Free Printable AP Calculus: Avoid the Traps

1.

How can you tell if a particle is slowing down?

a)

If velocity is negative at t=___

b)

If acceleration is negative at t=___

c)

If velocity and acceleration have the same sign at t=___

d)

If velocity and acceleration have opposite signs at t=___

2.

What is $\int_{ }^{ }e^{\frac{x}{4}}dx$ ?

a)

$4e^{\frac{x}{4}}+C$

b)

$\frac{1}{4}e^{\frac{x}{4}}+C$

c)

$e^{\frac{x}{4}}+C$

d)

$\frac{1}{4}e^x+C$

3.

If $g\left(x\right)=\int_a^xf\left(t\right)dt$ , what is the relationship between g(x) and f(x)?

a)

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

b)

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

c)

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

d)

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

4.

If f(1)=6, which of the following would give the value of f(6)?

a)

$\int_1^6f'\left(x\right)dx$

b)

$f\left(1\right)+\int_1^6f'\left(x\right)dx$

c)

$\int_1^6f\left(x\right)dx$

d)

$f\left(1\right)+\int_1^6f\left(x\right)dx$

5.

Which justification is acceptable for using L'Hopital's Rule to find $\lim_{x\rightarrow a}\frac{f\left(x\right)}{g\left(x\right)}$ ?

a)

$\lim_{x\rightarrow a}\frac{f\left(x\right)}{g\left(x\right)}=\frac{0}{0}$

b)

$\lim_{x\rightarrow a}\frac{f\left(x\right)}{g\left(x\right)}=\frac{\infty}{\infty}$

c)

$\lim_{x\rightarrow a}f\left(x\right)=0$ and $\lim_{x\rightarrow a}g\left(x\right)=0$

d)

$\frac{f\left(a\right)}{g\left(a\right)}=\frac{0}{0}$

6.

Let $f'\left(x\right)=3\cos\left(x^2\right)$ . To find where f is increasing, what should you do with $3\cos\left(x^2\right)$ ?

a)

Integrate

b)

Differentiate

c)

Use the chain rule

d)

Look where it is positive

7.

When is a tangent line approximation to f(x) at x=a an overestimation?

a)

When f is increasing around x=a

b)

When f is decreasing around x=a

c)

When f is concave up around x=a

d)

When f is concave down around x=a

8.

Which of the following would make x=k an asymptote of f(x)?

a)

$\lim_{x\rightarrow\infty}f\left(x\right)=k$

b)

$\lim_{x\rightarrow k}f\left(x\right)=0$

c)

$\lim_{x\rightarrow k}f\left(x\right)=\infty$

d)

$\lim_{x\rightarrow\infty}f\left(x\right)=\infty$

9.

Which of the following theorem(s) requires a function to be differentiable?

a)

IVT

b)

EVT

c)

MVT

d)

L'Hop

10.

Which of the following integrals correctly uses natural logs?

a)

$\int_{ }^{ }\frac{1}{x+3}dx=\ln\left|x+3\right|+C$

b)

$\int_{ }^{ }\frac{1}{x^2+1}dx=\ln\left|x^2+1\right|+C$

c)

$\int_{ }^{ }\frac{1}{\sqrt{1-x^2}}dx=\ln\left|\sqrt{1-x^2}\right|+C$

d)

$\int_{ }^{ }\frac{1}{\left(x-1\right)^2}dx=\ln\left|\left(x-1\right)^2\right|+C$

11.

Which of the following represents the average value of f(x) on the interval [a,b]?

a)

$\frac{f\left(b\right)-f\left(a\right)}{b-a}$

b)

$\frac{f\left(a\right)+f\left(b\right)}{2}$

c)

$\frac{1}{b-a}\int_a^bf\left(x\right)dx$

d)

$f\left(\frac{a+b}{2}\right)$

12.

Which of the following statements about natural logs is true?

a)

$\frac{d}{dx}\left(\frac{1}{x}\right)=\ln\left|x\right|$

b)

$\int_{ }^{ }\frac{1}{x}dx=\ln x+C$

c)

$\frac{d}{dx}\ln x=\frac{1}{x}$

d)

$\int_{ }^{ }\ln xdx=\frac{1}{x}+C$

13.

Complete the justification: The graph of f is concave down on (a,b) when:

a)

f is negative on (a,b)

b)

f' is negative on (a,b)

c)

f" is negative on (a,b)

14.

Which of the following is the correct derivative of $xy$ with respect to x?

a)

$y$

b)

$x\left(\frac{dy}{dx}\right)+y$

c)

$x+\frac{dy}{dx}$

d)

$y\left(\frac{dy}{dx}\right)$

15.

