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Calculus 2019-2020: Summer Study Guide Name: _____________________________

Mr. Gary Pinkerton (gpinkerton@bdcs.org)

Bishop Dunne Catholic School

Calculus

Summer Math Study Guide

After you have practiced the skills on Khan Academy (list available on bdcs.org/summer2019/read),

complete the following study guide. Be sure to show all work and describe your reasoning, as this

study guide should be a resource for you at the beginning of the school year. If you have any

questions, be sure to contact me at gpinkerton@bdcs.org. I will reply within 48 hours Monday-Friday.

NOTE: The last 5 skills listed are only required for students entering AP Calculus AB. However, due

to possible changes in schedule during the summer, they will be recommended for all Calculus

students. Students in College Prep Calculus may ignore those 5 recommendations.

1. Find Inverse Functions

a. Describe in your own words how to solve algebraically for an inverse function.

Solve algebraically for the inverse of the following functions.

b. 𝑓(𝑥) =2𝑥+8

5 c. 𝑔(𝑥) = √−3𝑥 + 10

2. Transforming Functions

Match the functions below to the corresponding transformations:

______ a. 𝑔(𝑥) = 𝑓(𝑥 + 2) 1. Shift right

______ b. ℎ(𝑥) = 𝑓(𝑥) − 4 2. Shift left

______ c. 𝑘(𝑥) = 𝑓(𝑥 − 3) 3. Shift up

______ d. 𝑞(𝑥) = 3𝑓(𝑥) 4. Shift down

______ e. 𝑟(𝑥) = 𝑓(−𝑥) 5. Vertical stretch

______ f. 𝑝(𝑥) =1

2𝑓(𝑥) 6. Vertical compression

______ g. 𝑣(𝑥) = 𝑓(𝑥) + 7 7. Reflection over x-axis

______ h. 𝑤(𝑥) = −𝑓(𝑥) 8. Reflection over y-axis

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3. Evaluate Composite Functions

Given 𝑓(𝑥) = 𝑥2 + 8𝑥 + 4 and 𝑔(𝑥) = √𝑥 − 1, evaluate the following.

a. 𝑓(𝑔(5)) b. 𝑔(𝑓(3))

4. Evaluate Composite Functions: Graphs & Tables

Evaluate the following function values using the information below.

a. 𝑔(ℎ(−3)) = b. (𝑔 ∘ 𝑓)(−2)

𝑥 −10 −8 −3 2 3 5

𝑔(𝑥) −17 −13 −3 7 9 6

𝑥 −4 −3 −1 0 3 4

ℎ(𝑥) 6 2 −6 −10 −14 −26

5. Find Composite Functions

a. Describe in your own words what the expression 𝑓(𝑔(𝑥)) means.

Given that 𝑓(𝑥) = 3𝑥 − 1 and 𝑔(𝑥) = 𝑥2 + 4, calculate the following. Show work.

b. 𝑓(𝑔(𝑥)) c. 𝑔(𝑓(𝑥))

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6. Positive & Negative and Increasing & Decreasing Intervals

Given the graph below, identify the positive, negative, increasing, and decreasing intervals. Explain

why each interval is described by that word. Use approximations, if necessary.

a. Positive intervals: _________________________

b. Negative intervals: _________________________

c. Increasing intervals: _________________________

d. Decreasing intervals: _________________________

7. Trig Unit Circle Review & Trig Ratios of Special Angles

a. On the Unit Circle, the cosine function is the ______ value and the sine function is the ______

value.

b. Complete the Unit Circle below. Give the x and y-values as well as the angle measure in degrees

and radians.

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8. Reciprocal Trig Ratios

Define a) each reciprocal trig function as a ratio of sides. Then b) determine its value for ∠𝐵 in the

triangle provided:

a. Secant: ________

sec 𝐵 = ________

b. Cosecant: ________

csc 𝐵 = ________

c. Cotangent: ________

cot 𝐵 = ________

9. Solve for a Side in Right Triangles

Solve for the missing side. Show ALL work. Round decimal answers to 3 places.

a. 𝐴𝐶̅̅ ̅̅ = ________ b. 𝐶𝐴̅̅ ̅̅ = ________

10. Find Trig Ratios using the Pythagorean Theorem

Solve for the ratios below using the given information.

a. If cos 𝜃 =4

5 determine the value of cot 𝜃. b. If csc 𝛼 =

7

3 determine the value of cos 𝛼.

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11. Right Triangle Word Problems

a. You’re building a wood cabin. The cabin is 42 feet wide. You buy a bunch of wooden beams 27 ft

long for the roof. In order to place the beams so that they meet exactly in the middle, what is the angle

of elevation in degrees of the roof beams? Round your answer to the nearest hundredth.

b. You’re in charge of building a bridge across a nearby river. You spot a tree on the opposite bank

and mark a spot directly across from it. You then walk a distance of 15 feet down the right and found

that the angle between your side and the distance to the tree is 76°. What is the width of the river?

