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Logarithmic and Exponential Functions

Mathletics Instant

Workbooks

Teacher Book - Series M-2

y

x

114 EXCEL HSC MATHEMATICS REVISION & EXAM WORKBOOK

TOPIC TEST

Time allowed: 1 hour Total marks = 100

SECTION I Multiple-choice questions 12 marksInstructions • This section consists of 12 multiple-choice questions

• Each question is worth 1 mark• Fill in only ONE CIRCLE• Calculators may be used

1 23 × 22 = ?

A 25 B 26 C 45 D 46

2 88 ÷ 82 = ?

A 14 B 18 C 84 D 86

3 7m0 + 70 = ?

A 1 B 2 C 7 D 8

4 p–3 = ?

A p3 B p3 C

13p

D none of these

5 xmn = ?

A x mn B x nm C

xx

m

nD none of these

6 m– 2

3 = ?

A

13m

B

123 m

C

mm

2

3D

mm

2

3

7 log42 = ?

A 12

B 1 C 2 D 4

8 2 loga3 – loga2 = ?

A loga7 B loga4.5 C 2 loga1.5 D cannot be simplified

9 The value of e2 correct to three decimal places is?

A 0.301 B 0.693 C 6.581 D 7.389

Logarithmic and exponential functions

iiLogarithmic and exponential functions

Mathletics Instant Workbooks – Series M 2 Copyright © 3P Learning

Instructions This part consists of 12 multiple-choice questions Each question is worth 1 mark Calculators may be used Fill in only ONE CIRCLE for each question

Time allowed: 30 minutes Total marks = 12

Logarithmic and exponential functionsTopic Test PART A

12

115CHAPTER 4 – Logarithmic and exponential functions

10 ddx

e x ( ) = ?2

A e2x B 2ex C 2e2x D 12 2e x

11 The diagram could be a sketch of the graph of:

A y = 2x

B y = 2–x

C y = log2x

D y = 2 ln x

12 log27 = ?

A ln 7ln 2

B ln 2ln 7

C 2 ln 7 D 7 ln 2

SECTION II 88 marksShow all necessary working

13 Simplify: 1 mark each

a 8x+1 × 25x ÷ 42–x b log645 + log620 – log625

14 Find x if: 1 mark each

a x8 = 1 679 616 b (1 – x)3 = 0.512 c logx16 = 4

15 Find, correct to three decimal places: 1 mark each

a log102.9 b 9.31875 c log211

0 1 2 x

y = f(x)

1

y

115CHAPTER 4 – Logarithmic and exponential functions

10 ddx

e x ( ) = ?2

A e2x B 2ex C 2e2x D 12 2e x

11 The diagram could be a sketch of the graph of:

A y = 2x

B y = 2–x

C y = log2x

D y = 2 ln x

12 log27 = ?

A ln 7ln 2

B ln 2ln 7

C 2 ln 7 D 7 ln 2

SECTION II 88 marksShow all necessary working

13 Simplify: 1 mark each

a 8x+1 × 25x ÷ 42–x b log645 + log620 – log625

14 Find x if: 1 mark each

a x8 = 1 679 616 b (1 – x)3 = 0.512 c logx16 = 4

15 Find, correct to three decimal places: 1 mark each

a log102.9 b 9.31875 c log211

0 1 2 x

y = f(x)

1

y

Topic Test PART BInstructions Show all necessary working Time allowed: 30 minutes Total marks = 88

