Free Printable AP Calculus Review

1.

What is the formal definition of the derivative?

a)

b)

c)

2.

What is the formal definition of the derivative at the point x = a?

a)

b)

c)

3.

What is the alternative definition of the derivative at the point x = a?

a)

b)

c)

4.

$\frac{d}{dx}\left[f\left(x\right)g\left(x\right)\right]=$

a)

$f'\left(x\right)g'\left(x\right)$

b)

$f\left(x\right)g'\left(x\right)+g\left(x\right)f'\left(x\right)$

c)

$f'\left(x\right)g\left(x\right)g'\left(x\right)$

5.

$\frac{d}{dx}\left[\frac{f\left(x\right)}{g\left(x\right)}\right]$

a)

$\frac{f'\left(x\right)}{g'\left(x\right)_{ }}$

b)

$\frac{g'\left(x\right)f\left(x\right)-f'\left(x\right)g\left(x\right)}{\left[g\left(x\right)\right]^2}$

c)

$\frac{g\left(x\right)f'\left(x\right)-f\left(x\right)g'\left(x\right)}{\left[g\left(x\right)\right]^2}$

6.

$\frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]$

a)

$f'\left(g'\left(x\right)\right)$

b)

$f\left(g'\left(x\right)\right)g'\left(x\right)$

c)

$f'\left(g\left(x\right)\right)g'\left(x\right)$

7.

$\frac{d}{dx}\left[\sin u\right]$

a)

$\cos u$

b)

$\sin u$

c)

$-\cos u$

d)

$-\sin\ u$

8.

$\frac{d}{dx}\left[\cos u\right]$

a)

$\cos u$

b)

$\sin u$

c)

$-\cos u$

d)

$-\sin\ u$

9.

$\int_{ }^{ }\sin u\ du$

a)

$\cos u$

b)

$\sin u$

c)

$-\cos u$

d)

$-\sin\ u$

10.

$\int_{ }^{ }-\cos u\ du$

a)

$\cos u+c$

b)

$\sin u+c$

c)

$-\cos u+c$

d)

$-\sin\ u+c$

11.

$\frac{d}{dx}\left[\tan u\right]$

a)

$\sec^2u$

b)

$-\sec^2u$

c)

$\csc^2u$

d)

$-\csc^2u$

12.

$\frac{d}{dx}\left[\cot u\right]$

a)

$\sec^2u$

b)

$-\sec^2u$

c)

$\csc^2u$

d)

$-\csc^2u$

13.

$\frac{d}{dx}\left[e^{3x}\right]$

a)

$e^{3x}$

b)

$\frac{1}{3}e^{3x}$

c)

$3e^{3x}$

14.

$\frac{d}{dx}\left[a^x\right]$

a)

$a^x$

b)

$\log_ax$

c)

$a^x\ln a$

d)

$x\ln a$

15.

$\frac{d}{dx}\left[\ln x\right]$

a)

$\frac{1}{x}$

b)

$x^{-2}$

c)

$\ln x$

d)

$e^x$

16.

$\frac{d}{dx}\left[\ln u\right]$

a)

$\frac{1}{u}$

b)

$\frac{du}{u}$

c)

$e^u$

17.

Speeding up or speed if increasing

a)

velocity and acceleration has same sign

b)

velocity and acceleration have opposite signs

c)

velocity is positive

d)

acceleration is positive

18.

Slowing down or speed if decreasing

a)

velocity and acceleration has same sign

b)

velocity and acceleration have opposite signs

c)

velocity is negative

d)

acceleration is negative

19.

$\int_{ }^{ }e^{\frac{x}{2}}$

a)

$e^{\frac{x}{2}}+c$

b)

$\frac{1}{2}e^{\frac{x}{2}}+c$

c)

$2e^{\frac{x}{2}}+c$

d)

$2e^x+c$

20.

If $f'\left(x\right)$ is positive, then....

a)

$f\left(x\right)$ is increasing

b)

$f\left(x\right)$ is decreasing

c)

$f\left(x\right)$ is concave up

d)

$f\left(x\right)$ is concave down

21.

If $f''\left(x\right)$ is positive, then....

SELECT ALL THAT APPLY

a)

$f'\left(x\right)$ is increasing

b)

$f'\left(x\right)$ is decreasing

c)

$f\left(x\right)$ is concave up

d)

$f\left(x\right)$ is concave down

22.

If $f''\left(x\right)$ is negative, then....

SELECT ALL THAT APPLY!

a)

$f'\left(x\right)$ is increasing

b)

$f'\left(x\right)$ is decreasing

c)

is concave up

d)

is concave down

23.

If $f'\left(x\right)$ changes from positive to negative at a, what do we know about the point at a?

a)

a is a realtive minimum

b)

a is a relative maximum

c)

a is an absolute minimum

d)

a is an absolute maximum

24.

$\int_{ }^{ }\sec u\tan u\ du$

a)

$\tan u+c$

b)

$\sec u+c$

c)

$\csc u+c$

d)

$\cot u+c$

25.

