PMI AP Calculus AB NJCTL.org

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REVIEW UNIT PROBLEM SETS

PROBLEM SET #1 – Slopes ***Calculators Not Allowed***

Calculate the slope of the line containing the following points:

1. (2,8) 𝑎𝑛𝑑 (−4,6) 2. (−4, −7) 𝑎𝑛𝑑 (3,0) 3. (−3, −6) 𝑎𝑛𝑑 (−1, −6) 4. (4, −2) 𝑎𝑛𝑑 (4,5)

5. (4

5, 6) 𝑎𝑛𝑑 (

3

5, 4)

6. (3

2, −4) 𝑎𝑛𝑑 (2,0)

7. (11

14,

3

7) 𝑎𝑛𝑑 (

9

14,

5

7)

8. (8

9,

2

3) 𝑎𝑛𝑑 (

5

6,

2

5 )

9. (−4, −3) 𝑎𝑛𝑑 (0, −11)

10. (−3

7,

3

8) 𝑎𝑛𝑑 (−

1

6,

5

6)

PMI AP Calculus AB NJCTL.org

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PROBLEM SET #2 – Equations of Lines ***Calculators Not Allowed***

For each of the following questions, write the equation of the line given the specific information. 1. Passes through (2,3) and 𝑚 = 2

2. Passes through (−2,4) and 𝑚 =1

2

3. Passes through (−4, −5) 𝑎𝑛𝑑 (2,7) 4. Passes through (3, −5) 𝑎𝑛𝑑 (−3,5)

5. Passes through (-1, 2) and 𝑚 = 0

6. Passes through (-1, 2) and the slope is undefined.

7. Passes through (-2, 2) and is parallel to 2𝑦 =4𝑥 − 12 8. Passes through (-3, 2) and is perpendicular

to 15𝑦 = 10𝑥 + 2

9. 𝑚 =3

5 𝑎𝑛𝑑 𝑏 = 0

10. 𝑚 = 0 𝑎𝑛𝑑 𝑏 = −1

7

PMI AP Calculus AB NJCTL.org

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Problem Set #3 – Functions & Graphing Functions ***Calculators Not Allowed***

1. a) True/False 3𝑥2 + 5𝑦 = 7 − 2𝑥 is a function. b) Why or why not? ______________________________________________________________ ___________________________________________________________________________

2. a) True/False 2𝑥2 + 3𝑦2 = 11 is a function. b) Why or why not? ______________________________________________________________ ___________________________________________________________________________

3. a) True/False The following table represents a function. b) Why or why not? ______________________________________________________________ ___________________________________________________________________________

4. a) True/False The following table represents a function. b) Why or why not? ______________________________________________________________ ___________________________________________________________________________

5. Evaluate 𝑔(2) if 𝑔(𝑥) = 3𝑥2 − 5𝑥 + 5

6. Evaluate 𝑔(−3) if 𝑔(𝑥) = −𝑥2 − 2𝑥 + 15

7. Evaluate 𝑓(𝑥 − 2) if 𝑓(𝑥) = −2𝑥2 − 3𝑥 + 11

𝑥 −2 −1 0 1 2 5

𝑦 5 3 6 3 2 −4

𝑥 −2 2 0 −2 2 5

𝑦 4 3 6 3 2 −3

PMI AP Calculus AB NJCTL.org

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8. Evaluate 𝑓(5 − 𝑥) if 𝑓(𝑥) =4𝑥−3

2−𝑥

9. Write the new equation of the function 𝑦 = √𝑥 with the following transformations: reflection over x-axis, vertical compression of 1/2, right 3 units, and up 4 units

10. Write the new equation of the function 𝑦 =1

𝑥 with the following transformations:

horizontal compression of 3, right 3 units, and down 2 units

11. Write the new equation of the function 𝑦 = 𝑥2 with the following transformations: reflection over x-axis, vertical stretch of 2, left 2 units, and down 3 units

12. Write the new equation of the function 𝑦 = ln 𝑥 with the following transformations: vertical stretch of 3, right 5 units, and up 2 units