iiiLogarithmic and exponential functions

Mathletics Instant Workbooks – Series M 2 Copyright © 3P Learning

Logarithmic and exponential functionsTopic Test PART A

Total marks achieved for PART A

ivLogarithmic and exponential functions

Mathletics Instant Workbooks – Series M 2 Copyright © 3P Learning

Logarithmic and exponential functionsTopic Test PART B

116 EXCEL HSC MATHEMATICS REVISION & EXAM WORKBOOK

16 If loga3 = 0.565 and loga2 = 0.356 find: 1 mark each

a loga6 b loga9 c loga1.5

17 Find the value of x, correct to three decimal places, if: 2 marks each

a 5x = 424 b 1 – 3x = 0.57 c 6e2x+1 = 192

18 Write down the exact value of: 1 mark each

a 9 ln e b ln e4 c eln8

19 Sketch the graph of: 2 marks each

a y = 8x b y = log8x

x

y

x

y

Logarithmic and exponential functionsTopic Test PART B

vLogarithmic and exponential functions

Mathletics Instant Workbooks – Series M 2 Copyright © 3P Learning 117CHAPTER 4 – Logarithmic and exponential functions

20 Differentiate: 1 mark each

a y = 7x b y = ex c y = ln x

d y = 3e–2x e y = ln (5x – 4) f y = ln (x2 + 6x)

g y = 5e7x–4 h y = 4 loge(6 – 3x) i y e x = 2

21 Find the derivative of: 3 marks each

a y = x3e2x b y = 2x logex

c

ex

x6

6 – 1d

ln

4 + 1x

x

Logarithmic and exponential functionsTopic Test PART B

viLogarithmic and exponential functions

Mathletics Instant Workbooks – Series M 2 Copyright © 3P Learning118 EXCEL HSC MATHEMATICS REVISION & EXAM WORKBOOK

22 Find: 2 marks each

a ∫ 3

xdx b

∫ e dxx8 c ∫ 4

2 – 32x

xdx

d ∫

e dxx–2

2 e

∫3

2 + 1

xdx f

∫14 5–3e dxx

23 Find the exact value of: 3 marks each

a 1

1 e

xdx∫ b

0

1

2 ∫ e dx

x

c 1

5

∫ dxe x

d 1

4

22 + 5

+ 5 ∫ x

x xdx

e 1

3 47 – 2

∫ xdx f

0

23 –4∫ e dxx

Total marks achieved for PART B

Logarithmic and exponential functionsTopic Test PART B

viiLogarithmic and exponential functions

Mathletics Instant Workbooks – Series M 2 Copyright © 3P Learning

88

119CHAPTER 4 – Logarithmic and exponential functions

x

y

y = e–x

y

0 1–1 x

24 Find the equation of the tangent to the curve y = 2ex+1 at the point where x = 0 3 marks

25 Find the coordinates of the stationary point of the curve y = x ln x 4 marks

26 Find the area bounded by the curve y

x = 1 , the x-axis and the lines x = 1 and x = 5 3 marks

27 Find the volume of the solid of revolution formed when that portion of the curve y = e–x between x = –1and x = 1 is rotated about the x-axis. 3 marks

119CHAPTER 4 – Logarithmic and exponential functions

x

y

y = e–x

y

0 1–1 x

24 Find the equation of the tangent to the curve y = 2ex+1 at the point where x = 0 3 marks

25 Find the coordinates of the stationary point of the curve y = x ln x 4 marks

26 Find the area bounded by the curve y

x = 1 , the x-axis and the lines x = 1 and x = 5 3 marks

27 Find the volume of the solid of revolution formed when that portion of the curve y = e–x between x = –1and x = 1 is rotated about the x-axis. 3 marks

119CHAPTER 4 – Logarithmic and exponential functions

x

y

y = e–x

y

0 1–1 x

24 Find the equation of the tangent to the curve y = 2ex+1 at the point where x = 0 3 marks

25 Find the coordinates of the stationary point of the curve y = x ln x 4 marks

26 Find the area bounded by the curve y

x = 1 , the x-axis and the lines x = 1 and x = 5 3 marks

27 Find the volume of the solid of revolution formed when that portion of the curve y = e–x between x = –1and x = 1 is rotated about the x-axis. 3 marks

