Chapter 5 Practice Masters Answers …BLM 5-13..

(page 1)

Calculus and Vectors 12: Teacher’s Resource Copyright ® 2008 McGraw-Hill Ryerson Limited BLM 5-13 Practice Masters Answers

Prerequisite Skills 1. Sketches may vary. i) a) 3 b) {x ∈ R} c) {f(x) ∈ R | f(x) > 2} d) f(x) = 2 ii) a) –2 b) {x ∈ R} c) {g(x) ∈ R | g(x) > –1} d) g(x)= –1 iii) a) 8 b) {x ∈ R} c) {h(x) ∈ R | h(x) > –8} d) y = –8 2. Sketches may vary. i) a) none b) {x ∈ R | x > 0} c) {f(x) ∈ R} d) x = 0 ii) a) none b) {x ∈ R | x > 1} c) {g(x) ∈ R} d) x = 1 iii) a) –2 b) {x ∈ R | x > –1} c) {y ∈ R} d) x = –1 3. a) y = 33x b) y = 38x c) y = 3x d) y = 36x

4. a) 1.771 b) 3.170 c) 2.622 d) 3.465 e) 6.615 f) 4 g) –2 h) –3.322

5. a) (12x11y10) b)

3y5

x3

z2

c)

x2

2

d) 26x e) 38 f) x14a

6. a) 5 b) 1 c) 23 d) 2 e) 5 f) 0.5

7. a) x = –3 b) x = –1 c) x =

3

4 d) x = 3

e) x = 9 f) x = 125 g) x = –3 h) x =

1

3

8. a)

A = 7501

2

!"#

$%&

t

2

b) i) 187.5 mg ii) 23.4 mg 9. approximately 7.8 years 5.1 Rates of Change and the Number e 1. a) C b) D c) B d) A 2. a) Sketches may vary. b)

h

5h

! 1

h

0.1 1.746 0.01 1.622 0.001 1.610 0.000 1 1.609

slope of the a secant line near point (0, 1) c) 1.61 d) The slope of the tangent line to y = 5x at (0, 1)

5.2 The Natural Logarithm 1. a) 3x b) 5 – x c) –2 d) 8 e) 3 – 8x f) 2x – 6 2. a) 0.549 b) 0 c) –3.000 d) 0.916 e) 1.221 f) 0.779 g) 1.820 h) 7.213 3. approximately 58 days 4. a) approximately 16.0°F b) after approximately 8.9 min 5. a) e.g., M(t) = 1000(0.90)t b) e.g., on day 10 c) e.g., 121.6 mg 6. a) approximately 4.67 years b) approximately 7.27 years c) approximately 28.41 years d) approximately 14.27 years 5.3 Derivatives of Exponential Functions 1. a)

!f (x) = ln 3( ) 3

x( )

b)

!h (x) = ln 1

6

"#$

%&'

1

6

"#$

%&'

x

c) !P (x) = "5ex

d) !g (x) = "ex

e)

dy

dx= ln

3

!

"#$

%&'

3

!

"#$

%&'

x

f)

dy

dx= ln 100( ) 100

x( )

2. (ln 3)34 3. e7 4. –2e

5. y =

ln 5

25x +

1+ 2 ln 5

25

6. y = 3e

4

x ! 9e4

7. a) 250 b) i) approximately 17.3 h ii) approximately 27.5 h iii) approximately 109.6 h c) 14.9 bacteria/h; 22.3 bacteria/h 8. a) $18 393.97; $6766.76 b) –$3678.79; –$1353.35 9. a) 42°F; this is the initial temperature of the milk b) 72°F; this is the temperature that the milk’s temperature approaches as t increases c) when t &= 28.1 min d) 0.34°F/min 10. a) e.g., 1 student b) approximately 200 students c) when t = 5; 50 students/day

Chapter 5 Practice Masters Answers …BLM 5-13..

