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AP Calculus Review - Limits, Derivatives, and Integrals

Total questions: 62

Worksheet time: 2hrs 54mins

Name
Class
Date
1.
Evaluate the limit
a)
a
b)
b
c)
c
d)
d
2.
Evaluate the limit
a)
a
b)
b
c)
c
d)
d
3.
Find the equation of the line tangent to the function at the given point
a)
a
b)
b
c)
c
d)
d
4.
What is the derivative of csc(x)?
a)
csc(x)cot(x)
b)
-csc(x)cot(x)
c)
-csc2(x) 
d)
-cot2(x)
5.
Find h'(3)
a)
-2
b)
0
c)
1
d)
3
6.
When we "take the derivative" of a function what are we finding?
a)
What's a derivative?
b)
The rate at which our struggles in Calculus are increasing.
c)
The slope of the secant line
d)
The slope of the tangent line
7.
Find dy/dx by Implicit Differentiation 
x3 +y3  = 36
a)
6 -x
b)
3x2 +3y2 
c)
−x2/y2
d)
0
8.

Find the value for that makes f(x) continuous.

a)

0

b)

1

c)

1.5

d)

2

e)

No such value exists

9.
Find the average rate of change of Pete's height from 3 years old to 5 years old.
a)
2 inches per year
b)
8 inches per year
c)
2 years
d)
4 inches per year
10.
If a function has a derivative that is negative, what does that tell you?
a)
The function is increasing
b)
The function is decreasing
c)
The concavity of the function is up
d)
The concavity of the function is down
11.

Given the table containing some values of differentiable functions and their derivatives, solve:
If  h(x)=f(x)g(x)h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}  Find h′(4)h'\left(4\right)  

a)

 00  

b)

 22  

c)

 56\frac{5}{6}  

d)

 −3-3  

12.

Use L'Hospital to evaluate the limit.

a)

 00  

b)

 e99\frac{e^9}{9}  

c)

 3e93e^9  

d)

 6e96e^9 

e)

nonexistent

13.

Use L'Hospital to evaluate the limit.

a)

-1/2

b)

0

c)

1/2

d)

1

e)

nonexistent

14.

Is the above function differentiable at

x = 0? Why?

a)

Yes. The derivative would be equal to zero since there is a vertical tangent.

b)

No. The derivative would not exist at x = 0 since there is a vertical tangent.

c)

No. The derivative does not exist since lim⁡x→0 x13\lim_{x\rightarrow0}\ x^{\frac{1}{3}} does not exist.

d)

Yes since lim⁡h→0 (x+h)13−x13h\lim_{h\rightarrow0}\ \frac{\left(x+h\right)^{\frac{1}{3}}-x^{\frac{1}{3}}}{h} exists.

15.
Where is the graph continuous yet NOT differentiable?
a)
x = a, b, c, d
b)
x = b, c, d
c)
x = a, b, 
d)
x = b, d
16.
True or False: Continuity implies differentiability 
a)
True
b)
False
17.
True or False: Differentiability implies continuity 
a)
True
b)
False
18.

When applying calculus, the second derivative of position helps find...

a)

the distance traveled by an object.

b)

The velocity of a particle at any given point

c)

acceleration of an object at any given time

19.
What is the derivative of xn?
a)
(n-1)xn
b)
nxn+1
c)
(n+1)xn-1
d)
nxn-1
20.

A farmer wants to construct a rectangular pigpen using 400 ft of fencing. The pen will be built next to an existing stone wall, so only three sides of fencing need to be constructed to enclose the pen. What dimensions should the farmer use to construct the pen with the largest possible area?

a)

100ft x 200ft

b)

102ft x 196 ft

c)

50 ft x 300 ft

d)

50 ft x 175 ft

21.
Find the point(s) of inflection for f(x) = 2(x)1/5 + 3
a)
None
b)
x = 0
c)
x = 0, 2
d)
x = -2, 0, 2
22.
What are the intervals of the graph increasing for f(x) = 2x4- 4x2 + 1
a)
(-1,0)
b)
(0,1)
c)
(-∞,-1) and (1,∞)
d)
(0,1) and (-1,0)
23.

