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MATH 2313 – Calculus I Quiz #1 – 1.1–1.3 Spring 2018 (21300)

Name: Row: Date: January 25, 2018

Instructions.

• There are two sides to this assignment.

• You may use a scientific calculator.

• Clearly mark the final answer.

• Use proper notation.

• Correct answers without adequate support (or with incorrect work) will receive minimal

credit.

1. Use the graph provided to estimate the following limits or function values.

x

y =

f(x)

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6

−2

−1

0

1

2

3

4

●● ●

●

● ●

(a) limx!0�

f(x) =

(b) limx!0+

f(x) =

(c) limx!0

f(x) =

(d) f(0) =

(e) limx!3

f(x) =

(f) f(3) =

(g) limx!�1

f(x) =

(h) limx!1

f(x) =

2. Evaluate limx!3

x2+ x� 9.

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MATH 2313 – Calculus I Quiz #1 – 1.1–1.3 Spring 2018 (21300)

3. Evaluate the following limits for the function f(x) =x2 � 9

x+ 1. Be as descriptive as

possible with answers involving ‘DNE’, 1, or �1.

(a) limx!3

f(x) =

(b) limx!�1�

f(x) =

(c) limx!�1+

f(x) =

(d) limx!�1

f(x) =

4. Evaluate limx!+1

f(x) = limx!+1

2x8+ 7x3 � 8.1x2 � 7

�7x9 � 4x6 + x3 � 0.1x� 9.For y

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MATH 2313 – Calculus I Test #1 – 1.1–1.3, 1.5, 2.1–2.3 Spring 2018

Name: Date: February 12, 2018

Instructions.


• Use proper notation – symbols, equals signs, . . . .

• If you are instructed to use a specific technique for a problem, use that technique.

• Correct answers without support (or with incorrect work) will receive minimal credit.

• If you have any questions or need extra paper, please come to the front of the room.

1. True or False Clearly mark each statement as True (‘T’) or False (‘F’), please do not

leave any of these blank!

(a) We can always evaluate limits by direct substitution.

(b) To compute the limit of f(x) =1

xas x ! 1, we know arithmetically that

1

1 = 0.

(c) The slope of the secant line and average rate of change describe the same quantity.

(d) Polynomials are continuous everywhere.

(e) To evaluate certain limits it can be helpful or necessary to rationalize radicals or

factor and divide common terms.

(f) For constants c and a, limx!a

c = a.

(g) If f(a) = L, then we know that limx!a

f(x) = L.

(h) When evaluating limits,0

0is always 0.

(i) limx!3�

x = 3�

(j) The derivative of a sum can be simplified as the sum of the derivatives.

(k) The limit of a product can be simplified as the product of the limits.

(l) The derivative f 0(x) gives the slope of the graph of f(x) at a point.

2. Find each of the following:

(a) limx!5

3 =

(b) limx!3

5x+ 2 =

(c)d

dx

⇣3

⌘=

(d)d

dx

⇣5x+ 2

⌘=

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3. Reasoning from graphs. Use the graph in Part a) to answer the following questions

about limits and continuity.

(a) Identify the following limits and function values from the graph on the left. If

you identify that a limit does not exist, justify your answer in the space below

the table.

−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

x

f(x)

●

● ●

●

●

i) limx!�2�

f(x) =

ii) f(�2) =

iii) limx!0�

f(x) =

iv) limx!0+

f(x) =

v) limx!0

f(x) =

vi) f(2) =

vii) limx!2

f(x) =

(b) Using the graph in Part a), identify the location and the type of any discontinuities

visible from the graph of f(x).

4. Computing limits. Show your work! If you identify that a limit does not exist,

justify your answer and provide as much detail about the limit as possible. You mustuse correct notation to earn full credit.

(a) limx!�4

x

x� 4=

(b) limx!4�

x

x� 4=

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(c) limx!2

x2 � 4

x� 2=

(d) limx!2

x2 � 4

x� 4=

(e) limx!4

x� 4px� 2

=

(f) limx!1

r3x6 + 12

27x6 + 4x5 + 1=For y

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5. Additional limits at infinity. State each of the following.

(a) limx!�1

3

x=

(b) limx!�1

3x =

(c) limx!1

px =

(d) limx!1

2

x2=

6. Limits and continuity. Consider the function f(x) =

8><

>:

1

1 + x, x < 1

1

2, x � 1

(a) Find the left- and right-hand limits at x = 1.

(b) Is f(x) continuous at x = 1? Why or why not?

(c) Would you expect f(x) to be continuous elsewhere (i.e., for all other values of x)?Why or why not?

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7. Rates of change. Use the function given to make the assigned calculations.

(a) Compute the average rate of change, labeled ravg, of the function g(x) = 3x2� 2xbetween x = 0 and x = 1.