Which of the following derivatives does not require the Chain Rule?

a)

$\frac{d}{dx}\left(\sin^2x\right)$

b)

$\frac{d}{dx}\left(e^{3x}\right)$

c)

$\frac{d}{dx}\left(\frac{1}{x^3}\right)$

d)

$\frac{d}{dx}\left(\sqrt{x-1}\right)$

16.

In which of these situations might you use an integral?

a)

You know an amount and you want the rate of change

b)

You know a rate and you want the change in amount

c)

You want to see how fast the rate is changing

d)

You want to see if a function is continuous

17.

If the graph of f is increasing, which of the following must be an overestimate?

a)

Left Riemann Sum

b)

Right Riemann Sum

c)

Midpoint Riemann Sum

d)

Trapezoidal Sum

18.

When calculating the absolute maximum of f on [a,b], which points need to be considered?

a)

critical points

b)

points of inflection and end points

c)

critical points and points of inflection

d)

critical points and end points

19.

Which of the following is justification that f has a relative minimum at x=a?

a)

f'(x)=0 at x=a

b)

f' changes from increasing to decreasing

c)

f' changes from negative to positive at x=a

d)

f" changes from negative to positive at x=a

20.

Suppose $f\left(1\right)=-3$ and $f\left(2\right)=4$ . Which of the following must be true?

a)

There is a c in (1,2) such that f(c)=7

b)

There is a c in (1,2) such that f(c)=2

c)

There is a c in (1,2) such that f'(c)=7

d)

None of the answers must be true

21.

What is $\int_{ }^{ }5dt$ ?

a)

0+C

b)

5+C

c)

0

d)

5t+C

22.

Suppose the substitution $u=2x+1$ is made in $\int_1^5g\left(2x+1\right)dx$ . What will the new integral be?

a)

$\frac{1}{2}\int_1^5g\left(u\right)du$

b)

$2\int_1^5g\left(u\right)du$

c)

$\frac{1}{2}\int_3^{11}g\left(u\right)du$

d)

$2\int_3^{11}g\left(u\right)du$

23.

Which expression gives the total distance traveled by a particle along the x-axis during $a\le t\le b$ ?

a)

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

b)

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

c)

$x\left(a\right)+\int_a^bv\left(t\right)dt$

d)

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

24.

A particle moves along the x-axis on [a,b]. What is its average velocity?

a)

$\frac{v\left(b\right)-v\left(a\right)}{b-a}$

b)

$\frac{x\left(a\right)+x\left(b\right)}{2}$

c)

$\frac{1}{b-a}\int_a^bv\left(t\right)dt$

d)

$\frac{v\left(a\right)+v\left(b\right)}{2}$

25.

If f(t) represents the rate at which something's price is changing, when is the price increasing?

a)

When $f\left(t\right)>0$

b)

When $f'\left(t\right)>0$

c)

When $f"\left(t\right)>0$

d)

When f is concave up

26.

What is $\frac{d}{dx}\int_{-3}^{x^2}\left(\cos t+5\right)dt$ ?

a)

$\cos x^2+5$

b)

$2x\cos x^2+5$

c)

$2x\left(\cos x^2+5\right)$

d)

$2x\cos x^2$

27.

$\lim_{h\rightarrow0}\frac{\sin\left(5+h\right)-\sin5}{h}$

a)

sin(5)

b)

-sin(5)

c)

cos(5)

d)

-cos(5)

28.

What is the first step to solve this problem?

a)

Take the derivative and set it equal to 0

b)

Take the integral

c)

Set this equation equal to 0

d)

Find the second derivative and set it equal to 0

29.

A particle is moving, and it is known that v(3)=4 and a(3)=1. Which is true at t=3?

a)

Speeding up because v(3) and a(3) have the same sign

b)

Slowing down up because v(3) and a(3) have the same sign

c)

Speeding up because v(3) and a(3) have opposite signs

d)

Slowing down because v(3) and a(3) have opposite signs

30.

a)

f only

b)

g only

c)

f and g

d)

f, g, and h

31.

If $F\left(x\right)=\int_0^x\sqrt{t^3+1}dt$ , then F'(2)=

a)

-3

b)

-2

c)

2

d)

3

32.

The graph of the function f is shown and has a vertical tangent at the point (2,0) and horizontal tangents at the points (1, -1) and (3, 1). For what values of x on -2<x<4 is f not differentiable?

a)

0 only

b)

0 and 2

c)

0, 1, and 3

d)

0, 1, 2, and 3

33.

What rule/formula would you use to solve this problem?
"What is the instantaneous rate of change at x=2 of the function $f\left(x\right)=\frac{x^2-2}{x-1}$ ?"

a)

Product Rule

b)

Quotient Rule

c)

Chain Rule

d)

Slope Formula

34.