$\int_{ }^{ }\left(\frac{du}{u}\right)$

a)

$u^{-2}+c$

b)

$u^{-1}+c$

c)

$\ln u+c$

26.

$\int_{ }^{ }\left(\frac{2x}{x^2+1}\right)dx$

a)

$\frac{1}{2}\ln\left(2x\right)+c$

b)

$\frac{1}{2}\ln\left(x^2+1\right)+c$

c)

$\ln\left(x^2+1\right)+c$

27.

Net distance or displacement

a)

$\int_a^bv\left(t\right)dt$

b)

$\int_a^b\left|v\left(t\right)\right|dt$

28.

Total distance

a)

$\int_a^bv\left(t\right)dt$

b)

$\int_a^b\left|v\left(t\right)\right|dt$

29.

If f is continuous on [a,b], and k is between f(a) and f(b), then there exists at least one c between a and b such that f(c) = k.

a)

Intermediate Value Theorem

b)

Extreme Value Theorem

c)

Rolle's Theorem

d)

Mean Value Theorem

30.

If f is continuous on [a,b], and differentiable on (a, b), then there exists at least one c between a and b such that $f'\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$ .

a)

Intermediate Value Theorem

b)

Extreme Value Theorem

c)

Rolle's Theorem

d)

Mean Value Theorem

31.

If f is continuous on [a,b], then there exists at least one maximum value and one minimum value on the interval.

a)

Intermediate Value Theorem

b)

Extreme Value Theorem

c)

Rolle's Theorem

d)

Mean Value Theorem

32.

$\frac{d}{dx}\left[\sin^{-1}u\right]=$

a)

$\frac{du}{1+u^2}$

b)

$\frac{-du}{\sqrt{1-u^2}}$

c)

$\frac{du}{\sqrt{1-u^2}}$

d)

$\frac{-du}{1+u^2}$

33.

$\frac{d}{dx}\left[\tan^{-1}u\right]=$

a)

$\frac{du}{1+u^2}$

b)

$\frac{-du}{\sqrt{1-u^2}}$

c)

$\frac{du}{\sqrt{1-u^2}}$

d)

$\frac{-du}{1+u^2}$

34.

$\frac{d}{dx}\left[\cot^{-1}u\right]=$

a)

$\frac{du}{1+u^2}$

b)

$\frac{-du}{\sqrt{1-u^2}}$

c)

$\frac{du}{\sqrt{1-u^2}}$

d)

$\frac{-du}{1+u^2}$

35.

$\frac{d}{dx}\left[\cos^{-1}u\right]=$

a)

$\frac{du}{1+u^2}$

b)

$\frac{-du}{\sqrt{1-u^2}}$

c)

$\frac{du}{\sqrt{1-u^2}}$

d)

$\frac{-du}{1+u^2}$

36.

$\int_{ }^{ }\left[\frac{1}{1+x^2}dx\right]$

a)

$\sin^{-1}x+c$

b)

$\cos^{-1}x+c$

c)

$\tan^{-1}x+c$

d)

$\cot^{-1}x+c$

37.

$\int_{ }^{ }\left[\frac{1}{\sqrt{1-x^2}}dx\right]$

a)

$\sin^{-1}x+c$

b)

$\cos^{-1}x+c$

c)

$\tan^{-1}x+c$

d)

$\cot^{-1}x+c$

38.

$\int_{ }^{ }\left[\frac{-1}{\sqrt{1-x^2}}dx\right]$

a)

$\sin^{-1}x+c$

b)

$\cos^{-1}x+c$

c)

$\tan^{-1}x+c$

d)

$\cot^{-1}x+c$

39.

$\int_{ }^{ }\left[\frac{-1}{1+x^2}dx\right]$

a)

$\sin^{-1}x+c$

b)

$\cos^{-1}x+c$

c)

$\tan^{-1}x+c$

d)

$\cot^{-1}x+c$

40.

$\int_{ }^{ }-\csc^2x\ dx$

a)

$\tan x+c$

b)

$\cot x+c$

c)

$\sec x+c$

d)

$\csc x+c$

41.

$\frac{d}{dx}\left[\sec u\right]=$

a)

$\sec u\tan u\ du$

b)

$-\csc u\cot u\ du$

c)

$\sec^2u\ du$

d)

$-\csc^2u\ du$

42.

$\frac{d}{dx}\left[f^{-1}\left(x\right)\right]=$

a)

$\frac{1}{f'\left(x\right)}$

b)

$\frac{1}{f^{-1}\left(f'\left(x\right)\right)}$

c)

$\frac{1}{f'\left(f^{-1}\left(x\right)\right)}$

d)

$f'\left(f^{-1}\left(x\right)\right)$

43.

Given a function $f\left(x\right)$ . If $f'\left(a\right)=0$ and $f''\left(x\right)<0$ what can be concluded about $a$ ?

a)

$a$ is a relative minimum of $f\left(x\right)$

b)

$a$ is a relative maximum of $f\left(x\right)$

c)

$a$ is a point of inflection of $f\left(x\right)$

d)

There is not enough information to conclude anything.

44.