PMI AP Calculus AB NJCTL.org

Problem Set #4 – Piecewise Functions ***Calculators Not Allowed***

1. Graph the following piecewise function:

𝑓(𝑥) = {𝑥 − 1 − 5 ≤ 𝑥 < −1−2 − 1 ≤ 𝑥 < 2−𝑥 + 3 2 < 𝑥 ≤ 6

2. Graph the following piecewise function:

𝑓(𝑥) = {2𝑥 + 1 𝑥 < 1

−𝑥2 + 5 𝑥 ≥ 1

PMI AP Calculus AB NJCTL.org

3. Graph the following piecewise function:

𝑓(𝑥) = {−2𝑥 − 6 𝑥 < −4

𝑥 + 4 − 4 ≤ 𝑥 < 2 𝑥2 − 3 𝑥 ≥ 2

4. Graph the following piecewise function:

𝑓(𝑥) = {|𝑥 + 3| − 1 − 6 ≤ 𝑥 < 1

−𝑥 − 2 𝑥 > 1

PMI AP Calculus AB NJCTL.org

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5. Given:

𝑓(𝑥) = { 𝑥2 − 5 𝑥 < −5

11 − 5 ≤ 𝑥 < 1−3𝑥2 + 10 𝑥 ≥ 1

a) Find 𝑓(−7)

b) Find 𝑓(−5)

c) Find 𝑓(0)

d) Find 𝑓(1)

e) Find 𝑓(3)

6. Given:

𝑓(𝑥) = { |𝑥 − 5| + 3 𝑥 < −2

2𝑥3 − 4 𝑥 ≥ −2

a) Find 𝑓(−5)

b) Find 𝑓(−2)

c) Find 𝑓(0)

d) Find 𝑓(1)

PMI AP Calculus AB NJCTL.org

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Problem Set #5 – Function Composition ***Calculators Not Allowed***

Use the following functions to answer questions 1 – 16.

𝒇(𝒙) = 𝟑𝒙 𝒈(𝒙) = −𝟐𝒙 − 𝟏 𝒉(𝒙) = |𝒙 − 𝟑| 𝒌(𝒙) = −𝟒𝒙𝟐

1. 𝑔(𝑓(2)) =

2. 𝑓 ∘ 𝑔(3) =

3. ℎ (𝑓(𝑔(0))) =

4. 𝑘 ∘ 𝑔 ∘ ℎ(−2) =

5. 𝑔 ∘ 𝑘(𝑥) =

6. 5𝑓(𝑥) − 3𝑔(𝑥) =

7. 𝑔 ∘ 𝑓 ∘ 𝑘(𝑥) =

8. 𝑓(𝑥)

𝑔(𝑥)=

9. 𝑓(𝑘(3)) =

10.ℎ ∘ 𝑓(−7) =

11. 𝑘 (𝑓(ℎ(−5))) =

12. ℎ ∘ 𝑘 ∘ 𝑔(−1) =

13. 𝑘 ∘ 𝑓(𝑥) =

14. −2𝑔(𝑥) + 4𝑓(𝑥) =

15. 𝑔 ∘ 𝑘 ∘ 𝑓(𝑥) =

16. 𝑔(𝑥)

𝑘(𝑥)=

PMI AP Calculus AB NJCTL.org

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Problem Set #6 – Function Roots ***Calculators Not Allowed***

Find any real roots, if they exist, for questions 1 – 12.

1. 𝑦 = 𝑥2 − 2𝑥 − 8

2. 𝑓(𝑥) = 𝑥2 + 4𝑥 − 32

3. 𝑟(𝑡) = 𝑡3 − 11𝑡2 + 18𝑡

4. 𝑦 = −3𝑥2 − 10𝑥 + 8

5. 𝑟(𝑡) = 𝑡3 − 5𝑡2 + 12𝑡

6. 𝑟(𝑡) = 𝑡2 − 6𝑡 + 17

7. 𝑦 = 2𝑥2 − 𝑥 − 10

8. 𝑓(𝑥) = −𝑥2 + 4𝑥 + 12

9. 𝑓(𝑥) = 5𝑥2 + 5𝑥 + 12

10. 𝑦 = 3𝑥2 − 8𝑥 − 2

11.𝑞(𝑧) = 5𝑧3 + 2𝑧2 − 7𝑧

12. 𝑦 = 3𝑥3 + 6𝑥2 − 𝑥

PMI AP Calculus AB NJCTL.org

Problem Set #7 – Domain & Range ***Calculators Not Allowed***

1.

Domain: ____________________ Range: ____________________ 2.

Domain: ____________________ Range: ____________________ 3.