A bug begins to crawl up a vertical wire at t=0. The velocity of the bug at time t on [0,8] is given by the function whose graph is shown. When does the bug change direction?

a)

t=2

b)

t=6

c)

t=4

d)

t=7

35.

What is the first step to solving this:
"If $f\left(x\right)=-x^3+x+\frac{1}{x}$ , then $f'\left(-1\right)=$ "

a)

Plug in -1

b)

Find f'(x)

c)

Integrate f(x)

36.

Evaluate $\lim_{x\rightarrow\infty}\frac{29x-10+3x^2}{18x^2-3}$

a)

$\frac{29}{18}$

b)

$\frac{1}{6}$

c)

$\frac{10}{3}$

d)

does not exist

37.

$\frac{d}{dx}\left(2^x\right)=$

a)

$x2^{\left(x-1\right)}$

b)

$2^{\left(x-1\right)}$

c)

$2^x\ln2$

d)

$\frac{2^x}{\ln2}$

38.

What rule does this derivative require?
"If $f\left(x\right)=\left(x-1\right)^2\sin x$ , then f'(0)="

a)

Product

b)

Quotient

c)

Chain

d)

L'Hoptial

39.

Given the graph of f(x), how do you find f'(2)?

a)

Find the area from 0 to 2

b)

Pick the function value on the graph at x=2

c)

Find the slope of the line at x=2

d)

Find the average rate of change from 0 to 2

40.

Given $y=2\cos\left(5x^3\right)$ , find $\frac{dy}{dx}$ .

a)

$2\cos\left(15x^2\right)$

b)

$-30x^2\sin\left(5x^3\right)$

c)

$-2\sin\left(15x^2\right)$

d)

$-30\sin\left(5x^3\right)$

41.

Find the derivative of $y=\ln\left(4x^3+4x^2\right)$

a)

$y'=\frac{1}{4x^3+4x^2}$

b)

$y'=\frac{12x^2+8x}{4x^3+4x^2}$

c)

$\frac{1}{\left(4x^3+4x^2\right)\left(12x^2+8x\right)}$

d)

$\ln\left|4x^3+4x^2\right|$

42.

Find the derivative of $f\left(x\right)=e^{3x}$

a)

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

b)

$3e^{3x}$

c)

$e^{3x}$

d)

$e^{\left(3x-1\right)}$

43.

Given x(0)=4 and v(t). Which expression gives x(8)?

a)

$x\left(0\right)+\int_0^8v\left(t\right)dt$

b)

$\int_0^8v\left(t\right)dt$

c)

$x\left(0\right)+\int_4^8v\left(t\right)dt$

d)

$\int_4^8v\left(t\right)dt$

44.

Given rate-in f(t) and rate-out g(t) and 200 pieces of candy at t=0. How much candy at t=4?

a)

$200+f\left(4\right)-g\left(4\right)$

b)

$f'\left(4\right)-g'\left(4\right)$

c)

$200+\int_0^4\left[f\left(t\right)-g\left(t\right)\right]dt$

d)

$\int_0^4\left[f\left(t\right)-g\left(t\right)\right]dt$

45.

$\int_{ }^{ }\cos\left(2x\right)dx$

a)

$\frac{1}{2}\sin\left(2x\right)+C$

b)

$2\sin\left(2x\right)+C$

c)

$\sin\left(2x\right)+C$

d)

$\frac{1}{2}\cos\left(2x\right)+C$

46.

$\int_{ }^{ }e^{3x}dx$

a)

$3e^{3x}+C$

b)

$e^{\left(3x+1\right)}+C$

c)

$e^{3x}+C$

d)

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

47.

The average rate of change of f(x) on [a,b] is

a)

$\frac{1}{b-a}\int_a^bf\left(x\right)dx$

b)

$f'\left(b\right)-f'\left(a\right)$

c)

$\frac{f\left(b\right)-f\left(a\right)}{b-a}$

d)

$\frac{f\left(b\right)+f\left(a\right)}{2}$

48.

To find the area between two curves f(x) and g(x) on [a,b]:

a)

$\int_a^b\left(top-bottom\right)dx$

b)

$\frac{1}{2}\int_a^b\left(top-bottom\right)dx$

c)

$\frac{1}{2}\int_a^b\left(top-bottom\right)^2dx$

d)

$\pi\int_a^b\left(top-bottom\right)^2dx$

49.

P(t) is measured in gallons per minute. What are the units of P'(t)?

a)

gallons per minute per minute

b)

gallons

c)

gallons per minute

d)

gallon*minutes

50.

$\int_{ }^{ }\frac{1}{x^2}dx$

a)

$\ln\left|x^2\right|+C$

b)

$\frac{1}{2x}$

c)

$-\frac{1}{x}+C$

d)

$\ln\left|x\right|+C$

51.