230 EXCEL PRELIMINARY GENERAL MATHEMATICS REVISION & EXAM WORKBOOK

Answers

Page 81 1 a 0 b 1 c 0 2 a cos x b –sin x c sec2x d 3 cos 3x e –4 sin x f 12

sec2

2 x g 10 cos 5x h –sin 2x i π cos πx

j 2 cos (2x + 3) k 15sec 3 –

42 x

π

l 1 + sin x m 2x cos (x2) n 1 + sec2(x – 1) o 5 sin (2 – 5x)

Page 82 1 a –4 sin 2x b – 1

4cos

2x c 4π2 sin πx 2 a 6 b –1.5 c 0 3 a 1.5 b 16 c

3

6

Page 83 1 a –x sin x + cos x b x2 cos x + 2x sin x c 2 tan x sec2x d –2x sin (x2) e

2 cos – sin 22

x x x

x f

x x xx

sin + cos – 12 2

g tan x sec x x

x= sin

cos2

h

–cot cosec =–

cos

sin2x x

x

x

Page 84 1 3 – + 3 –

3 = 0x y π 2

x y + – 1

2 –

4 = 0π 3

x =

6 or

56

π π 4 (cos x = 2 has no solutions)

Page 85 1 a –cos x + C b sin x + C c tan x + C d 3 sin x + C e – 1

2 cos 2 + x C f

18

tan 4 + x C g 4 cos

2 + x C

h tan (x + 1) + C i x – sin x + C j 13

sin(3 – 2) + x C k – 1 cos +

ππ x C l

tan2

+ x

C 2 a 2 b 1

Page 86 1 a 1 b –1 + 3 c 0 d 0 e 2 f π 2

2

Page 87 1 2 2 units2 2 a 0, 0.25, 0.75, 1 b π4

3 a x cos x + sin x b π2

Page 88 1 4 units2 2 2 units 2 3 π units3

Pages 89-94 1 B 2 D 3 B 4 C 5 A 6 C 7 B 8 A 9 D 10 B 11 a

49π

b π5

12 a 15° b 330° 13 0.9511 14 a

32

b – 1

3 15 a

π6

or

116π

b

34

or 74

π π c

π π π π3

, 23

, 43

or 53

16 π cm 17 96π cm2 18 3.4 km2

19 ab c

20 a 8 cos 4x b 32

sin 2x c

2sec 2 –

32 x

π

d x cos x + sin x 21 y x = – +

2π π 22 a

– 1

2cos 2 + x C b

6 sin

2 + x C

23 a

π – 2 24

b

2 33

24 2 units2 25 π units3

Page 95 1 a x7 b a8 c 2p9 d x3y7 e a10 f 81m8n4 g 1 h 6 i x6 j 3t4 k

y

x 2 l

14a

m 2a8b5 n 20n5 o 2g11h12 2 a 8

b 10 000 c 3 d 1 e 2 f 4 g 125 h 16 3 a 15

b 18

c

1256

d

1100 000

e 136

f 13

g 12

h

1100

Page 96 1 a 2 097 152 b 39.0625 c 12 167 d 0.007 8125 e 1567 f 13 g 0.0081 h 0.0081 2 a 54x b 713 c 311x d 22x

e 310m+1 f 2–7n 3 a k = 4 b m = 2 c p = –5 d a = 1 1

6 e x = 5 f q = 1 4 a 3.26 b 1.36 c 1.06

Page 97 1 a ac b loga y c loga x – loga y d 1 e 0 f n loga x 2 a 3 b 5 c 1 d 0 e 2 f 5 g 3 h 8 3 a 1 b 1 c 12

d 1 e 2 f 2 4 a loga72 b logm6 c logn9

y = 2 sin x

y

2

0

–2

–52π –2π

–3π2

–π

–π2

π2

π

3π2

2π

5π2

3π

7π2

x

y = cos 2x

y

1

0

–1

–32π

–54π –π

–3π4

–π2

–π4

π4

π2

3π4

π

5π4

3π2

x

y y = 4 tan x

4

–4

–32π –π

–π2

0

π2

π

3π2

2π

5π2

x

231ANSWERS

Answers

Page 98 1 loga y 2 a 1.230 b 2.312 c –0.456 d 0.217 3 a 2.37 b 9.48 c –4.74 4 a 0.96 b 0.867 c 0.382 d 1.053