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Calculus and Vectors 12: Teacher’s Resource Copyright ® 2008 McGraw-Hill Ryerson Limited BLM 5-13 Practice Masters Answers

5.4 Differentiation Rules for Exponential Functions

1. a)

dy

dx= 3e

3x

b)

dy

dx= !3e

4!3x

c)

dy

dx= 5x

4

ex

5

d)

dy

dx= 2xe

2 x+ 2x

2

e2 x

e)

dy

dx=

xex! e

x

x2

f) !f (x) = 30e3x

(1+ 5e3x

)

g) !g (x) = 5xe

x(ln 5 + 1)

h) !h (x) = 2x(ln 5)e5

x2

5x

2

i)

dy

dx=

1! 2xe1! x

2

2 x + e1! x

2

j)

dy

dx=

ex(1+ e

2 x)

(1! e2 x

)2

k) !k (x) = sec

2x( )e

tan x

l)

dy

dx= e

2 x(2sin3x + 3cos3x)

m) !m (x) =

1

e

x " 1

n) !n (x) = "200e"10 x

(1+ 5e"10

)2

2. local minimum value of y = –1, when x = 0 3. a) i) approximately 23.1 years ii) approximately 36.6 years b) i) 72.9 mice/year ii) 114.3 mice/year c) 162.0 mice/year 4. a) !p (t) = "(ake

" kt)(1" ae

" kt)

b)

c) e.g., approximately 7.4 days

5. a) approximately 89 bacteria

b) approximately 5.2 days c) P(t) = 40(10

t

5.76 ) d) approximately 2184 e) approximately 5 6. a) P0 = 1500; a = 64 b) approximately 2274 c) i) approximately 399 253/h ii) approximately 25 552 177/hr 5.5 Making Connections: Exponential Models

1. a) e.g., N = 2000 5( )

t

3 b) 5848 bacteria c) 3137 bacteria/h d) after approximately 6.8 h

2. a) M(t) = (300)2

! t

138 b) approximately 81 days c) –1.0 mg per day 3. a) !s (t) = "12(ln 0.8)(0.8)

t b) i) 2.1 km/h ii) 1.1 km/h 4. when t = 1.98 months 5. a) !a (t) = "180(2

" t

)(t ln 2 " 1) b) 13.9 degrees/s 6. a) 9.23 mg b) –0.22 mg/year c) 65.6 years 7. a)

b) linear;

log

2f = N + 3.781

c) f = (2)N + 3.781 d) 1760 Hz Chapter 5 Review 1. Sketches may vary. For example,

Chapter 5 Practice Masters Answers …BLM 5-13..

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Calculus and Vectors 12: Teacher’s Resource Copyright ® 2008 McGraw-Hill Ryerson Limited BLM 5-13 Practice Masters Answers

a)

b)

c)

d)

2. a) domain: {x ∈ R} range: {y ∈ R | y > 0} b) domain: {x ∈ R} range: {y ∈ R | y > 0} 3. a) 0.007 b) 2.163 c) π d) 4.57 4. a) 1.396 b) 1.041 c) 0.968 d) 12.281 5. a) y′ = –5ex b) y′ = 7xln 7

6. y = 8x ln

1

2+ 8 ! 8ln

1

2

7. a) e.g., N = 5000(2)

t

2 b) 320 000 bacteria

c) 15.3 h d) !N = 2500(ln 2)(2)

t

2

8. a)

dy

dx= 3x

2

+ 2ex

b) !f (x) = (8x " 7)e4 x

2" 7 x +11

c) !g (x) = "2e"2 x

x2+ 2xe

"2 x

d)

!h (x) =1" e

x

2 x " ex

9. a) 178.5 mg b) 2.77 h c) –100.1 mg/h 10. 24 days 11. t = 14.97 min Chapter 5 Test

1. a)

dy

dx= 6e

6 x b)

dy

dx= 2xe

x2+ 3

c)

dy

dx= 16xe

1+ 4 x2

d)

dy

dx= (4x + 1)e

4 x

e)

dy

dx= (2x

!2! 3x

!4)e

x2!1

f)

dy

dx=

(6x ! 17)e3x

(2x ! 5)2

g)

dy

dx= 5e

5x! 5e

!5x

h)

dy

dx=

e! x

(2 + e! x

)2

2. a) 125 people b) 16 000 people c) 2773 people/day d) approximately 30.0 days 3. a) –2.64°C/min b) 7.69 min 4. a) approximately 23.4 years b) !V = 14.779(1.03)

t c) 23 dollars/year 5. a) 14.65% b) 11.8 mg/h 6. a) 0.6 A b) i) 0.001 A/s ii) 0.0001 A/s 7. a) e.g., A = 1390.3(0.88)t b) approximately 300.0 mg c) i) –82.5 mg/h ii) –38.3 mg/h