(b) Find the instantaneous rate of change, labeled rinst, of the function at x = 1.

Show your work.

(c) Write the equation for the tangent line to g(x), called gtan(x), at x = 1.

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8. Limit definition of derivative. State and apply the limit definition of the derivative

to compute the derivative for one of the functions listed below. You must use correct

notation to earn full credit.

(a) State the limit definition for the derivative of f(x).

(b) Apply the limit definition to one of the following functions (maximum credit for

this problem is listed in parenthesis for each function).

f(x) = 2px+ 5 (maximum credit 100%)

g(x) = 3x2 � 2x (maximum credit 80%)

h(x) = x2+ 5 (maximum credit 60%)

Clearly indicate your choice of function before continuing.

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9. Rules of di↵erentiation. Use the rules of derivatives to directly compute derivatives

of the following functions. Use proper notation and labels.

(a) g(x) = 3x2 � 2x

(b) h(x) = x2+ 5

(c)d

dx

⇣⇡2+ x� 2x�2

+3

x3

⌘

(d)d

dx

⇣px� x�1/2

⌘

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MATH 2313 – Calculus Quiz #2 Spring 2018

Name: Date:

Directions. Please complete this quiz using at most a scientific calculator. There are twosides to this quiz.

1. Compute the following derivatives. Use proper notation and mark your final answer.

Beyond applying the appropriate derivative rules and and correct derivatives, you do

not need to simplify your final answers.

(a) Find f 0(x) for f(x) = �1

4(x4

+ 3px+ cos(x))

(b) Find g0(x) for g(x) =

✓3x3 � 1

x2

◆✓px+

1

27

◆

(c) Finddh

dxfor h(x) =

2px� 2

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MATH 2313 – Calculus Quiz #2 Spring 2018

2. Find the following limits of trigonometric functions. Show work for (c) and (d).

(a) limx!0

sin(x)

x

(b) limx!0

1� cos(x)

x

(c) limx!1

sin

⇣3⇡x2

+ 2x

4⇡x2 + 7

⌘

(d) limx!0

sin(7x)

5x

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MATH 2313 – Calculus I Test #2 – 1.6, 2.3 – 2.7 Spring 2018

Name: Date: March 13, 2018

Instructions.


• Use proper notation – symbols, equals signs, . . . .

• Clearly mark the final answer or print it in the box provided.

• There are two sides to each page – extra paper is available.

• Correct answers without support (or with incorrect work) will receive minimal credit.


1. True or False Clearly mark each statement as True or False. Do not leave any of

these blank!

(a) x3= 3x2

(b) The derivative of a quotient is the quotient of the derivatives, for example

d

dx

✓f(x)

g(x)

◆=

f 0(x)

g0(x)

(c) Implicit di↵erentiation is an application of the chain rule.

(d) The chain rule states thatd

dx

⇣f(g(x))

⌘= f(x)g0(x) + g(x)f 0

(x).

(e) For some x values, the functions sin(x) and cos(x) are not continuous.

(f) We can compute derivatives of tan(x), cot(x), sec(x), and csc(x) using

the quotient rule, once we know derivatives of sin(x) and cos(x).

(g) The limit limx!1

cos

✓1

x

◆does not exist since we cannot plug in infinity to

calculate the value.

(h) limx!0

sin(x)

x= 1

(i) The slope of the tangent line to a function at a point is the instantaneous

rate of change at that point.

(j) If f(x) = c for any constant c, we know f 0(x) = 0.

2. Trigonometry derivatives. Compute 4 of the 6 derivatives below.

d

dx

⇣cos(x)

⌘=

d

dx

⇣csc(x)

⌘=

d

dx

⇣cot(x)

⌘=

d

dx

⇣sin(x)

⌘=

d

dx

⇣sec(x)

⌘=

d

dx

⇣tan(x)

⌘=

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3. Computing derivatives: I. Compute the derivatives as instructed. Show all impor-tant steps in your work below. You must use correct notation to earn full credit.

(a) f(x) = x tan(x)

(b) g(x) = (1� 4x2)(3x� 2x2

+ x�4)

(c) h(x) =4 sin(x)

x2 + sec(x)or f(x) =

x2+ cos(x)

5 + x�2 + csc(x)(Label your choice!)

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4. Computing derivatives: II. Compute the derivative of each function. Show allimportant steps in your work below. You must use correct notation to earn full credit.

(a) g(x) = 3p

x2 + sin(x)

(b) h(x) = sin(csc(x))

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5. Implicit di↵erentiation: I. Consider the equation defined by xy + y + 1 = x.

(a) Use implicit di↵erentiation to finddy

dx.

(b) Evaluate the derivative to finddy

dxat the point (2, 1/3) on the graph of the function.

