The Calculus Primer (2011)

Answers to Problems

Ex. 1—1:

1. −1, 8, 23, 44.

2. The positive integers and zero.

3. 7, 11, 15, 19, ··· ; 1, 2, 3, ··· ; 7, 11, 15, 19, · · · .

4. u = 0, 1, 2, ··· 9; t = 0, 1, 2, ··· 9; h = 1, 2, 3, ··· 9.

7. Functions: a, c, e, g, i, k, o, p, q, r; non-functions: b, d, f, h, j, l, m, n, s, t, u.

Ex. 1—2:

1. a₀x⁵ + a₁x⁴ + a₂x³ + a₃x² + a₄x⁵ + a₅

2. a₀x⁹ + a₂x⁷ + a₄x⁵ + a₆x³ + a₈x

3. f(m) = am² − bm + c; f(5) = 25a − 5b + c

4. F(−l) = −7; F(0) = −6

5. f(x + h) = 2x² + (4h + 5)x + (2h² + 5h − 3); absolute term = 2h² + 5h − 3

7. x³ − x² + 3x − 3; x³ − 3x²h + 3xh² − h³ − 4x² + 8xh − 4h² + 8x − 8h − 8

11. Range of y: all y’s − 6.

12. (a) {2, 5, 8, 11, 14, 17, 20}
(b) {0, 2, 8, 18, 32, 50, 72}
(c) {0, 3, 6, 9, 12, 15, 18}

Ex. 1—3:

1. 9

2. ∞

3. 0

4. ∞

5. ∞

6. 2

7. −6

8. ∞

9.

10. −2

11.

12. 2

Ex. 2—1:

1. 2

2. −2x

3. 4x + 3

4. 20x

5. 2x − 1

6. 6x − 2

7. 6x²

8. 3x² − 2

9. 2x − 1

10. 12x + 1

11. 2

Ex. 2—2:

1. 12

2. 8

3. 5

4. −8

5. 6

6. 33

7. 12

8.

Ex. 3—1:

Ex. 3—2:

1. 2x − 1

2. 9x² + 8x + 3

3. 8x³ − 9x² − 2x + 3

4. 6x²(x³ − 1)

5. 6x(x² + l)²

6. 4t³ − 3t² + 4t − 2

7. 48x³ + 15x² − 4x

8. 3x² − 7

9. 3(2t − 3)(t² − 3t + 4)²

10. (x + 2)(x + 3)²(5x + 12)

Ex. 3—3:

Ex. 3—4:

Ex. 3—5:

Ex. 3—6:

Ex. 4—1:

1. 20.6π sq. in./in.

2. 576π cu. in./in.

3. 9 units

4. 48 cal./amp.

5. − 1 units

6. 2997 cu. ft. per sec./ft.

7. −494 B.T.U. per unit concentration

8. 16,000 H.P.; 2400 H.P./knot

9. 48 B.T.U./degree

Ex. 4—2:

1. (a) 20x³ + 18x(b) 100 − 40t₃(c) 40x⁻⁶(d) 8πr

3. (a) 360x² + 1200x⁻⁶ (b) 120t

4. (a) s = 14, v = 15, a = 12
(b) s = 5, v = 0, a = −10
(c) s = −4, v = −6, a = 0

5. (a) 322 ft./sec.;(b) 35.9 ft./sec.(c) 32.2 ft./sec./sec.
(d) 257.6 ft.

7. (a) mi./min./min.;(b) 2 miles

8. (a) 1760 ft./min./min.;(b) 3520 ft.

9. (a) v = v₁ − 32.2t; a = −32.2
(b) v = 103.4 ft./sec. upward; a = 32.2 ft./sec./sec. downward
(c) v = 186.4 ft./sec. downward; a = 32.2 ft./sec./sec. downward

10. (a) 717 ft./sec.;(b) 37.2 + sec. (approx.)

Ex. 4—3:

1. x = 2 gives min. y = −4

2. x = −3 gives min. y = −1

3. x = +2 gives min. y = −16; x = −2 gives max. y = +16

4. x = 0 gives max. y = 15; x = +3 gives min. y = −66; x = −3 gives min. y = −66

5. Neither max. nor min.

6. 20; 20

7.