Find the derivative f(x) = xex

a)

f'(x) = ex

b)

f'(x) = xex + xex

c)

f'(x) = ex - xex

d)

f'(x) = ex + xex

24.

All of the follow are TRUE about Ms Kwong except...

Todos los siguientes son VERDADEROS sobre la Sra. Kwong, excepto ...

a)

She loves grocery shopping.

A ella le encanta ir de compras.

b)

She only wants to use an Android phone.

Ella solo quiere usar un teléfono Android.

c)

She loves to sing and acted in a school play in high school.

Le encanta cantar y actuó en una obra de teatro en la escuela secundaria.

d)

She is in her late 20s (> 25 years old).

Ella tiene alrededor de 20 años (> 25 años).

25.

Would you rather go back to kindergarten or go straight to having a job? Why?

4 lines
26.

Given the attached graph of the object's POSITION, identify intervals where the object is moving right.

a)

(0,1), (17,18)

b)

(7,9), (13,18)

c)

(6,8), (11,18)

d)

(4,7), (9,13)

27.
Rachel is standing atop a 13 ft ladder. The ladder is leaning against a vertical wall. The ladder starts sliding away from the wall at a rate of 3 ft/sec. How fast is the ladder sliding down the wall when the tip of the ladder is 5 ft high?
a)
3 ft/sec
b)
-7.2 ft/sec
c)
7.2 ft/sec
d)
12
28.
sin π/4
a)
√2∕2
b)
1/2
c)
√3/2
d)
0
29.
cos 5π/6
a)
-1/2
b)
1/2
c)
√3/2
d)
-√3/2
30.

I'd love your feedback! Based on the past 3 weeks, which of the following did you like learning from the most? Please select all that apply.

a)

Edpuzzle videos

b)

DeltaMath

c)

Uploading skill/mock practice

d)

Office hours / video live discussion

e)

Emails / gchats / private comments from teacher on your work

31.

What do you like listening to when you study?

4 lines
32.

May is Asian American Pacific Islander Heritage Month!


This person was the first woman to serve as the district attorney of the state of California. Her mother is from India.

a)

Representative Alexandria Ocasio-Cortez

b)

Senator Tammy Duckworth

c)

Senator Kamala Harris

d)

Secretary Elaine Chow

33.

Why was May selected to celebrate Asian American Pacific Islander Heritage Month? Pick 2 only.

a)

Japanese Americans were forced to leave their homes and live in internment camps in May 1942.

b)

The immigration of the first Japanese to the United States was on May 7, 1843.

c)

The Vietnam - American war ended in May 1975.

d)

The transcontinental railroad (which was mostly built by Chinese Americans) was completed on May 10, 1869.

34.

May is Asian American Pacific Islander Heritage Month!


Eric Yuan is the founder of which video conferencing app?

a)

Instagram Live

b)

WebEx

c)

Google Meet / Hangouts

d)

Zoom

35.

May is Asian American Pacific Islander Heritage Month!


In 2016, this musician received the Record of the Year Grammy.

a)

Beyoncé

b)

Kelly Clarkson

c)

Bruno Mars

d)

Adele

36.
a)
Average Rate of Change
b)
Instantaneous Rate of Change
c)
Intermediate Value Theorem
d)
Average Value of f
37.
 If (a,b) is a local minimum, then what will be true about f''(a)?
a)
It's postive
b)
It's negative
c)
It's zero
d)
Cannot be determined
38.
a)
speed
b)
acceleration
c)
displacement
d)
total distance
39.
a)
Mean value theorem
b)
Rolle's theorem
c)
Intermediate Value Theorem
d)
Fundamental Theorem of Calculus
40.
a)
f(a) - f(b)
b)
f(b) - f(a)
c)
f '(b) - f '(a)
d)
f '(a) - f '(b)
41.
a)
position 
b)
velocity
c)
acceleration
d)
total distance
42.