Round your answer to the nearest hundredth.

12. Evaluating Expressions using Basic Trigonometric Identities

Evaluate the expressions below without the use of a calculator.

a. sin(25°) cos(65°) + cos(25°) sin(65°) b. 2 tan(18°) cot(72°) − 2 sec(18°) csc(72°)

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13. Graph Sinusoidal Functions

Given an equation in the form of 𝑦 = 𝑎 sin(𝑏(𝑥 − 𝑐)) + 𝑑, describe what each term means on the

graph and how you would calculate the following:

a. Amplitude: _______________________________________________________________

b. Period: _______________________________________________________________

c. Midline: _______________________________________________________________

Graph the following trig functions. Give the amplitude, period, and midline.

d. 𝑓(𝑥) = 3sin(4𝑥) + 1

Amplitude: ___________

Period: ___________

Midline: ___________

e. 𝑓(𝑥) = −2cos (𝜋𝑥) − 1

Amplitude: ___________

Period: ___________

Midline: ___________

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** AP Course Summer Assignment Extension **

14. Before we get to specific skills, please answer the following:

a. How is a limit different than a normal function value?

b. What does the → symbol mean mathematically?

c. In your own words, describe what the statement lim𝑥→4

𝑓(𝑥) = 3 means.

15. Estimating Limit Values from Graphs (** AP only **)

Evaluate the limits below. Describe your reasoning.

a. lim𝑥→2

𝑔(𝑥) b. lim𝑥→3

𝑔(𝑥)

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16. One-Sided Limits from Graphs (** AP only **)

a. What does the notation 𝑥 → 3+ and 𝑥 → 3− mean? How are they different than 𝑥 → 3?

Evaluate the limits below. Describe your reasoning.

b. lim𝑥→3−

ℎ(𝑥) c. lim𝑥→6+

ℎ(𝑥)

17. Limits at Infinity of Quotients (** AP only **)

What are the three outcomes for limits that approach infinity?

a. Bottom wins: ________________________________

b. Top wins: ________________________________

c. Evenly matched: ________________________________

Calculate the following limits.

d. lim𝑥→∞

3𝑥2−8

5𝑥+1 e. lim

𝑥→−∞

2𝑥3−12𝑥+7

5𝑥3+11𝑥2+3 f. lim

𝑥→∞

10𝑥+4

5𝑥2−6

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18. Limits by Factoring (** AP only **)

a. What types of limit problems require factoring?

b. How does factoring help you solve these problems?

Calculate the following limits. Show your steps.

c. lim𝑥→−3

4𝑥+12

2𝑥+6 d. lim

𝑥→5

𝑥2−2𝑥−15

𝑥2−4𝑥−5 e. lim

𝑥→2

𝑥2−2𝑥

𝑥2−4

19. Differentiate Polynomials (** AP only **)

Calculate 𝑓 ’(𝑥) for the functions below.

a. 𝑓(𝑥) = 7𝑥 + 9 b. 𝑓(𝑥) = 3𝑥2 + 11𝑥 − 10 c. 𝑓(𝑥) = 𝑥4 – 9𝑥3 + 𝑥

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Answer Key

1.

a. To solve for an inverse function algebraically, switch x and y, then isolate y.

b. 𝑓−1(𝑥) =5𝑥−8

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c. 𝑔−1(𝑥) =−𝑥2+ 10

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2.

a. 2. Horizontal shift left two units

b. 4. Vertical shift down four units

c. 1. Horizontal shift right three units

d. 5. Vertical stretch by a factor of three

e. 8. Horizontal reflection/reflection over the y-axis

f. 6. Vertical compression by a factor of one-half

g. 3. Vertical shift up seven units

h. 7. Vertical reflection/reflection over the x-axis

3.

a. 𝑓 (𝑔(5)) = 22

b. 𝑔 (𝑓(3)) = 6

4.

a. To evaluate 𝑔(ℎ(−3)), locate −3 in the input row for ℎ(𝑥) and determine the

corresponding output as 2. Then, locate 2 on the input row for 𝑔(𝑥) and determine the

corresponding output of 7. Therefore, 𝑔(ℎ(−3)) = 7.

b. To evaluate (𝑔 ∘ 𝑓)(−2), locate -2 on the x-axis and determine the corresponding

output of 𝑓(𝑥) as 0. Then, locate 0 on the x-axis and determine the corresponding

output of 𝑔(𝑥) as 8. Therefore, (𝑔 ∘ 𝑓)(−2) = 8.