(c) Write the equation for ytan(x), the tangent line to the curve, at the point (2, 1/3).

(d) Findd2y

dx2.

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6. Implicit di↵erentiation: II. Finddy

dxfor the equation 5y2 + cos(x) + sin(y) = x.

7. Computing limits. Find the limits as instructed below. Show all important steps inyour work below. You must use correct notation to earn full credit.

(a) limx!1

sin

✓3⇡x2

+ 6x

6x2 + 2x+ 2

◆=

(b) limx!0

cos

✓3⇡

4+ x

◆=

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7. (cont) Computing limits. Find the limits as instructed below. Show all importantsteps in your work below. You must use correct notation to earn full credit.

(c) limx!0

sin(7x)

9x=

(d) Find the value of one of the following limits

limx!0

sin2(x)

(1 + cos(x))xor lim

x!0

sin(x)

1� cos(x)

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8. Di↵erentiation and antidi↵erentiation. Consider the function

f(x) = 1 +1

3x2 � 3 sin(x)

The first two parts of this problem are required, the third is bonus.

(a) REQUIRED. Find f 0(x).

(b) REQUIRED. Find f 00(x).

(c) BONUS. Find the indefinite integral

Zf(x) dx.

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MATH 2313 – Calculus I Quiz #3 Spring 2018

Name: Date: April 3, 2018

Instructions. Be sure to show and label your work and identify your final answer.

1. Compute the following indefinite integrals.

(a)

Z1

3+

x

2� x2

+ sec(x) tan(x) dx

(b)

Zx0.2

+ x�1.2+

3px dx

2. Evaluate either

Zsin(sin(x)) cos(x) dx or

Z17x2

�4x3

+ 7�9

dx. Show your work!

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MATH 2313 – Calculus I Quiz #3 Spring 2018

3. Consider the sum

11X

k=1

�2k � 1

�.

(a) Write out the first four terms in the sum.

(b) Compute the value of the original sum (not just of the four terms above) using

algebraic rules for sums and the formulas for closed forms of special sums.

Show your work!

(Bonus) Explain why

7X

k=3

k2cannot be evaluated directly (i.e., by theorems), then rewrite

the sum in such a way that it could be evaluated directly using formulas for closed

forms of special sums.

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MATH 2313 – Calculus I Test 3 (Section 2.8, Chapter 4) Spring 2018

Name: Date: April 17, 2018

Instructions.• Attempt every problem.

• You may use a scientific calculator – no books, notes, or other aids.

• Show your work and support your final answers using techniques from this class.


• Attach labeled work on extra paper with staples.

1. Compute each of the following indefinite integrals.

(a)

Z1 + x� 2

x2dx

(b)

Zsec(x) tan(x) + cos(x) dx

(c)

Z�x�2

+2

x3� x0.5

+ x7/3 dx

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2. Choose one of the following related rates problems. The maximum point value is listed

for each with the di�culty increasing from top to bottom.

(a) (up to 75% credit) After tossing a rock in a pond, you notice that the ripples

spread out in the shape of a circle. How quickly is the area of the circle changing

at the instant that the area is 1m2and the radius is increasing at a rate of 0.5

m

s?

(b) (up to 100% credit) While finally pulling down your holiday lights from a 12 ft

ladder, you realize that the top of the ladder starts to slide down the wall as the

base of the ladder starts moving away from the wall. Your friend (who was sup-

posed to brace the ladder) is instead distracted by the shape of the triangle made

by the wall, ladder, and ground. How quickly is the area of the triangle changing

if the base of the ladder is 8 ft from the wall and moving away at 1.5 ft per second

and the top of the ladder is sliding down the wall at ⇡ 1.34 ft per second.

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(cont.) Extra space for related rates.

3. Find the function y(x) passing through (⇡, 0) whose slope at every point is

dy

dx= x+ cos(x)

Show your work!

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We've skipped this for now.


4. Compute two of the following three indefinite integrals:

Z(2x+1) sin(x2

+x) dx or

Z4x2

(3x3 + 2)4dx or

Z(cos(4x))5 sin(4x) dx

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5. Compute the following definite integrals.

(a)

Z 2

�2

1 + x2 dx

(b)

Z 3

1

2xp2x2 � 1 dx

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6. Consider the integral

Z 2

�1

x

2dx.

(a) Sketch the associated net signed area.

(b) Evaluate the definite integral.

(c) State the average value formula and use it to find the average value of the function

f(x) =x

2over the interval [�1, 2].

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7. Evaluate the following sums (do not write out the terms). Recall that

nX

k=1

k2=

n(n+ 1)(2n+ 1)

6

(a)

19X

j=2

j2

(b)

40X

j=1

1

40(2j � 1)

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