Find F'(x) given  F(x)=∫0x2csc⁡2x dxF\left(x\right)=\int_0^{x^2}\csc^2x\ dx  

a)

 csc⁡2(x2)\csc^2\left(x^2\right)  

b)

 csc⁡2(x)\csc^2\left(x\right)  

c)

 2xcsc⁡2(x)2x\csc^2\left(x\right)  

d)

 2xcsc⁡2(x2)2x\csc^2\left(x^2\right)  

43.

What does this picture represent?

a)

Left Riemann Sum

b)

Right Riemann Sum

c)

Middle Riemann Sum

d)

Trapezoidal Sum

44.

What does picture represent?

a)

Left Riemann Sum

b)

Right Riemann Sum

c)

Middle Riemann Sum

d)

Trapezoidal Sum

45.
For a function that is strictly decreasing, a right hand Riemann Sum is which of the following:
a)
Overestimate
b)
Underestimate
c)
Exact Solution
d)
Unable to Determine
46.

Based on the table, use a Right Riemann sum and 4 sub-intervals to estimate the Area under the curve. (Choose the correct set-up.)

a)

5(3) + 1(4) + 2(5) + 1(7)

b)

5(4) + 1(5) + 2(7) + 1(6)

c)

5(3) + 6(4) + 8(5) + 9(7)

d)

0(3) + 5(4) + 6(5) + 8(7)

47.
a)
Average Value
b)
Net Area
c)
Total Area
d)
Average Rate of Change
48.
∫ 1/x dx
a)
ln x
b)
ln x + C
c)
-1/x²
d)
-1/x² + C
49.

Find the integral:

a)

1/8(4x2 + 3)4 + c

b)

1/2(4x2 + 3)4 + c

c)

1/32(4x2 + 3)4 + c

d)

1/4(4x2 + 3)4 + c

50.
∫ e2x dx
a)
e2x +C
b)
2e2x + C
c)
(1/2)ex + C
d)
(1/2)e2x +C
51.

Evaluate the definite integral. I encourage you to practice using the CALCULATOR trick to evaluate.

a)

8 pi

b)

16 pi

c)

128 pi

d)

Not possible

52.

Evaluate in terms of area

∫02 f(x) dx

a)

4

b)

-4

c)

4 + 2π

d)

-4 - 2π

53.

Which of the following cases would lead to an object slowing down?

a)

positive acceleration, positive velocity

b)

positive acceleration, negative velocity

c)

negative acceleration, positive velocity

d)

negative acceleration, negative velocity

54.

The slope of the position-time graph tells us the

a)

acceleration

b)

displacement

c)

velocity

d)

direction of travel

55.

If f '(x) changes from positive to negative, f(x) has __________ at x.

a)

a maximum

b)

a minimum

c)

no extrema

56.

Given the sign chart for f '(x), f(x) has ...

a)

a local maximum at x = -2.

b)

a local maximum at x = 4.

c)

local maxima at x = -2 and x = 4.

d)

no extrema.

57.

Given the sign chart for f '(x), f(x) has ...

a)

a local maximum at x = -2.

b)

a local minimum at x = -2 and

a local maximum at x = 4.

c)

a local maximum at x = -2 and

a local minimum at x = 4.

d)

no extrema.

58.

Given the graph of f '(x), f(x) has ...

a)

a local min at x = -6

b)

a local min at x = 2

c)

a local min at x = 2 and

a local max at x = -6

d)

local mins at x = -2, 5 and

a local max at x = 3

59.

Given the sign chart for f '(x) and table of f(x), f(x) has ...

a)

a local maximum at (-10,5).

b)

an absolute minimum at (-2,0).

c)

a local minimum at (20,7).

d)

all of the above

60.
For a function f(x), f'(-3) = 5 indicates f(x) is ___________ at x=-3.
a)
increasing
b)
decreasing
c)
concave up
d)
concave down
61.

Find the critical value(s) of

f(x) = x2 + 2x + 1.

a)

x = -1

b)

x = 2

c)

x = -1, 0

d)

x = 0

62.

The concavity of a function is described by its _______________.

a)

first derivative

b)

second derivative

c)

third derivative

d)

the zeros

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