If $F\left(x\right)=\int_a^{g\left(x\right)}f\left(t\right)dt$ where a is a constant, then

a)

$F'\left(x\right)=f\left(g\left(x\right)\right)$

b)

$F'\left(x\right)=f\left(g\left(x\right)\right)g'\left(x\right)$

c)

$F'\left(x\right)=f\left(g\left(x\right)\right)f'\left(x\right)$

d)

$F\left(x\right)=f\left(t\right)$

45.

What type of graph is the following parametric equations? $x=3-3t,\ y=2t$

a)

line

b)

parabola

c)

ellipse

d)

hyperbola

e)

square root

46.

What is this formula?

a)

Velocity Formula

b)

Vector Formula

c)

Magnitude Formula

d)

Arc Length Formula

47.

What is this formula?

a)

Polar Arc Length

b)

Area under a polar leaf

c)

Slope of a polar function

48.

What is this formula?

a)

Distance Travelled

b)

Distance Travelled (Vector)

c)

Final Position

d)

Arc Length of a Vector equation

49.

What is this formula

a)

Average Rate of Change Formula

b)

Average Value of a Function Formula

c)

Arc Length Formula

50.

What is this formula?

a)

Polar Arc Length

b)

Area under a polar leaf

c)

Slope of a polar function

51.

What is this?

a)

Geometric Series Test

b)

Nth Term Test

c)

Power Series

d)

Geometric Series Sum

52.

This is an example of what?

a)

Telescoping Series

b)

Nth Term test

c)

P-Series Test

d)

Ratio Test

e)

Alternating Series Test

53.

What makes a Point of Inflection

a)

Where the concavity changes sign

b)

Where the slope changes sign

c)

Where the slope =0

d)

Where the function changes sign

54.

What is this?

a)

Derivative of Arctangent

b)

Final Position Formula

c)

Volume by Cross Sections Formula

d)

Arc Length

55.

What is this formula?

a)

Polar Arc Length

b)

Area under a polar leaf

c)

Slope of a polar function

56.

What series matches this function?

a)

b)

c)

d)

e)

57.

What series matches this function?

a)

b)

c)

d)

e)

58.

What series matches this function?

a)

b)

c)

d)

e)

59.

What is the harmonic series, does it converge, what if it alternates?

a)

$\sum_{n=0}^{\infty}\frac{1}{n}$ ,yes, no

b)

$\sum_{n=0}^{\infty}x^n$ ,no, yes

c)

$\sum_{n=0}^{\infty}\frac{1}{n}$ ,no, yes

d)

$\sum_{n=0}^{\infty}x^n$ ,yes, no

e)

$\sum_{n=0}^{\infty}\frac{1}{n}$ , no, no

60.

This is an example of what?

a)

Limit Comparison Test

b)

Integral test

c)

Direct Comparison Test

d)

P-Series

e)

Nth Term Test

61.

This is an example of what?

a)

Direct Comparison Test

b)

Ratio Test

c)

Alternating Series Test

d)

Telescoping Series

e)

Geometric Series Test

62.

What series matches this function?

a)

b)

c)

d)

e)

63.

This is an example of what?

a)

Geometric Series Test

b)

Direct Comparison Test

c)

Limit Comparison

d)

Integral Test

e)

P-Series

64.

Define Interval of Convergence

a)

The range of y-values within which the series will converge

b)

The range of x-values within which the series will diverge

c)

The range of x-values within which the series will converge

d)

The range of y-values within which the series will diverge

65.

A geometric series converges when...

a)

$\left|r\right|\ge1$

b)

$\left|r\right|<1$

c)

$\left|r\right|=1$

66.

The nth term test says:

If $\lim_{x\rightarrow\infty}a_{n\ne0,\ }$ then

a)

$a_{n\ }$ converges

b)

$a_{n\ }$ diverges

67.

A p-series of the form $\sum_{n=1}^{\infty}\frac{1}{n^p}$ , converges when...

a)

$p>1$

b)

$p<1$

c)

$p=1$

68.

Which statement is true about the SERIES $\sum_{n=1}^{\infty}\frac{6}{n\sqrt[]{n}}$ ?

a)

It is a convergent geometric series.

b)

It is a convergent P-series.

c)

It is a divergent P-series.

d)

It is a divergent geometric series.

69.

What is the sum of the converging geometric series below?

$\sum_{n=1}^{\infty}\frac{7^{n+1}}{10^n}$

(a)

70.

Let f be a positive, continuous, decreasing function such that

a_n=f\left(n\right)

. If

\sum_{n=1}^{\infty}a_n

converges to k, which of the following must be true?

a)

$\lim_{n\rightarrow\infty\ }a_n=k$

b)

$\int_1^nf\left(x\right)dx=k$

c)

$\int_1^{\infty}f\left(x\right)dx$ diverges

d)

$\int_1^{\infty}f\left(x\right)dx\$ converges

e)

$\int_1^{\infty}f\left(x\right)dx=k$

71.

$\sum_{n=1}^{\infty}2\left(\frac{3}{2}\right)^n$

a)

Converges

b)

Diverges

AP Calculus Review

Total questions: 71

Worksheet time: 2hrs 22mins

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