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Chapter 8 โ Introduction to Calculus Answer Key
CK-12 Math Analysis Concepts 1
8.1 Definition of a Limit
Answers
1. lim๐ฅโ๐โ
4๐ฅ3 + 3๐ฅ2 โ 4๐ฅ โ 1
2. lim๐งโ๐โ
๐(๐ง)
3. lim๐ฆโ๐โ
๐(๐ฆ)
4. lim๐งโโ1+
โ(๐ง)
5. lim๐ฆโ๐โ
โ(๐ฆ)
6. lim๐งโ๐
โ(๐ง)
7. -0.35355
8. -1
9. 1.8508
10. -0.02066
11. The limit does not exist
12. -2
13. -0.05
14. the limit does not exist
15. -0.05774
16. 1.5574
17. For each element > 0 there exists a difference > 0,
such that if 0 < |๐ฆ โ 2| < difference, then |๐ก๐๐ ๐ฆโ ๐ฟ| < element
18. The answer for each element > 0 there exists a difference > 0,
such that if 0 < |๐ฅ โ 1| < difference, then |๐(๐ฅ) โ ๐| < element
19. The answer for each element > 0 there exists a difference > 0,
such that ๐๐ 0 < |๐ฅ โ (โ๐ฅ)| < difference, then | โ ๐ฅ3 + 3๐ฅ2 + 2๐ฅ + 4 โ ๐ฟ| < element
Chapter 8 โ Introduction to Calculus Answer Key
CK-12 Math Analysis Concepts 2
8.2 One Sided Limits
Answers
1. 5
2. -3
3. -8, 2
4. 2
5. -2.5, 5
6. Substituting ๐ฅ = 2 into โ๐ฅ โ 4, we get an answer of -6.
7. From the left we are looking at 1. Substituting ๐ฅ = โ3 into 1, we get 1.
8. Substituting ๐ฅ = 0 into โ๐ฅ + 4, we get an answer of 4.
9. From the right we are looking at -5. Substituting ๐ฅ = โ1 into -5, we get -5.
10. Substituting ๐ฅ = 1 into 4๐ฅ + 3, we get an answer of 7.
11. From the left we are looking at ๐ฅ + 1. Substituting ๐ฅ = 3 into ๐ฅ + 1, we get 4.
12. Substituting ๐ฅ = 0 into ๐ฅ โ 4, we get an answer of -4.
13. From the right we are looking at 4๐ฅ + 4. Substituting ๐ฅ = 2 into 4๐ฅ + 4, we get 12.
14. Substituting ๐ฅ = 2 into 4๐ฅ + 1, we get an answer of 9.
15. From the left we are looking at 4๐ฅ + 1. Substituting ๐ฅ = โ2 into 4๐ฅ + 1, we get -7.
16. From the left we are looking at โ3๐ฅ. Substituting ๐ฅ = 3 into โ3๐ฅ, we get -9.
17. Substituting ๐ฅ = โ5 into โ3๐ฅ + 2, we get an answer of 17.
18. From the left we are looking at 3๐ฅ โ 3. Substituting ๐ฅ = 2 into 3๐ฅ โ 3, we get 3.
Chapter 8 โ Introduction to Calculus Answer Key
CK-12 Math Analysis Concepts 3
8.3 Infinite Limits
Answers
1. โโ
2. +โ
3. โโ
4. 1
5. โโ
6. 11
9
7. 13
8. โ2
17
9. 15
10. โ โ
11. โ
12. โ โ
13. 0
14. โโ
15. โโ
Chapter 8 โ Introduction to Calculus Answer Key
CK-12 Math Analysis Concepts 4
8.4 Polynomial Function Limits
Answers
1. -12
2. 2
3. 4
4. -2
5. 4
6. 3
7. 0
8. -94
9. -7
10. -44
11. โ2
12. 10
13. 10
14. โโ3๐
15. -3
16. -2354
17. โ26
Chapter 8 โ Introduction to Calculus Answer Key
CK-12 Math Analysis Concepts 5
8.5 Rational Function Limits
Answers
1. -6
2. the limit does not exist
3. 0.17284
4. -3
5. 2.75
6. -0.04
7. the limit does not exist
8. 0
9. 0.05159
10. 17
11. -18
12. 0.01561
13. 0.25
14. the limit does not exist
15. 1.5
16. 2
17. 3
Chapter 8 โ Introduction to Calculus Answer Key
CK-12 Math Analysis Concepts 6
8.6 Applications of One-Sided Limits
Answers
1. Yes
2. No
3. No
4. Yes
5. Yes
6. 0
7. 9
8. -6
9. 3
10. 9
11. limit does not exist
12. -8
13. -3
14. -3
15. -7
16. 9
17. limit does not exist
18. 4
19. -2
20. +โ
21. Use a graph, see it here: https://www.desmos.com/drive/calculator/esekwoanq8
Chapter 8 โ Introduction to Calculus Answer Key
CK-12 Math Analysis Concepts 7
8.7 Tangents to a Curve
Answers
1. The secant line
2. Tangent
3. The distance between the two points used to find the tangent line
4. โhโ โ the distance between the points
5. The limit of the function ๐(๐ฅ+โ)โ๐(๐ฅ)
โ as โ โ 0 describes the slope of the tangent.
6. ๐ฆ = ๐ฅ โ 2
7. ๐ฆ = โ5๐ฅ + 8
8. ๐ฆ = โ3๐ฅ + 7
9. ๐ฆ = 3๐ฅ โ 8
10. ๐ฆ = 5๐ฅ + 22
11. ๐ฆ = โ20๐ฅ + 16
12. ๐ฆ = โ2๐ฅ
13. ๐ฆ = 19๐ฅ โ 5
14. ๐ฆ = 8๐ฅ + 3
15. ๐ฆ = 10๐ฅ
16. ๐ฆ = โ19๐ฅ โ 7
17. ๐ฅ = ๐ฆ
18. ๐ฆ = โ2๐ฅ + 3
19. ๐ฆ = 3
20. ๐ฆ = 36๐ฅ + 19
Chapter 8 โ Introduction to Calculus Answer Key
CK-12 Math Analysis Concepts 8
8.8 Instantaneous Rates of Change
Answers
1. 2376
44=
54
1
2. 646
19=
34
1
3. 10208
44=
232
1
4. 5341
49= 109
5. 9720
24= 405
6. 210
7. 55
8. 80
9. 105
10. 140
11. ๐โฒ(๐ฅ) = 12๐ฅ, ๐ฆ = 36๐ฅ โ 54
12. ๐โฒ(๐ฅ) =1
2โ(๐ฅ+2), ๐ฆ =
1
โ(10) (
1
2๐ฅ + 6)
13. ๐โฒ(๐ฅ) = 9๐ฅ2, ๐ฆ = 9๐ฅ + 4
14. ๐โฒ(๐ฅ) =โ1
(๐ฅ+2)2, ๐ฆ = โ๐ฅ
15. ๐โฒ(๐ฅ) = 2๐๐ฅ, ๐ฆ = 2๐๐๐ฅ โ ๐(๐(๐๐ + 1)
16. ๐โฒ(๐ฅ) =1
3๐ฅ23
โถ ๐ฆ =1
3๐ฅ +
2
3
17. ๐โฒ(0) = 0, ๐(๐ฅ) = 4 + 3๐ฅ
18. 10
Chapter 8 โ Introduction to Calculus Answer Key
CK-12 Math Analysis Concepts 9
19. ๐โฒ(๐ฅ) is the instantaneous rate of change of ๐ฝ with respect to ๐ฅ, that is, change in the
production cost with respect to the number of jars produced. So the rate of change
in the production cost with respect to the number of jars produced is 9999๐๐๐๐๐๐๐
๐๐๐. So
we get the instantaneous rate of change in the production cost with respect to the
number of jars produced is 9999๐๐๐๐๐๐๐
๐๐๐
20. ๐โฒ(๐ฅ) is the instantaneous rate of change of ๐ with respect to ๐ฅ, that is, change in the
temperature of the pie with respect to the number of minutes that have passed. So the rate of change in the temperature of the pie with respect to the number of minutes that have passed is 102 degrees/minute. So we get the instantaneous rate of change in the temperature of the pie with respect to the number of minutes that
have passed is 102๐๐๐๐๐๐๐
๐๐๐๐ข๐ก๐.
