review unit problem sets problem set #1 slopes
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PMI AP Calculus AB NJCTL.org
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REVIEW UNIT PROBLEM SETS
PROBLEM SET #1 β Slopes ***Calculators Not Allowed***
Calculate the slope of the line containing the following points:
1. (2,8) πππ (β4,6) 2. (β4, β7) πππ (3,0) 3. (β3, β6) πππ (β1, β6) 4. (4, β2) πππ (4,5)
5. (4
5, 6) πππ (
3
5, 4)
6. (3
2, β4) πππ (2,0)
7. (11
14,
3
7) πππ (
9
14,
5
7)
8. (8
9,
2
3) πππ (
5
6,
2
5 )
9. (β4, β3) πππ (0, β11)
10. (β3
7,
3
8) πππ (β
1
6,
5
6)
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PROBLEM SET #2 β 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 π = 2
2. Passes through (β2,4) and π =1
2
3. Passes through (β4, β5) πππ (2,7) 4. Passes through (3, β5) πππ (β3,5)
5. Passes through (-1, 2) and π = 0
6. Passes through (-1, 2) and the slope is undefined.
7. Passes through (-2, 2) and is parallel to 2π¦ =4π₯ β 12 8. Passes through (-3, 2) and is perpendicular
to 15π¦ = 10π₯ + 2
9. π =3
5 πππ π = 0
10. π = 0 πππ π = β1
7
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Problem Set #3 β Functions & Graphing Functions ***Calculators Not Allowed***
1. a) True/False 3π₯2 + 5π¦ = 7 β 2π₯ is a function. b) Why or why not? ______________________________________________________________ ___________________________________________________________________________
2. a) True/False 2π₯2 + 3π¦2 = 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 π(2) if π(π₯) = 3π₯2 β 5π₯ + 5
6. Evaluate π(β3) if π(π₯) = βπ₯2 β 2π₯ + 15
7. Evaluate π(π₯ β 2) if π(π₯) = β2π₯2 β 3π₯ + 11
π₯ β2 β1 0 1 2 5
π¦ 5 3 6 3 2 β4
π₯ β2 2 0 β2 2 5
π¦ 4 3 6 3 2 β3
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8. Evaluate π(5 β π₯) if π(π₯) =4π₯β3
2βπ₯
9. Write the new equation of the function π¦ = βπ₯ 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 π¦ =1
π₯ with the following transformations:
horizontal compression of 3, right 3 units, and down 2 units
11. Write the new equation of the function π¦ = π₯2 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 π¦ = ln π₯ with the following transformations: vertical stretch of 3, right 5 units, and up 2 units
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Problem Set #4 β Piecewise Functions ***Calculators Not Allowed***
1. Graph the following piecewise function:
π(π₯) = {π₯ β 1 β 5 β€ π₯ < β1β2 β 1 β€ π₯ < 2βπ₯ + 3 2 < π₯ β€ 6
2. Graph the following piecewise function:
π(π₯) = {2π₯ + 1 π₯ < 1
βπ₯2 + 5 π₯ β₯ 1
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3. Graph the following piecewise function:
π(π₯) = {β2π₯ β 6 π₯ < β4
π₯ + 4 β 4 β€ π₯ < 2 π₯2 β 3 π₯ β₯ 2
4. Graph the following piecewise function:
π(π₯) = {|π₯ + 3| β 1 β 6 β€ π₯ < 1
βπ₯ β 2 π₯ > 1
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5. Given:
π(π₯) = { π₯2 β 5 π₯ < β5
11 β 5 β€ π₯ < 1β3π₯2 + 10 π₯ β₯ 1
a) Find π(β7)
b) Find π(β5)
c) Find π(0)
d) Find π(1)
e) Find π(3)
6. Given:
π(π₯) = { |π₯ β 5| + 3 π₯ < β2
2π₯3 β 4 π₯ β₯ β2
a) Find π(β5)
b) Find π(β2)
c) Find π(0)
d) Find π(1)
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Problem Set #5 β Function Composition ***Calculators Not Allowed***
Use the following functions to answer questions 1 β 16.