A water pump adds water at a rate of p(t), measured in gallons/minute. What is $\int_0^4p\left(t\right)dt$ ?

a)

Amount of water added to the tank at 4 minutes

b)

Amount of water added to the tank from 0 to 4 minutes

c)

Rate at which water is added to the tank over 4 minutes

d)

Total amount of water in the tank after 4 minutes

52.

When solving differential equations, what is the first step?

a)

integrate

b)

derive

c)

separate the variables

d)

use the initial condition

53.

$\int_{ }^{ }\frac{1}{1+x^2}dx$

a)

$\arctan x+C$

b)

$\arcsin x+C$

c)

$\ln\left|1+x^2\right|+C$

d)

$\frac{\ln\left|1+x^2\right|}{2x}+C$

54.

Given $h\left(x\right)=g\left(f\left(x\right)\right)$ , find h'(x)

a)

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

b)

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

c)

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

d)

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

55.

Given $h\left(x\right)=f\left(x\right)g\left(x\right)$ , what is h'(x)?

a)

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

b)

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

c)

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

d)

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

56.

$\int_{ }^{ }\left(2x^2-3x^3+4x-1\right)dx$

a)

$\frac{2}{3}x^3-\frac{3}{4}x^4+2x^2-x+C$

b)

$\frac{2}{3}x^3-x^4+2x^2-x+C$

c)

$\frac{1}{3}x^3-\frac{1}{4}x^4+4x^2-x+C$

d)

$4x-9x^2+4+C$

57.

What is f(4) if f(6)=2 and f'(x) is given?

a)

$f\left(2\right)+\int_4^6f'\left(x\right)dx$

b)

$f\left(2\right)+\int_6^4f'\left(x\right)dx$

c)

$f\left(6\right)+\int_4^6f'\left(x\right)dx$

d)

$f\left(6\right)+\int_6^4f'\left(x\right)dx$

58.

How do you justify a point of inflection?

a)

$f"\left(x\right)$ changes sign

b)

$f'\left(x\right)$ changes from increasing to decreasing (or vice versa)

c)

$f'\left(x\right)$ changes sign

d)

$f"\left(x\right)$ changes from increasing to decreasing (or vice versa)

59.

Which of these steps does not have to be done to find an absolute max of f(x)?

a)

set f'(x)=0

b)

Test end points and critical values

c)

Find inflection points of f(x)

d)

Evaluate endpoints or critical values into f(x)

60.

When is the graph of f(x) concave up?

a)

When f'(x) is positive

b)

When f''(x) is positive

c)

When f'(x) is increasing

d)

When f"(x) is increasing

61.

How do we find the volume of a solid of revolution by the method of washers?

a)

$\pi\int_{ }^{ }\left(R^2-r^2\right)dx$

b)

$\pi\int_{ }^{ }\left(R-r\right)^2dx$

c)

$\pi\int_{ }^{ }\left(R-r\right)dx$

d)

$\int_{ }^{ }\left(R^2-r^2\right)dx$

62.

How do you find the displacement of an object?

a)

$v\left(b\right)-v\left(a\right)$

b)

$x\left(b\right)-x\left(a\right)$

c)

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

d)

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

63.

When does a particle change direction?

a)

v(t)=0

b)

v(t) changes sign

c)

x(t) changes sign

d)

a(t) changes sign

64.

What is the justification for a relative minimum?

a)

f' changes from positive to negative

b)

f' changes from negative to positive

c)

f' changes from increasing to decreasing

d)

f' changes from decreasing to increasing

65.

$\int_3^5f'\left(x\right)dx$

a)

$f'\left(5\right)-f'\left(3\right)$

b)

the net area between $f'\left(x\right)$ and the x-axis from x=3 to x=5

c)

$f\left(5\right)-f\left(3\right)$

d)

the net area between $f\left(x\right)$ and the x-axis from x=3 to x=5

66.

When using theorems, what must you do first?

a)

State the theorem used

b)

Show the conditions hold

c)

State the conclusion

67.

When is f(x) increasing?

a)

f'(x) is increasing

b)

f'(x) is decreasing

c)

f'(x) is positive

d)

f'(x) is negative

68.

What is the relationship between continuity and differentiability?

a)

They imply each other

b)

Continuity implies differentiability

c)

Differentiability implies continuity

d)

They have no relationship

69.

When you see the words "with respect to time" you should be thinking what?

a)

d/dx

b)

d/dt

c)

integrate

d)

panic

70.

Which theorem(s) requires ONLY continuity?

a)

EVT

b)

MVT

c)

IVT

d)

L'Hopital

AP Calculus: Avoid the Traps

Total questions: 70

Worksheet time: 53mins

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