21. ๐โฒ(๐ฅ) is the instantaneous rate of change of ๐ with respect to ๐ฅ, that is, change in the
quantity of the virus with respect to the number of hours that have passed. So we get ๐ฃ๐๐๐ข๐
โ๐๐ข๐.
22. ๐โฒ(๐ฅ) is the instantaneous rate of change of ๐ with respect to ๐ฅ, that is, change in the
number of cold cases in the US with respect to the date in November.
23. Change in households affected by hurricanes is: 2483 โ 76 = 2407. Change in days is 34 โ 5 = 29 2407/ 29 = 83 households affected per day on average.
24. 135 ๐๐๐๐๐๐๐
๐๐๐๐ข๐ก๐.
25. So the change in degrees is 6107 โ 80 = 6027
And the change in minutes is 54 โ 5 = 49
So the answer is 123 ๐๐๐๐๐๐๐
๐๐๐๐ข๐ก๐
Chapter 8 โ Introduction to Calculus Answer Key
CK-12 Math Analysis Concepts 10
8.9 Constant Derivatives and the Power Rule
Answers
1. ๐2 = ๐๐ฅ๐โ1
2. ๐ฆโฒ = 35๐ฅ6
3. ๐ฆโ = โ3
4. ๐โฒ(๐ฅ)1
3
5. ๐ฆโฒ = 4๐ฅ3 โ 6๐ฅ2 โ5
2โ(๐ฅ)
6. ๐ฆโฒ = 20๐ฅ (5๐ฅ2 โ 3)
7. โ29.4784
8. 0 ๐๐๐ ๐๐๐ ๐ฅ
9. 0
10. 0
11. โ0.37
12. ๐โฒ(๐ฅ) = โ3๐ฅโ4 for all x
13. ๐ขโฒ(๐ฅ) = .96๐ฅโ0.49 for all x
14. ๐โฒ(๐ฅ) = โ0.49๐ฅโ1.49 for all x
15. ๐ โฒ(๐ฅ) = โ5๐3 ๐ฅโ5๐3โ1 for all x
Chapter 8 โ Introduction to Calculus Answer Key
CK-12 Math Analysis Concepts 11
8.10 Derivative pf Sums and Differences
Answers
1. ๐ฆโฒ =3
2๐ฅ2 โ 2๐ฅ
2. ๐ฆโฒ = 3โ2 ๐ฅ2โ โ2๐ฅ + 2
3. ๐ฆโฒ = 2๐ฅ + 1
4. ๐ฆโฒ = โ3
๐ฅ4โ
7
๐ฅ8
5. ๐ฆโฒ =1
2โ๐ฅ โ
1
2๐ฅ32
6. ๐(๐ฅ) = 18๐ฅ โ 24
7. โ9.3๐ฅ9 + (โ5
12๐3 ๐ฅโ
17
12) for all x
8. 8๐ฅ + 4
9. 50๐ฅ โ 30
10. (โ๐ฅ + 2)(๐๐ฅ)
11.
12.
13. 27๐ฅ2 + 12๐ฅ โ 15
14.
15. 3 = ๐(โ2)
16. ๐โฒ(๐ฅ) = 45
17.