π(π) = ππ π(π) = βππ β π π(π) = |π β π| π(π) = βπππ
1. π(π(2)) =
2. π β π(3) =
3. β (π(π(0))) =
4. π β π β β(β2) =
5. π β π(π₯) =
6. 5π(π₯) β 3π(π₯) =
7. π β π β π(π₯) =
8. π(π₯)
π(π₯)=
9. π(π(3)) =
10.β β π(β7) =
11. π (π(β(β5))) =
12. β β π β π(β1) =
13. π β π(π₯) =
14. β2π(π₯) + 4π(π₯) =
15. π β π β π(π₯) =
16. π(π₯)
π(π₯)=
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Problem Set #6 β Function Roots ***Calculators Not Allowed***
Find any real roots, if they exist, for questions 1 β 12.
1. π¦ = π₯2 β 2π₯ β 8
2. π(π₯) = π₯2 + 4π₯ β 32
3. π(π‘) = π‘3 β 11π‘2 + 18π‘
4. π¦ = β3π₯2 β 10π₯ + 8
5. π(π‘) = π‘3 β 5π‘2 + 12π‘
6. π(π‘) = π‘2 β 6π‘ + 17
7. π¦ = 2π₯2 β π₯ β 10
8. π(π₯) = βπ₯2 + 4π₯ + 12
9. π(π₯) = 5π₯2 + 5π₯ + 12
10. π¦ = 3π₯2 β 8π₯ β 2
11.π(π§) = 5π§3 + 2π§2 β 7π§
12. π¦ = 3π₯3 + 6π₯2 β π₯
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Problem Set #7 β Domain & Range ***Calculators Not Allowed***
1.
Domain: ____________________ Range: ____________________ 2.
Domain: ____________________ Range: ____________________ 3.
Domain: ____________________ Range: ____________________
4.
Domain: ____________________ Range: ____________________ 5. Domain: ____________________ Range: ____________________ 6.
Domain: ____________________ Range: ____________________
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7. π¦ = β2π₯ + 12 Domain: ____________________ Range: ____________________
8. π¦ = π₯2 + 4π₯ β 32 Domain: ____________________ Range: ____________________
9. π¦ = β3π₯2 + 6π₯ + 5 Domain: ____________________ Range: ____________________
10. π¦ = βπ₯ + 5 β 2 Domain: ____________________ Range: ____________________
11. π¦ = ββπ₯ + 7 + 5 Domain: ____________________ Range: ____________________
12. π¦ =5π₯+2
βπ₯+3
Domain only: ____________________
13. π¦ = ln(π₯ β 3) Domain: ____________________ Range: ____________________ 14. π¦ = 4 ln(π₯ + 2) β 1 Domain: ____________________ Range: ____________________
15. π¦ = βπ₯3 + 14 Domain: ____________________ Range: ____________________
16. π¦ = βπ₯ β 8 3
+ 4 Domain: ____________________ Range: ____________________
17. π¦ =2π₯
π₯2+2π₯β8
Domain: ____________________ Range: ____________________
18. π¦ =25
2π₯2+5π₯β3+ 4
Domain only: ____________________
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Problem Set #8 β Inverses ***Calculators Not Allowed***
State whether the following functions are inverses.
1. π(π₯) = (π₯ β 1)2
π(π₯) = 1 + π₯2
2. π(π§) =3
π§+ 5
π(π§) =3
π§β5
3. π(π₯) = βπ₯ β 3 +5
β(π₯) = (π₯ β 5)2 + 3
4. π(π‘) = 2π‘3 β 1
π(π‘) =βπ‘+1
3
2
Find the inverse of each function.