5 a x = 1

3 b x = 56

Page 99 1 logba 2 a 1 1

2 b

23

c 2 1

2 d

2 2

3 e 2 log23 3 a 2.0959 b 1.3917 c 3.1699 d 1.7297 e 0.6275 f 2.5052

g –0.4650 h –0.4150 i –1.1495

Page 100 1 a 12

log 112 b 12

log 65 c 13

log 323 2 a 2.579 b 6.229 c –0.802 d 1.738 e 2.425 f 1.395

Page 101 1 a b c 2 a

b c 3 a b y = x

Page 102 1 10x+h – 10x, 10h – 1, 10 – 1h

h 2 a 2.30 b 0.69 c 1.10 3 a 2.30 b 0.69 c 1.10 4 a 2.72 5 1

6 a (ln 5)5x b (ln 7)7x c (ln 4)4x d (ln 11)11x e (ln 6)6x f (ln 9)9x g (ln 8)8x h (ln 15)15x

Page 103 1 a 1 b 0 c 1 d 2 e 2 f 7 g 7 h 5 2 a 7.3891 b 54.5982 c 296.8263 d 0.3679 e 0.2231 f 2.0541g 1.2809 h –1.4397 i 8.3178 j 81.3421 k 2.7183 l 3.3944 3 a 0.470 b 6.686 c 0.671 d –0.077 e 0.128 f –1.363Page 104 1 a ex b 3ex c 2e2x d 4ex e 10e5x f –e–x g 1 – ex h 12e2x+5 i –32e–8x j 18x2 – 9e3x k ex + e–x l 63e–9x

2 a ex(x + 1) b 2xe2x(x + 1) c e–x(7 – 3x) d 5e7x(7x2 – 61x + 5)

Page 105 1 a 4ex(ex + 5)3 b 3(4x – ex)2(4 – ex) 2 a

1 – xe x

b

1 – xe x

3 a

xe

x

x

( + 1)2 b

3 ( – 2 – 5)

( – 5)

2

2 2

e x x

x

x

Page 106 1 a ex + C b 5ex + C c 13

+ 3e Cx d 12

+ 2 +3e Cx e –4e–x + C f – 1

2 + 3–2e Cx g ex + x2 + C h

e C

x4

8 +

i x x e Cx

32 –2

3 – 4 + 3 + 2 a e2 – 1 b 6(e – 1) c

14

( – 1)12e d e e2

( – 1)4 e 14

(1 – )–4e f e(e2 – 1) g e2 + e–2 – e – e–1

h 7 i 13

– 13

+ 1 12

6 3e e

Page 107 1 a 1x

b 1x

c 1x

d

77 + 5x

e

–21 – 2x

f

55 + 3x

g 2x

h 5x

i 9x

j

2

+ 52

x

x k

6 x

x3 – 42 l

3 – 14

– 72

x

x x

2 a

2 ln xx

b

63 – 1x

Page 108 1 a 1e

b

22 – 1e

c

6

+ 12

e

e 2 a 1 + ln x b x3(1 + 4 loge x) c

1 – 2 ln3

x

x d

x x x

x x

ln – – 1

(ln )2

Page 109 1 a ln x + C b 6 ln x + C c 3 ln (x + 2) + C d ln (x2 + 5) + C e ln (x3 – 2) + C f ln (3x – 7) + C

g 4 ln (x2 – 3) + C h 14

ln (4 – 1) + x C i – 7

2 ln (1 – 2 ) + x C 2 a ln 4 b

12

c ln 4 d ln 6.8

Page 110 1 2x – ey = 0 2 e2x – 2ey – e2 + 4 = 0 3 (0, 1) 4 1e

Page 111 1 a (1, 0) b maximum at (0, 1) c

–1, 2e

d i –∞ ii 0 e

Page 112 1 a 2 units2 b (3 ln 3 – 2) units2 2 y = ex + e–x + 2Page 113 1 4 ln 2 units2 2 y = 3x2 – ln(2x – 1) + 4 3 2 ln 2 units2