18. 282
19. ๐(1)
20. ๐โฒ(๐ฅ) = โ3
Chapter 8 โ Introduction to Calculus Answer Key
CK-12 Math Analysis Concepts 12
8.11 Quotient Rule and Higher Derivatives
Answers
1. ๐(0) = 14
2. ๐โฒ(๐ฅ) = โ1
32
3. (3๐ฅ๐๐ฅ + ๐๐ฅ)(9๐ฅ2 + 24๐ฅ + 16)
4. ๐ฅ๐๐๐ ๐ฅ โ 4๐๐๐ ๐ฅ โ ๐ ๐๐ ๐ฅ
๐ฅ2โ8๐ฅ+16
5. ๐ ๐๐๐ฅ โ ๐ฅ๐๐๐ ๐ฅ
๐ ๐๐2๐ฅ
6. โ24๐ฅ + 6
7. 2
8. 3๐ฅ4๐๐ฅ + 24๐ฅ3๐๐ฅ + 36๐ฅ2๐๐ฅ
9. 2๐ฅ5๐ ๐๐ ๐ฅ โ 20๐ฅ4๐๐๐ ๐ฅ โ 40๐ฅ3๐ ๐๐ ๐ฅ
10. 3๐ฅ5๐๐ฅ + 30๐ฅ4๐๐ฅ + 60๐ฅ3๐๐ฅ
11. ๐ฆโฒ =โ3
2โ๐ฅ (โ๐ฅ + 3)
2
12. ๐ฆโฒ =โ4๐ฅ2โ2๐ฅโ36
(๐ฅ2โ9)2
13. ๐๐น
๐๐= โ2๐บ
๐๐
๐3
14.
15. โ120
Chapter 8 โ Introduction to Calculus Answer Key
CK-12 Math Analysis Concepts 13
8.12 Area Under the Curve
Answers
1. 8
3
2. 4
3. -4
4. 0
5. 18
6. = ๐น(5) โ ๐น(4) = 3(5) โ 3(4) = 15 โ 12 = 3
7. =๐น(5) โ ๐น(1) = (3
252 + 5) โ [
3
2(1)2 + (1)] =
85
2 โ
5
2= 40
8. =๐น(4) โ ๐น(3) = ๐๐(4) โ ๐๐(3) = 0.2877
9. = ๐น(6) โ ๐น(5) = [(6)2 + 4(6)] โ [(5)2 + 4(5)] = 60 โ 45 = 15
10. = 11645
12โ
110
3=
3735
4
11. = ๐น(7) โ ๐น(3) = [๐๐(7)] โ [๐๐(3)] = 0.8473
12. = ๐น(6) โ ๐น(5) = [(6)3 + (6)2] โ [(5)3 + (5)2] = 252 โ 150 = 102
13. = ๐น(6) โ ๐น(2) = [4(6)] โ [4(2)] = 24 โ 8 = 16
14. = 475
3โ
23
3=
452
3
15
Area is 1
6
Chapter 8 โ Introduction to Calculus Answer Key
CK-12 Math Analysis Concepts 14
8.13 Fundamental Theorem of Calculus
Answers
1. 45
2
2. 1
5
3. โ3
2
4. โ9
2
5. 18 6. ๐น(0) โ ๐น(โ1) = [โ3(0)] โ [โ3(โ1)] = 0 โ 3 = โ3 7. ๐น(3) โ ๐น(โ1) = [(3)] โ [(โ1)] = 3 โ โ1 = 4
8. ๐น(๐
2) โ ๐น(โ๐) = [โ4๐ ๐๐ (
๐
2)] โ [โ4๐ ๐๐(โ๐)] = โ4 โ 0 = โ4
9. ๐น(2) โ ๐น(0) = [โ2] โ [0] = โ2 10. ๐น(7) โ ๐น(2) = [๐๐(7)] โ [๐๐(2)] = 1.2528
11. ๐น(0) โ ๐น(โ2) = [1
2(0)2 + 5(0)] โ [
1
2(โ2)2 + 5(โ2)] = 0 โ โ8 = 8
12. ๐น(3๐
2) โ ๐น(โ๐) = [โ6๐๐๐ (
3๐
2)] โ [โ6๐๐๐ (โ๐)] = 0 โ 6 = โ6
13. ๐น(7) โ ๐น(6) = [๐๐(7)] โ [๐๐(6)] = 0.1542 14. a) ยผ
b) 0
15. 4๐
3๐ 3