5.

a. The expression 𝑓(𝑔(𝑥)) means using 𝑔(𝑥) as the input of 𝑓(𝑥)

b. 𝑓(𝑔(𝑥)) = 3𝑥2 + 11

c. 𝑔(𝑓(𝑥)) = 9𝑥2 − 6𝑥 + 5

6. Positive intervals are x-values for which the function has an output greater than zero. Negative

intervals are x-values for which the function has an output less than zero.

a. Positive intervals: {(−3.5, −1), (1, 3.5)}

b. Negative intervals: {(−5, −3.5), (−1,1), (3.5,5)}

Increasing intervals are x-values for which the function has a positive rate of change.

Decreasing intervals are x-values for which the function has a negative rate of change.

c. Increasing intervals: {(−4.5, −2.25), (0,2.25), (4.5,5)}

d. Decreasing intervals: {(−5, −4.5), (−2.25,0), (2.25,4.5)}

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7.

a. The cosine function is the x-value and the sine function is the y-value.

b.

8.

a. Secant is the reciprocal of cosine; ℎ𝑦𝑝

𝑎𝑑𝑗 sec 𝐵 =

25

24

b. Cosecant is the reciprocal of sine; ℎ𝑦𝑝

𝑜𝑝𝑝 csc 𝐵 =

25

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c. Cotangent is the reciprocal of tangent; 𝑎𝑑𝑗

𝑜𝑝𝑝 cot 𝐵 =

24

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9.

a. 𝐴𝐶̅̅ ̅̅ = 2.868

b. 𝐶𝐴̅̅ ̅̅ = 4.284

10.

a. cot 𝜃 =4

3

b. cos 𝛼 =2√10

7

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11. a. 𝜃 = 38.942°

b. 𝑑 = 60.162 ft

12. a. 1

b. -2

13. a. Amplitude: the vertical distance from the midline to a maximum or minimum value;

calculated by |a|

b. Period: the horizontal distance required for the function to complete one full

revolution; calculated by 2𝜋

𝑏

c. Midline: the imaginary horizontal line that passes through the middle of the function’s

maximum and minimum points; calculated by y = d

d. i. Amplitude: 3

ii. Period: 𝜋

2

iii. Midline: 𝑦 = 1

e. i. Amplitude: 2

ii. Period: 2

iii. Midline: 𝑦 = −1

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14. a. A limit in math is a term that describes where a function is going at an x-value,

regardless of where the function actually is at that value.

b. The → symbol indicates where the x-value is approaching.

c. The statement lim𝑥→4

𝑓(𝑥) = 3 is read “the limit of 𝑓(𝑥) as x approaches four is three”

and means that as the function gets closer to an x-value of four from either side, the y-

value gets closer and closer to three.

15. a. lim

𝑥→2𝑔(𝑥) = 1.5; Even though the function equals −2.4 at 𝑥 = 2, the function

approaches 1.5 as it gets closer to 2 from either side.

b. lim𝑥→3

𝑔(𝑥) does not exist. The left and right-hand sides approach different values, so the

function does not agree about where it is going.

16. a. The notation 𝑥 → 3+ means “as x approaches three from the right”. The notation 𝑥 →

3− means “as x approaches three from the left”. These are different than 𝑥 → 3, which

can only be answered if the left- and right-hand limits agree.

b. lim𝑥→3−

ℎ(𝑥) = 6; Even though the function equals 2 at 𝑥 = −3, from the left-hand side

the function approaches a y-value of 6 as x gets closer to −3.

c. lim𝑥→6+

ℎ(𝑥) = 2.7; Even thought the function is undefined at 𝑥 = 6, from the right-hand

side the function approaches a y-value of approximately 2.7 as x gets closer to 6.

17. a. If the exponent on bottom is bigger, the limit is zero.

b. If the exponent on top is bigger, the limit is unbounded.

c. If the exponents are the same, divide leading coefficients.

d. DNE, unbounded, or ∞

e. 2

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f. 0

18. a. Limits of rational functions often require factoring, particularly when the limit is of a

hole in that function.

b. This method allows you to simplify the function and calculate the y-coordinate of the

hole, which answers the question.

c. lim𝑥→−3

4𝑥+12

2𝑥+6= 2

d. lim𝑥→5

𝑥2−2𝑥−15

𝑥2−4𝑥−5=

4

3

e. lim𝑥→2

𝑥2−2𝑥

𝑥2−4=

1

2

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19. a. 𝑓 ’(𝑥) = 7 b. 𝑓 ’(𝑥) = 6𝑥 + 11 c. 𝑓 ‘(𝑥) = 4𝑥3 – 27𝑥2 + 1

Dear Students,

It has truly been an honor being your teacher, and I know you’ll go on to do great things whether or

not I’m there with you next year.

See you at graduation,

Mrs. Smith

PS- Congrats on finishing this study guide! 14 pages—what was Mr. Braun thinking??