5. β(π₯) = 4βπ₯3
+ 2
6. π(π‘) = β5π‘ + 11
7. π(π₯) = 7π₯2 β 4
8. π(π§) = (π§ β 3)5 + 2
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Problem Set #9 β Trigonometry ***Calculators Not Allowed***
Evaluate each of the following.
1. csc7π
6
2. tanπ
3
3. sin7π
4
4. π πππ
6
5. πππ‘π
11. ππ π15π
4
12. πππ‘2π
4
13. π ππ 4π
3
14. ππ π4π
3
15. πππ 11π
6
6. cscπ
4
7. sinπ
2
8. cos5π
3
9. ππ π14π
6
10. π‘ππ2π
3
16. πππ‘4π
3
17. π‘πππ
2
18. πππ‘5π
4
19. ππ π3π
4
20. πππ 5π
2
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Evaluate:
21. πππ β1 (ββ3
2)
22. sinβ1( 0)
27. 3 + 2 cos2 (3π
2)
28. cotβ1(β1)
23. ππ πβ1 (2β3
3)
24.tanβ1(ββ3
3)
25.sinβ1(1)
29. π ππβ1( β2)
30. 2 β 3 sin2 (π
2)
31. cosβ1 (β1
2)
26. πππ β1(0)
32.cscβ1(1)
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Problem Set #10 β Exponents ***Calculators Not Allowed***
Simplify:
1. 15π11πβ5
10π4πβ12
2. 21π3π14
42π7πβ3
3. (3π₯2 β 5π₯ + 2)(π₯2 + 3π₯ β 1)
4. (2π¦3 + 3π¦2 β 4)(π¦2 + 7π¦ β 3)
5. ((2π4π2)3
(4π9πβ5)2)β3
6. ((3π3π4)4
(6πβ8π12)2)β2
7. (β5π₯3π¦β6π§4)β3
8. (4πβ2π4π)β2
9. (2π₯3π¦3π§)2(15π₯10π¦4π§0)
10. (β5π5π β2π‘4)2(3ππ 5π‘2)
11. (5π β 2π)2
12. (π β 4)3
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Problem Set #11 β Logarithms ***Calculators Allowed***
Solve the following equations:
1. logπ₯ 16 = 4
2. logπ₯ 125 = 3
3. 33π₯+2 = 108
4. 24π₯β3 = 12 5. log(7π₯ + 3) = log (2π₯ + 23)
6. log(2π₯ + 3) = log (12π₯ β 1)
7. 53π₯ = 26
8. 42π₯ = 54
9. log2(π + 3) + log2(π) = log2 10
10. log4(π + 5) β log4(π) = log4 10
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REVIEW PROBLEM SET ANSWER KEYS
Problem Set #1 β Slopes
1. 1
3
2. 1
3. 0
4. π’ππππ
5. 10
6. 8
7. β2
8. 24
5
9. β2
10. 7
4
Problem Set #2 β Eqns. of Lines
1. π¦ β 3 = 2(π₯ β 2) ππ π¦ = 2π₯ β 1
2. π¦ β 4 =1
2(π₯ + 2) ππ π¦ =
1
2π₯ + 5
3. π¦ + 5 = 2(π₯ + 4) ππ π¦ = 2π₯ + 3
4. π¦ + 5 = β5
3(π₯ β 3) ππ ππ π¦ = β
5
3π₯
5. π¦ = 2
6. π₯ = β1
7. π¦ = 2π₯ + 6
8. π¦ β 2 = β3
2(π₯ + 3) ππ π¦ = β
3
2π₯ β
5
2
9. π¦ =3
5π₯
10. π¦ = β1
7
Problem Set #3 β 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. β2π₯2 + 5π₯ + 9
8. 4π₯β17
3βπ₯
9. π¦ = β1
2βπ₯ β 3 + 4
10. π¦ =1
3π₯β9β 2
11. π¦ = β2(π₯ + 2)2 β 3