Domain: ____________________ Range: ____________________

4.

Domain: ____________________ Range: ____________________ 5. Domain: ____________________ Range: ____________________ 6.

Domain: ____________________ Range: ____________________

PMI AP Calculus AB NJCTL.org

7. 𝑦 = −2𝑥 + 12 Domain: ____________________ Range: ____________________

8. 𝑦 = 𝑥2 + 4𝑥 − 32 Domain: ____________________ Range: ____________________

9. 𝑦 = −3𝑥2 + 6𝑥 + 5 Domain: ____________________ Range: ____________________

10. 𝑦 = √𝑥 + 5 − 2 Domain: ____________________ Range: ____________________

11. 𝑦 = −√𝑥 + 7 + 5 Domain: ____________________ Range: ____________________

12. 𝑦 =5𝑥+2

√𝑥+3

Domain only: ____________________

13. 𝑦 = ln(𝑥 − 3) Domain: ____________________ Range: ____________________ 14. 𝑦 = 4 ln(𝑥 + 2) − 1 Domain: ____________________ Range: ____________________

15. 𝑦 = −𝑥3 + 14 Domain: ____________________ Range: ____________________

16. 𝑦 = √𝑥 − 8 3

+ 4 Domain: ____________________ Range: ____________________

17. 𝑦 =2𝑥

𝑥2+2𝑥−8

Domain: ____________________ Range: ____________________

18. 𝑦 =25

2𝑥2+5𝑥−3+ 4

Domain only: ____________________

PMI AP Calculus AB NJCTL.org

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Problem Set #8 – Inverses ***Calculators Not Allowed***

State whether the following functions are inverses.

1. 𝑓(𝑥) = (𝑥 − 1)2

𝑔(𝑥) = 1 + 𝑥2

2. 𝑔(𝑧) =3

𝑧+ 5

𝑓(𝑧) =3

𝑧−5

3. 𝑔(𝑥) = √𝑥 − 3 +5

ℎ(𝑥) = (𝑥 − 5)2 + 3

4. 𝑘(𝑡) = 2𝑡3 − 1

𝑚(𝑡) =√𝑡+1

3

2

Find the inverse of each function.

5. ℎ(𝑥) = 4√𝑥3

+ 2

6. 𝑘(𝑡) = −5𝑡 + 11

7. 𝑚(𝑥) = 7𝑥2 − 4

8. 𝑔(𝑧) = (𝑧 − 3)5 + 2

PMI AP Calculus AB NJCTL.org

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Problem Set #9 – Trigonometry ***Calculators Not Allowed***

Evaluate each of the following.

1. csc7𝜋

6

2. tan𝜋

3

3. sin7𝜋

4

4. 𝑠𝑒𝑐𝜋

6

5. 𝑐𝑜𝑡𝜋

11. 𝑐𝑠𝑐15𝜋

4

12. 𝑐𝑜𝑡2𝜋

4

13. 𝑠𝑖𝑛 4𝜋

3

14. 𝑐𝑠𝑐4𝜋

3

15. 𝑐𝑜𝑠11𝜋

6

6. csc𝜋

4

7. sin𝜋

2

8. cos5𝜋

3

9. 𝑐𝑠𝑐14𝜋

6

10. 𝑡𝑎𝑛2𝜋

3

16. 𝑐𝑜𝑡4𝜋

3

17. 𝑡𝑎𝑛𝜋

2

18. 𝑐𝑜𝑡5𝜋

4

19. 𝑐𝑠𝑐3𝜋

4

20. 𝑐𝑜𝑠5𝜋

2

PMI AP Calculus AB NJCTL.org

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Evaluate:

21. 𝑐𝑜𝑠−1 (−√3

2)

22. sin−1( 0)

27. 3 + 2 cos2 (3𝜋

2)

28. cot−1(−1)

23. 𝑐𝑠𝑐−1 (2√3

3)

24.tan−1(−√3

3)

25.sin−1(1)

29. 𝑠𝑒𝑐−1( √2)

30. 2 − 3 sin2 (𝜋

2)

31. cos−1 (−1

2)

26. 𝑐𝑜𝑠−1(0)

32.csc−1(1)

PMI AP Calculus AB NJCTL.org

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Problem Set #10 – Exponents ***Calculators Not Allowed***

Simplify:

1. 15𝑚11𝑘−5

10𝑚4𝑘−12

2. 21𝑒3𝑓14

42𝑒7𝑓−3

3. (3𝑥2 − 5𝑥 + 2)(𝑥2 + 3𝑥 − 1)

4. (2𝑦3 + 3𝑦2 − 4)(𝑦2 + 7𝑦 − 3)

5. ((2𝑎4𝑏2)3

(4𝑎9𝑏−5)2)−3

6. ((3𝑚3𝑛4)4

(6𝑚−8𝑛12)2)−2

7. (−5𝑥3𝑦−6𝑧4)−3

8. (4𝑚−2𝑘4𝑝)−2

9. (2𝑥3𝑦3𝑧)2(15𝑥10𝑦4𝑧0)

10. (−5𝑟5𝑠−2𝑡4)2(3𝑟𝑠5𝑡2)

11. (5𝑎 − 2𝑏)2

12. (𝑐 − 4)3

PMI AP Calculus AB NJCTL.org

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Problem Set #11 – Logarithms ***Calculators Allowed***

Solve the following equations:

1. log𝑥 16 = 4

2. log𝑥 125 = 3

3. 33𝑥+2 = 108

4. 24𝑥−3 = 12 5. log(7𝑥 + 3) = log (2𝑥 + 23)

6. log(2𝑥 + 3) = log (12𝑥 − 1)

7. 53𝑥 = 26

8. 42𝑥 = 54

9. log2(𝑟 + 3) + log2(𝑟) = log2 10

10. log4(𝑟 + 5) − log4(𝑟) = log4 10

PMI AP Calculus AB NJCTL.org

REVIEW PROBLEM SET ANSWER KEYS

Problem Set #1 – Slopes

1. 1

3

2. 1

3. 0

4. 𝑢𝑛𝑑𝑒𝑓

5. 10

6. 8

7. −2

8. 24

5

9. −2

10. 7

4

Problem Set #2 – Eqns. of Lines

1. 𝑦 − 3 = 2(𝑥 − 2) 𝑜𝑟 𝑦 = 2𝑥 − 1

2. 𝑦 − 4 =1

2(𝑥 + 2) 𝑜𝑟 𝑦 =

1

2𝑥 + 5

3. 𝑦 + 5 = 2(𝑥 + 4) 𝑜𝑟 𝑦 = 2𝑥 + 3

4. 𝑦 + 5 = −5

3(𝑥 − 3) 𝑜𝑟 𝑜𝑟 𝑦 = −

5

3𝑥

5. 𝑦 = 2

6. 𝑥 = −1

7. 𝑦 = 2𝑥 + 6

8. 𝑦 − 2 = −3

2(𝑥 + 3) 𝑜𝑟 𝑦 = −

3

2𝑥 −

5

2

9. 𝑦 =3

5𝑥

10. 𝑦 = −1

7

Problem Set #3 – Functions/Graphing

1. a) TRUE b) Each x-value corresponds to

only one y-value.