Pages 114-119 1 A 2 D 3 D 4 C 5 A 6 B 7 A 8 B 9 D 10 C 11 C 12 A 13 a 210x–1 b 2 14 a 6 b 0.2 c 2 15 a 0.462b 1.563 c 3.459 16 a 0.921 b 1.13 c 0.209 17 a 3.759 b –0.768 c 1.233 18 a 9 b 4 c 8 19 a b (see next page) 20 a (ln 7)7x

y

1

0 1 5 x

y = log5x

y

1y = log2x

0 1 2 x

y

1

0 1 10 x

y = log10x

y

2

10 1 x

y = 2x y

7 y = 7x

10 1 x

y = 3–x

y

3

–1 0 x1

y = 3x

y = log3x

y

31

0 1 3 x

y

1

–1, 2

e

1 x

y = ex(1 – x)

1e

Answers – Logarithmic and exponential functions

viiiLogarithmic and exponential functions

Mathletics Instant Workbooks – Series M 2 Copyright © 3P Learning

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

231ANSWERS

Answers

Page 98 1 loga y 2 a 1.230 b 2.312 c –0.456 d 0.217 3 a 2.37 b 9.48 c –4.74 4 a 0.96 b 0.867 c 0.382 d 1.053

5 a x = 1

3 b x = 56

Page 99 1 logba 2 a 1 1

2 b

23

c 2 1

2 d

2 2

3 e 2 log23 3 a 2.0959 b 1.3917 c 3.1699 d 1.7297 e 0.6275 f 2.5052

g –0.4650 h –0.4150 i –1.1495

Page 100 1 a 12

log 112 b 12

log 65 c 13

log 323 2 a 2.579 b 6.229 c –0.802 d 1.738 e 2.425 f 1.395

Page 101 1 a b c 2 a

b c 3 a b y = x

Page 102 1 10x+h – 10x, 10h – 1, 10 – 1h

h 2 a 2.30 b 0.69 c 1.10 3 a 2.30 b 0.69 c 1.10 4 a 2.72 5 1

6 a (ln 5)5x b (ln 7)7x c (ln 4)4x d (ln 11)11x e (ln 6)6x f (ln 9)9x g (ln 8)8x h (ln 15)15x

Page 103 1 a 1 b 0 c 1 d 2 e 2 f 7 g 7 h 5 2 a 7.3891 b 54.5982 c 296.8263 d 0.3679 e 0.2231 f 2.0541g 1.2809 h –1.4397 i 8.3178 j 81.3421 k 2.7183 l 3.3944 3 a 0.470 b 6.686 c 0.671 d –0.077 e 0.128 f –1.363Page 104 1 a ex b 3ex c 2e2x d 4ex e 10e5x f –e–x g 1 – ex h 12e2x+5 i –32e–8x j 18x2 – 9e3x k ex + e–x l 63e–9x

2 a ex(x + 1) b 2xe2x(x + 1) c e–x(7 – 3x) d 5e7x(7x2 – 61x + 5)

Page 105 1 a 4ex(ex + 5)3 b 3(4x – ex)2(4 – ex) 2 a

1 – xe x

b

1 – xe x

3 a

xe

x

x

( + 1)2 b

3 ( – 2 – 5)

( – 5)