8. 1

Ex. 4—4:

Ex. 4—5:

2. a = (4t + 8) ft./sec./sec.; 28 ft./sec./sec.

3. Max., x = −2; min., x = +4

4. r = 3

5. v = 0; a = −20 ft./sec./sec.

6. Max., x = −1; min., x = +5/3

7. Max., x = 0; min., x = +1 and x = −1

8. 7 mi./hr./hr.

9. Length = 30 in.; circumference = 60 in.

10. Square; area = 2k²

Ex. 5—1:

Ex. 5—2:

1. 4 log k(k^4x)

2. 2x (log a)

3. ke^kx

Ex. 5—3:

1. (x + 1)(x^x) + log x (x^x+1)

2. x(x² + 1)^x−1 + [log (x² + 1)](x² + 1)^x

3. −x^3x(3 + log x³)

4. (l + 2 log x)

Ex. 5—4:

Ex. 5—5:

1. 3 cos 3x

2. − 5a sin 5ax

3. sin x(2 cos² x − sin² x)

4. −2a sin 2θ

5. b cos (a + bx)

6. e^{tan x}·sec² x

7. −2 tan x

8. (sin x)^x·[(log sin x + x cot x)]

9. x^{cos x−1}·[cos x − x log x sin x] 10. e^x(cot x + log sin x)

10. ex(cotx + log sinx)

Ex. 5—6:

Ex. 5—7:

Ex. 5—8:

Ex. 6—1:

Ex. 6—2:

1. 3

2. Tangent, 4x + y = 0; normal, 4y − x = 34

3. Subtangent, 2x; subnormal, 2p

4. 3x + 4y = 25;

5. y = x

6. nk

8. Intercept on x-axis = 2x; intercept on y-axis = 2y; area = (2x₁)(2y₁) = 2x₁y₁ = a².

Ex. 6—3:

1. (2, − 114); (−3,−294)

2. None

4. (0,0)

5. None

6. (0,0)

7. None

9. (a,b)

10. (−1,0)

Ex. 6—5:

Ex. 6—6:

Ex. 6—7:

Ex. 7—1:

1. dy = (3x² + 10x) dx

2. dy = 15(5x + 2)² dx

3. dy = x(x² − 1)^−½ dx

4. dy = (kte^tx) dx

Ex. 7—2:

1. .72 sq. cm.; 3.6 cu. cm.

2. .04π in.; .2π sq. in.

3. 1.60π sq. in.; 8π cu. in.

4. .4π cu. cm.

5. 17.059

6. 3.917

7. .025 cm.

Ex. 7—3:

Ex. 8—1:

Ex. 8—2:

Ex. 8—3:

1. α = 2, β = 3

2. β = β = log 2 −5

3. α = β = 0

4. α = 5, β = −2

5. β = , β = 0

Ex. 8—4:

Ex. 9—1:

1. −1

2. n

3.

4. 0

5.

6. 1

7.

8.

9. 2

10. 2

Ex. 9—2:

1. 0

2. 0

3. 0

4. 1

5. 0

6. 0

7. 0

8. ∞

9.

10. 0

11. 0

12. 0

13. −3

14. ∞

15. −

Ex. 9—3:

1. 1

2. 1

3. 1

4. e

5. e³

6. 1

7. e^mn

8.

9. e

10. e

Ex. 9—4:

1. 3x + 2y − 10 = 0

2. 2y = 2x + 3

Ex. 10—1:

Ex. 10—2:

Ex. 10—4:

1. (a) 1;(b) n;(c) 0(b) dz = 2ay³x dx + 3ax²y² dy

2. (a) dz = 2ax dx + 3by² dy(c) dz = yx^y−1 dx + x^y log x dy

Ex. 11—1:

Ex. 11—2:

1. Divergent

2. Convergent

3. Divergent

4. Convergent

5. Convergent

6. Divergent

7. Divergent

8. Convergent

9. Convergent

10. Convergent

Ex. 11—3:

1. Convergent

2. No test

3. Convergent

4. Convergent

5. Convergent

6. No test

7. Divergent

8. Convergent

9. Convergent

10. Divergent

Ex. 11—4:

1. −1 < x < 1

2. −1 < x < 1

3. −1 x 1

4. −1 x < 1

5. −2 < x < 2

6. 1 > x −1

7. −1 x < 1

8. All values of x

Ex. 11—5:

1. Convergent for all values of x

2. Convergent for all values of x

3. Convergent for all values of x

4. Convergent for − 1 x < 1

5. Convergent for all values of x

6. Convergent for all values of x

Ex. 11—6:

Ex. 11— 8:

Ex. 12—1:

Ex. 12—2:

Ex. 12—3:

1. 3 log x + C

2. log x + C

3. 5 log x + C

4. 2 log x + C

5. log (2x + 3) + C

6. − log (2 − 3z) + C

7. log (x² + 2) + C

8. log (x² − 4) + C

9. log (z³ − 1) + C

10. log z + + C

11. y² − log y² + C

12. [x² + log (x² − 1)] + C

13. log (x³ − 1) + C

14. x³ − log (x³ + 2)² + C

15. log (x³ − 2x + 1) + C

Ex. 12—4:

Ex. 12—5:

Ex. 12—6:

Ex. 12—7:

Ex. 13—1:

1. x sin x + cos x + C

2. x log x − x + C

3. e^2x(x − ) + C

4. e^x(x² − 2x + 2) + C

5. e^x (sin x + cos x) + C

6. x(log²x − 2 log x + 2) + C

7. x² sin x + 2x cos x − 2 sin x + C

8. x⁴(log x − ) + C

9. θ tan θ − log sec θ + C

10. x arc tan x − log (1 + x²) + C

11. −e^−x(x² + 2x + 2) + C

12. x³(log x − ) + C

13. θ tan θ + log cos θ − θ² + C

14. z tan z + log cos z + C

15. e^x(2 sin 2x + cos 2x) + C

16. sin x (log sin x − 1) + C

17. x sin 2x − x² cos 2x + cos 2x + C

18. x sin 2x + cos 2x + C

Ex. 13—2:

Ex. 13—3:

Ex. 13—5:

Ex. 13—6:

1. 8 log x + 4 log (x + 2) + C

2. 2 log x + log (x − 3) + C

3. 2 1og (x − 4) + 5 1og (x + 3) + C

4. 4 log (x − 5) − 3 log (x + 5) + C

5. 2 log (x − 1) − 3 log (x − 2) + C

6. log x + 2 log (x + 4) + 3 log (x − 1) + C

7. log (2x + 1) − 41og (x − 3) + C

8. 3 log x + 7 log (x + 4) − log (3x − 1) + C

Ex. 13—7:

Ex. 13—8:

Ex. 14—1:

Ex. 14—2:

1. 112

2. A = 10 ; B = 5

3. 96

4. 2

5. 27

6. A = 4; B = 12

7. 77

8. a² (log b − log a)

9. a²/6

10. 21

Ex. 14—3:

1. 1

2.

3.

4. 1

5. e

6. No meaning

7.

8. No meaning

Ex. 14—4:

1. 8

2.

3.

4.

5.

6.

7. 4 − 2 log 2

8. 1

9. 36

10.

11. 2

12.

Ex. 14—5:

Ex. 15—1:

Ex. 15—2:

1. 12 log 3

2. 6

3. πab

4. 3πa²

5.

6.

7. a²

8.

9. πa²

10.

Ex. 15—3:

Ex. 15—4:

Ex. 16—1:

Ex. 16—2:

Ex. 16—3:

Ex. 16—4:

1. Trap. rule, 1.812; direct integration, 1.792

2. Trap. rule, 332.0; direct integration, 330.7

3. Simpson’s rule, 6554.7; direct integration, 6553.6

4. Simpson’s rule, 1.247; direct integration, 1.248

5. Trap. rule, 0.683; direct integration, 0.684

6. Simpson’s rule, 9.295; direct integration, 9.279
(Four-place logarithmic and trigonometric tables were used in the above verifications.)

Ex. 16—5: