AP Calculus AB practice questions

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Question 1

Evaluate lim (x→2) of (x² − 4)/(x − 2).

Accepted Answer

4

Answer

0

Answer

2

Answer

The limit does not exist

Question 2

Evaluate lim (x→∞) of (3x² + 2)/(5x² − x).

Accepted Answer

3/5

Answer

0

Answer

∞

Answer

−2

Question 3

Which condition is required for f to be continuous at x = a?

Accepted Answer

lim (x→a) f(x) exists and equals f(a)

Answer

f(a) is defined

Answer

f is differentiable at a

Answer

lim (x→a) f(x) exists

Question 4

What is the derivative of f(x) = x³?

Accepted Answer

3x²

Answer

x²

Answer

3x³

Answer

x⁴/4

Question 5

What is the derivative of sin x?

Accepted Answer

cos x

Answer

−cos x

Answer

−sin x

Answer

tan x

Question 6

Which statement is always true?

Accepted Answer

If f is differentiable at a, then f is continuous at a.

Answer

If f is continuous at a, then f is differentiable at a.

Answer

If f is continuous at a, then f′(a) = 0.

Answer

If f(a) is defined, then f is differentiable at a.

Question 7

What is d/dx of sin(3x)?

Accepted Answer

3cos(3x)

Answer

cos(3x)

Answer

−3cos(3x)

Answer

3sin(3x)

Question 8

For the circle x² + y² = 25, what is dy/dx?

Accepted Answer

−x/y

Answer

x/y

Answer

−y/x

Answer

2x + 2y

Question 9

What is the derivative of e^(2x)?

Accepted Answer

2e^(2x)

Answer

e^(2x)

Answer

2xe^(2x − 1)

Answer

e^(2x)/2

Question 10

A particle's position is s(t) = t² − 4t. What is its velocity at t = 3?

Accepted Answer

2

Answer

−3

Answer

6

Answer

−1

Question 11

The radius of a circle grows at 2 cm/s. How fast is the area growing when r = 5 cm?

Accepted Answer

20π cm²/s

Answer

10π cm²/s

Answer

25π cm²/s

Answer

4π cm²/s

Question 12

Use a tangent line approximation of f(x) = √x at x = 4 to estimate √4.1.

Accepted Answer

2.025

Answer

2.05

Answer

2.1

Answer

2.01

Question 13

f′(x) changes from positive to negative at x = c. What does f have at x = c?

Accepted Answer

A relative maximum

Answer

A relative minimum

Answer

A point of inflection

Answer

A vertical asymptote

Question 14

If f″(x) > 0 on an interval, the graph of f on that interval is

Accepted Answer

concave up

Answer

concave down

Answer

increasing

Answer

decreasing

Question 15

f is continuous on [a, b] and differentiable on (a, b). What does the Mean Value Theorem guarantee?

Accepted Answer

Some c in (a, b) has f′(c) = (f(b) − f(a))/(b − a).

Answer

f has a maximum at some c in (a, b).

Answer

Some c in (a, b) has f(c) = 0.

Answer

f′(c) = 0 for some c in (a, b).

Question 16

Find ∫ 2x dx.

Accepted Answer

x² + C

Answer

2x² + C

Answer

x²

Answer

2 + C

Question 17

Evaluate ∫ from 0 to 2 of 3x² dx.

Accepted Answer

8

Answer

12

Answer

6

Answer

4

Question 18

What is d/dx of ∫ from 0 to x of cos t dt?

Accepted Answer

cos x

Answer

sin x

Answer

−sin x

Answer

cos x − 1

Question 19

Which method would you use to find ∫ 2x·cos(x²) dx?

Accepted Answer

u-substitution with u = x²

Answer

Integration by parts

Answer

Partial fractions

Answer

The power rule only

Question 20

What is the general solution of dy/dx = ky?

Accepted Answer

y = Ce^(kx)

Answer

y = kx + C

Answer

y = Cx^k

Answer

y = e^(kx) + C

Question 21

For the differential equation dy/dx = xy, what is the slope of the slope field at the point (1, 2)?

Accepted Answer

2

Answer

3

Answer

1/2

Answer

1

Question 22

What is the average value of f on [a, b]?

Accepted Answer

(1/(b − a)) ∫ from a to b of f(x) dx

Answer

(f(a) + f(b))/2

Answer

∫ from a to b of f(x) dx

Answer

(f(b) − f(a))/(b − a)

Question 23

What is the area between y = x and y = x² from x = 0 to x = 1?

Accepted Answer

1/6

Answer

1/2

Answer

1/3

Answer

1

Question 24

The region under y = f(x) from a to b is rotated about the x-axis. Which integral gives the volume?

Accepted Answer

π ∫ from a to b of [f(x)]² dx

Answer

∫ from a to b of f(x) dx

Answer

2π ∫ from a to b of f(x) dx

Answer

π ∫ from a to b of f(x) dx

Definition of the derivative	f′(x) = lim (h→0) [f(x + h) − f(x)] / h
Power rule	d/dx xⁿ = nxⁿ⁻¹
Product rule	(fg)′ = f′g + fg′
Quotient rule	(f/g)′ = (f′g − fg′) / g²
Chain rule	d/dx f(g(x)) = f′(g(x)) · g′(x)
Derivatives of trig functions	(sin x)′ = cos x; (cos x)′ = −sin x; (tan x)′ = sec² x
Derivatives of eˣ and ln x	(eˣ)′ = eˣ; (ln x)′ = 1/x
First derivative test	f′ changes + to −: relative max. f′ changes − to +: relative min.
Mean Value Theorem	If f is continuous on [a, b] and differentiable on (a, b), some c has f′(c) = (f(b) − f(a))/(b − a).
Fundamental Theorem of Calculus	∫ₐᵇ f(x) dx = F(b) − F(a), and d/dx ∫ₐˣ f(t) dt = f(x).
Average value of a function	(1/(b − a)) ∫ₐᵇ f(x) dx
Disk method	V = π ∫ₐᵇ [R(x)]² dx
Exponential growth model	dy/dt = ky has solution y = y₀e^(kt).

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