2

2 2

e x x

x

x

Page 106 1 a ex + C b 5ex + C c 13

+ 3e Cx d 12

+ 2 +3e Cx e –4e–x + C f – 1

2 + 3–2e Cx g ex + x2 + C h

e C

x4

8 +

i x x e Cx

32 –2

3 – 4 + 3 + 2 a e2 – 1 b 6(e – 1) c

14

( – 1)12e d e e2

( – 1)4 e 14

(1 – )–4e f e(e2 – 1) g e2 + e–2 – e – e–1

h 7 i 13

– 13

+ 1 12

6 3e e

Page 107 1 a 1x

b 1x

c 1x

d

77 + 5x

e

–21 – 2x

f

55 + 3x

g 2x

h 5x

i 9x

j

2

+ 52

x

x k

6 x

x3 – 42 l

3 – 14

– 72

x

x x

2 a

2 ln xx

b

63 – 1x

Page 108 1 a 1e

b

22 – 1e

c

6

+ 12

e

e 2 a 1 + ln x b x3(1 + 4 loge x) c

1 – 2 ln3

x

x d

x x x

x x

ln – – 1

(ln )2

Page 109 1 a ln x + C b 6 ln x + C c 3 ln (x + 2) + C d ln (x2 + 5) + C e ln (x3 – 2) + C f ln (3x – 7) + C

g 4 ln (x2 – 3) + C h 14

ln (4 – 1) + x C i – 7

2 ln (1 – 2 ) + x C 2 a ln 4 b

12

c ln 4 d ln 6.8

Page 110 1 2x – ey = 0 2 e2x – 2ey – e2 + 4 = 0 3 (0, 1) 4 1e

Page 111 1 a (1, 0) b maximum at (0, 1) c

–1, 2e

d i –∞ ii 0 e

Page 112 1 a 2 units2 b (3 ln 3 – 2) units2 2 y = ex + e–x + 2Page 113 1 4 ln 2 units2 2 y = 3x2 – ln(2x – 1) + 4 3 2 ln 2 units2

Pages 114-119 1 A 2 D 3 D 4 C 5 A 6 B 7 A 8 B 9 D 10 C 11 C 12 A 13 a 210x–1 b 2 14 a 6 b 0.2 c 2 15 a 0.462b 1.563 c 3.459 16 a 0.921 b 1.13 c 0.209 17 a 3.759 b –0.768 c 1.233 18 a 9 b 4 c 8 19 a b (see next page) 20 a (ln 7)7x

y

1

0 1 5 x

y = log5x

y

1y = log2x

0 1 2 x

y

1

0 1 10 x

y = log10x

y

2

10 1 x

y = 2x y

7 y = 7x

10 1 x

y = 3–x

y

3

–1 0 x1

y = 3x

y = log3x

y

31

0 1 3 x

y

1

–1, 2

e

1 x

y = ex(1 – x)

1e

232 EXCEL PRELIMINARY GENERAL MATHEMATICS REVISION & EXAM WORKBOOK

Answers

b ex c

1x

d – 6e–2x e

55 – 4x

f

2 + 6

+ 62

x

x x g 35e7x–4 h

–126 – 3x

i 22

xe x

21 a x2e2x(2x + 3) b 2(1 + logex) c

12 (3 – 1)

(6 – 1)

6

2

e x

x

x

d

4 + 1 – 4x ln

(4 + 1)2

x x

x x 22 a 3 ln x + C

b 18

+ 8e Cx c ln (2x2 – 3) + C d – 1

4 + –2e Cx e

32

ln(2 + 1) + x C f – 1

12 + 5–3e Cx

23 a 1 b e – 1

2 c

– 1 + 1 or

– 15

4

5e ee

e

d ln 6 e 2 ln 5 f

13

( – 1)4

6

ee 24 y = 2e(x + 1) 25

1, –

1e e

26 ln 5 units2

27

π2

– 1 units22

3ee

Page 120 1 a 6t – 8 b –35t6 c 20(4t – 7)4 2 a 30t4 – 21t2 + 2 b 4t3 + 3t2 – 2t – 1 c 2 cos(2t + 5) 3 a 18t5 + 3t2 – 7

b 8e2t–1 c 6(3t + 1) 4 a 0 b –2–10t c 1t

5 a 6t – 20 b 0 c –3π2 cos πt

Page 121 1 a 2t2 + 3t b

83

– 6 + 7 + 18 23

32t

t t 2 a 4 –

6 + 4 + 22

3t t t b

72

– 12 + 682t

t

Page 122 1 a 35 b 10 c –6 d 4 2 a 96 b 449 1

3Page 123 1 a >,< b <, < c <, > d >, > 2 a The number of registered pets is increasing over theperiod. b The number of pet registrations is increasing at a decreasing rate. c (see right)