2. a) FALSE b) Does not pass vertical line

test; more than one y-value for each x-value

3. a) TRUE b) Each x-value corresponds to

only one y-value

4. a) FALSE b) Does not pass vertical line

test; more than one y-value for each x-value

5. 7

6. 12

7. −2𝑥2 + 5𝑥 + 9

8. 4𝑥−17

3−𝑥

9. 𝑦 = −1

2√𝑥 − 3 + 4

10. 𝑦 =1

3𝑥−9− 2

11. 𝑦 = −2(𝑥 + 2)2 − 3

12. 𝑦 = 3ln(𝑥 − 5) + 2

Problem Set #4 – Piecewise Functions

1. See graph

2. See graph

3. See graph

4. See graph

5. a) 44 b) 11 c) 11 d) 7 e) -17

6. a) 13 b) -20 c) -4 d) -2

PMI AP Calculus AB NJCTL.org

Problem Set #5 – Function Composition

1. −13

2. −21

3. 6

4. −484

5. 8𝑥2 − 1

6. 21𝑥 + 3

7. 24𝑥2 − 1

8. 3𝑥

−2𝑥−1

9. −108

10. 24

11. −2304

12. 7

13. −36𝑥2

14. 16𝑥 + 2

15. 72𝑥2 − 1

16. 2𝑥+1

4𝑥2

Problem Set #6 – Function Roots

1. 𝑥 = −2 & 𝑥 = 4

2. 𝑥 = −8 & 𝑥 = 4

3. 𝑡 = 0 & 𝑡 = 2 & 𝑡 = 9

4. 𝑥 =2

3 & 𝑥 = −4

5. 𝑡 = 0

6. no real roots

7. 𝑥 = −2 & 𝑥 =5

2

8. 𝑥 = −2 & 𝑥 = 6

9. no real roots

10. 𝑥 =4±√22

3

11. 𝑥 = 0, 𝑥 = −7

5 & 𝑥 = 1

12. 𝑥 =−3±2√3

3 𝑎𝑛𝑑 𝑥 = 0

Problem Set #7 – Domain & Range

1. Domain: (−∞, 1) ∪ [4, ∞)

Range: ℝ

2. Domain: ℝ

Range: (−∞, 3]

3. Domain: [−5,5]

Range: [−2,2]

4. Domain: ℝ

Range: 𝑦 = 3

5. Domain: (−∞, −3] ∪ (−2, ∞)

Range: (−∞, 3]

6. Domain: (−∞, 2] ∪ (3, ∞)

Range: 𝑦 = −2 𝑎𝑛𝑑 (−1, ∞)

7. Domain: ℝ

Range: ℝ

8. Domain: ℝ

Range: [−36, ∞)

9. Domain: ℝ

Range: (−∞, 8]

10. Domain: [−5, ∞)

Range: [−2, ∞)

11. Domain: [−7, ∞)

Range: (−∞, 5]

12. Domain: (−3, ∞)

13. Domain: (3, ∞)

Range: ℝ

14. Domain: (−2, ∞)

Range: ℝ

15. Domain: ℝ

Range: ℝ

16. Domain: ℝ

Range: ℝ

17. Domain: ℝ 𝑥 ≠ 2 𝑥 ≠ −4

Range: (−∞, 0) ∪ (0, ∞)

18. Domain: ℝ 𝑥 ≠1

2 𝑥 ≠ −3

Problem Set #8 – Inverses

1. No

2. Yes

3. Yes

4. No

5. ℎ−1(𝑥) = (𝑥−2

4)3

6. 𝑘−1(𝑡) =11−𝑡

5

7. 𝑚−1(𝑥) = √𝑥+4

7

8. 𝑔−1(𝑧) = √𝑧 − 25

+ 3

PMI AP Calculus AB NJCTL.org

Problem Set #9 – Trigonometry

1. −2

2. √3

3. −√2

2

4. 2√3

3

5. 𝑢𝑛𝑑𝑒𝑓

6. √2

7. 1

8. 1

2

9. 2√3

3

10. −√3

11. −√2

12. 0

13. −√3

2

14. −2√3

3

15. √3

2

16. √3

3

17. 𝑢𝑛𝑑𝑒𝑓

18. 1

19. √2

20. 0

21. 5𝜋

6

22. 0

23. 𝜋

3

24. −𝜋

6

25. 𝜋

2

26. 𝜋

2

27. 3

28. −𝜋

4

29. 𝜋

4

30. −1

31. 2𝜋

3

32. 𝜋

2

Problem Set #10 – Exponents

1. 3𝑚7𝑘7

2

2. 𝑓17

2𝑒4

3. 3𝑥4 + 4𝑥3 − 16𝑥2 + 11𝑥 − 2

4. 2𝑦5 + 17𝑦4 + 15𝑦3 − 13𝑦2 − 28𝑦 + 12

5. 8𝑎18

𝑏48

6. 16𝑛16

81𝑚56

7. −𝑦18

125𝑥9𝑧12

8. 𝑚4

16𝑘8𝑝2

9. 60𝑥16𝑦10𝑧2

10. 75𝑟11𝑠𝑡10

11. 25𝑎2 − 20𝑎𝑏 + 4𝑏2

12. 𝑐3 − 12𝑐2 + 48𝑐 + 64

Problem Set #11 – Logarithms

1. 𝑥 = 2

2. 𝑥 = 5

3. 𝑥 = 0.754 𝑜𝑟 0.753

4. 𝑥 = 1.646

5. 𝑥 = 4

6. 𝑥 = 0.4

7. 𝑥 = 0.861

8. 𝑥 = 2.322 𝑜𝑟 2.321

9. 𝑟 = 2

10. 𝑟 =5

9