If g is the function defined

then g(-3) =

A) -13/2

B) -11/2

C) -9/2

D) 11/2

Solution :

Problem 9 :

An equation of the line normal to the graph of y = sec x at the point (π, √2) is

A) y - √2 = √2(x - π/4)

B) y - √2 = -1/√2(x - π/4)

C) y - 1/√2 = -1√2(x - π/4)

D) y - 1/√2 = 1/√2(x - π/4)

Solution :

Problem 10 :

The first derivative f' of a function f is given by

f'(x) = ½- e^-x

On which of the following intervals is f increasing?

A) (-∞, ln 2)

B) (-ln 2, ln 2)

C) (ln 2, ∞)

D) (e, ∞)

Solution :

Problem 11 :

The graph of the function f is shown in the figure above. Which of the following statements about f is not true?

Solution :

Problem 12 :

The figure above shows the graph of the function f = ½(x - 2)² and the graph of g which tangent to the graph of f at the point (4, 2). If h(x) = f[g(x)], what is h'(4)?

A) -4

B) -2

C) 0

D) 2

Solution :

Problem 13 :

A) 0

B) 1

C) 2

D) 3

Solution :

Problem 14 :

A) -5

B) -3

C) 7

D) 14

Solution :

Problem 15 :

If f is a continuous function and F'(x) = f(x) for all real numbers x, then

A) 2[F(2) - F(1)]

B) 2[F(4) - F(1)]

C) ½[F(2) - F(1)]

D) ½[F(4) - F(1)]

Solution :

Problem 16 :

The graph of f' is shown in the figure above. Which of the following statements about f are true?

I. f has a relative minimum at x = a.

II. f has a relative maximum at x = b.

III. f is decreasing on the interval b < x < c.

A) None

B) I only

C) I and III only

D) II and III only

Solution :

Problem 17 :

The region enclosed by the graph of y = sin x and the lines y = 1/2, x = π/6 and x = 5π/6 is rotated about the x-axis. What is the volume of the solid generated?

Solution :

Problem 18 :

A) ln 6

B) ln √3

C) 1

D) ln √3 - 1

Solution :

Problem 19 :

If f(x) = ³√(x² + 7) ⋅ e^x, then what is the value of f'(1)?

A) 3e/4

B) 7e/6

C) 3e/2

D) 13e/6

Solution :

Problem 20 :

At a musical concert the audience stands inside a semicircular area of radius 50 yards. The stage is also a semicircular shape of radius 10 yards. if the denisty of the audience at r yards from the center of the stage is given by f(r) people per square yard, which of the following expressions gives the number of people at the concert?

Solution :

Problem 21 :

If four subdivisions of [1, 3] are used, what is the trapezoidal approximation of the following definite integral?

A) ½(cos 1 + √1.5 cos 1.5 + √2 cos 2 + √2.5 cos 2.5 + √3 cos 3)

B) ½(cos 1 + 2√1.5 cos 1.5 + 2√2 cos 2 + 2√2.5 cos 2.5 + √3 cos 3)

C) ¼(cos 1 + √1.5 cos 1.5 + √2 cos 2 + √2.5 cos 2.5 + √3 cos 3)

D) ¼(cos 1 + 2√1.5 cos 1.5 + 2√2 cos 2 + 2√2.5 cos 2.5 + √3 cos 3)

Solution :

Problem 22 :

Solution :

Problem 23 :

If the point (-1, 5) is a point of inflection of the curve

y = x³ + ax² + bx - 2,

what is the value of a + b?

A) -2

B) -1

C) 3

D) 6

Solution :

Problem 24 :

Solution :

Problem 25 :

The curves y = f(x) and y = g(x) shown in the figure above intersect at point (a, b). The area of the shaded region enclosed by these curves and the x-axis is given by