Page 124 1 a 0 b 0 or 40 c V t t = 800 –

4 + 10002

4 d 20 843.75 2 a 40 L/min b after 18 minutes c 810 litres

Page 125 1 a 4375 grams per second b 30 seconds c 140 kg d 6000 3 g/sPage 126 1 a 9113 b 20 c 442 943 2 a 51.8 b 7.7 c –3.0Page 127 2 a 400 b 0.0260 c 264 [nearest whole number] d –6.9 [1 d.p.]Page 128 1 a 59 g [nearest g] b 1.2 grams per year [1 d.p.] 2 b In the 10th hourPage 129 1 a 0.03466 b 1345 2 a 0.0080 [4 d.p.] b 174 hoursPage 130 1 a displacement b zero 2 a the particle is moving to the left b the velocity gives the direction as well as themagnitude; speed = | velocity | 3 a –12 m b at 2 seconds and at 9 seconds c v = 2t – 11 d 1 m s–1 4 a –11 m s–1 b 2.25 sc x = 9t – 2t2 + 7Page 131 1 velocity 2 a slowing down b slowing down c speeding up d speeding up 3 a a = 160 – 6t b 148 m s–2

4 a v = –2t + 6 b –4 m s–1 c x = –t2 + 6t – 2 d 3 m

Page 132 1 a 0 s and 12 s b 432 m 2 a x t t = 9 – 1

32 3 b –18 m s–2, 972 m

Page 133 1 a 45 m s–1 b 6 m s–2 c 8 s d 128 m 2 a v t

t = + 4

( + 1) + 12

2 b

x t

tt =

2 – 4

+ 1 + 12 + 1

2

Page 134 1 a i v = –6t + 13 ii x = –3t2 + 13t + 10 b i 2 1

6s ii 5 s

c d The particle is initially at a position 10 m to the right of the origin travelling right at a speed of

13 m s–1. It stops after 2 1

6 seconds, then moves left passing through the origin after 5 seconds, and

continues to travel left at increasing speed.

Page 135 1 a 1 m to the right of the origin b 12

s, 3 m c 2π2 m s–2 d (see right)

Pages 136-140 1 A 2 D 3 B 4 B 5 B 6 A 7 D 8 B 9 B 10 A 11 a –5 – 6t b –612 a –8 sin 4t b –32 cos 4t 13 a 3t2 – 6t + 7et b 6t – 6 + 7et

14 x

tt t =

83

– 3 + 3 – 12 13

32 15 x = –2t2 + 9t + 1 16 a 55 b 2 or 8 c 8 d 5 17 a 5 b 45 18 a 0.2 L/min b

V e t = – 1

5

19 a 10 000 b 0.1386 c 278 576 [nearest whole number] d 34th hour 20 a 13

m s–1 b – 1

9m s–2 21 a 4 s b 4 s c The particle

is stationary when t = 4, and because ̇̇x < 0, the maximum displacement occurs when t = 4. So the maximum displacement is 0 mand the particle never moves right of the origin.

y8 y = 8x

10 1 x

y

1y = log8x

0 1 8 x

P

25000

0 t

2c

x

10

0

21

65 t

x = 2 sin πt + 1

x

3

1

0

–11 2 3 4 t

19 a 19 b

Answers – Logarithmic and exponential functions

ixLogarithmic and exponential functions

Mathletics Instant Workbooks – Series M 2 Copyright © 3P Learning

(see below)

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