12. π¦ = 3ln(π₯ β 5) + 2
Problem Set #4 β 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
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Problem Set #5 β Function Composition
1. β13
2. β21
3. 6
4. β484
5. 8π₯2 β 1
6. 21π₯ + 3
7. 24π₯2 β 1
8. 3π₯
β2π₯β1
9. β108
10. 24
11. β2304
12. 7
13. β36π₯2
14. 16π₯ + 2
15. 72π₯2 β 1
16. 2π₯+1
4π₯2
Problem Set #6 β Function Roots
1. π₯ = β2 & π₯ = 4
2. π₯ = β8 & π₯ = 4
3. π‘ = 0 & π‘ = 2 & π‘ = 9
4. π₯ =2
3 & π₯ = β4
5. π‘ = 0
6. no real roots
7. π₯ = β2 & π₯ =5
2
8. π₯ = β2 & π₯ = 6
9. no real roots
10. π₯ =4Β±β22
3
11. π₯ = 0, π₯ = β7
5 & π₯ = 1
12. π₯ =β3Β±2β3
3 πππ π₯ = 0
Problem Set #7 β Domain & Range
1. Domain: (ββ, 1) βͺ [4, β)
Range: β
2. Domain: β
Range: (ββ, 3]
3. Domain: [β5,5]
Range: [β2,2]
4. Domain: β
Range: π¦ = 3
5. Domain: (ββ, β3] βͺ (β2, β)
Range: (ββ, 3]
6. Domain: (ββ, 2] βͺ (3, β)
Range: π¦ = β2 πππ (β1, β)
7. Domain: β
Range: β
8. Domain: β
Range: [β36, β)
9. Domain: β
Range: (ββ, 8]
10. Domain: [β5, β)
Range: [β2, β)
11. Domain: [β7, β)
Range: (ββ, 5]
12. Domain: (β3, β)
13. Domain: (3, β)
Range: β
14. Domain: (β2, β)
Range: β
15. Domain: β
Range: β
16. Domain: β
Range: β
17. Domain: β π₯ β 2 π₯ β β4
Range: (ββ, 0) βͺ (0, β)
18. Domain: β π₯ β 1
2 π₯ β β3
Problem Set #8 β Inverses
1. No
2. Yes
3. Yes
4. No
5. ββ1(π₯) = (π₯β2
4)3
6. πβ1(π‘) =11βπ‘
5
7. πβ1(π₯) = βπ₯+4
7
8. πβ1(π§) = βπ§ β 25
+ 3
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Problem Set #9 β Trigonometry
1. β2
2. β3
3. ββ2
2
4. 2β3
3
5. π’ππππ
6. β2
7. 1
8. 1
2
9. 2β3
3
10. ββ3
11. ββ2
12. 0
13. ββ3
2
14. β2β3
3
15. β3
2
16. β3
3
17. π’ππππ
18. 1
19. β2
20. 0
21. 5π
6
22. 0
23. π
3
24. βπ
6
25. π
2
26. π
2
27. 3
28. βπ
4
29. π
4
30. β1
31. 2π
3
32. π
2
Problem Set #10 β Exponents
1. 3π7π7
2
2. π17
2π4
3. 3π₯4 + 4π₯3 β 16π₯2 + 11π₯ β 2
4. 2π¦5 + 17π¦4 + 15π¦3 β 13π¦2 β 28π¦ + 12
5. 8π18
π48
6. 16π16
81π56
7. βπ¦18
125π₯9π§12
8. π4
16π8π2
9. 60π₯16π¦10π§2
10. 75π11π π‘10
11. 25π2 β 20ππ + 4π2
12. π3 β 12π2 + 48π + 64
Problem Set #11 β Logarithms
1. π₯ = 2
2. π₯ = 5
3. π₯ = 0.754 ππ 0.753
4. π₯ = 1.646
5. π₯ = 4
6. π₯ = 0.4
7. π₯ = 0.861
8. π₯ = 2.322 ππ 2.321
9. π = 2
10. π =5
9