Name: _______________________________ Date: ________________________

Calculus and Vectors 12: Teacher’s Resource Copyright ® 2008 McGraw-Hill Ryerson Limited BLM 5-1 Prerequisite Skills

Chapter 5 Prerequisite Skills …BLM 5-1. .

Graphing Exponential and Logarithmic Functions 1. Sketch a graph of each function. Then,

determine the: a) y-intercept b) domain c) range d) equation(s) of any asymptote(s)

i) f(x) = 3x + 2 ii) g(x) = (–1)2x – 1

iii) h(x) = 16

1

2

!"#

$%&

x

– 8

2. Sketch a graph of each function. Then,

determine the: a) y-intercept b) domain c) range d) equation(s) of any asymptote(s)

i) f(x) = log10x + 2 ii) g(x) = log2(x – 1) iii) y = log3(x + 1) – 2

Changing Bases of Exponential and Logarithmic Expressions 3. Rewrite each exponential function using a

base of 3. a) y = 27x b) y = 94x

c) y = 27

x

3

d) y =

1

9

!"#

$%&

'3x

4. Express each logarithm in terms of

common logarithms (base-10 logarithms), and then use a calculator to evaluate. Round answers to three decimal places. a) log37 b) log29 c) log568 d) log345

e) log298 f)

log1

2

1

16

g) log5

1

25

!"#

$%&

h) log0.510

Applying Exponent Laws and Laws of Logarithms 5. Simplify. Express answers with positive

exponents.

a) (2x3y4)2(3x5y2) b)

36x!2

y3

z!4

12xy!2

z!2

c)

3x!2

y3

12xy!1

"

#$%

&'10x

4

y!2

5x!1

y2

"

#$%

&' d) 23x × 8x

e)

3n+1

!9n"1

27n"3

f)

xa( ) x

6( )2

x!2a

6. Evaluate, by applying the laws of

logarithms. a) log39 + log327 b) 2log255 c) log2325 – log24 d) log4 + log25

e) log2512 – log216 f) log4 164

Solving Exponential and Logarithmic Equations 7. Solve for x.

a) 3x = 27x + 2 b) 6x + 1 = 1

c) 44x – 1 = 256 d) log7 + logx = log21 e) 2log7x = log781 f) log5x = 3

g) log4

1

64 = x h) logx

1

9

!"#

$%&

= 2

Constructing and Applying an Exponential Model 8. A new medication has a half-life of 2 days

once ingested. 750 mg of the medication is ingested. a) Determine a function that expresses

the mass of medication remaining in the body, A, as a function of time, t, in days.

b) Determine the mass of medication remaining in the body after

i) 4 days ii) 10 days

9. The current population of a city is 178 500 and its population is growing at a rate of 2.1% per year. In how many years will the population be 210 000?

Name: _______________________________ Date: ________________________

Calculus and Vectors 12: Teacher’s Resource Copyright ® 2008 McGraw-Hill Ryerson Limited BLM 5-3 Section 5.1 Rates of Change and the Number e

5.1 Rates of Change and the Number e …BLM 5-3. .

1. Match each graph with the graph that represents its rate of change as a function of x. Justify your choices. Graph of Original Function Graph of Derivative a) A

b) B

c) C

d) D

2. a) Sketch the graph of y = 5x.

b) Use a calculator to evaluate the

quantity

5h

! 1

h for h = 0.1, 0.01,

h = 0.001, and h = 0.000 1. What does the quantity represent?

c) Estimate the value of the limit

limh!0

5h

" 1

h.

d) What does the limit in part c) represent?

Name: _______________________________ Date: ________________________

Calculus and Vectors 12: Teacher’s Resource Copyright ® 2008 McGraw-Hill Ryerson Limited BLM 5-4 Section 5.2 The Natural Logarithm

5.2 The Natural Logarithm …BLM 5-4. . 1. Simplify.

a) ln(e3x) b) eln(5 – x) c) ln(e–2) d) 2eln(4)

e) ln(e3-8x) f) eln(2x – 6)

2. Solve for x, correct to three decimal places. a) e2x = 3 b) e5x = 1 c) 18 = e–x – 2

d) 5 =

1

1! 2e! x

e) 5ln(x) = 1

f) 4ln

1

x

!"#

$%&

= 1

g) 3x = e2

h) 2x = e5 3. A person who weighs 170 lbs wants to

loose 10 lbs. Their weight is given by the equation W(t) = 160 + 10e–0.012t, where W(t) is the person’s weight after being on a balanced diet for t days. How long will it take the person to lose 5 lbs?

4. A thermometer is moved from inside a

house where the temperature is 70ºF, to outside the house where the temperature is 10ºF. A function that models the temperature reading of the thermometer is T(t) = 10 + 60e–0.46t, where t is the time since the thermometer was moved, in minutes, and T(t) is temperature, in degrees Fahrenheit. a) What is the temperature reading 5 min

after the thermometer is moved? b) Determine when the thermometer

reading will be 11°F.

5. A chemist has a 1000 mg sample of a radioactive substance. She recorded the amount of substance remaining in the sample every day for a week. Her results are shown in the table.

Day Mass (mg) 0 1000.00 1 900.00 2 810.00 3 729.00 4 656.10 5 590.49 6 531.44 7 478.30

a) Find an exponential function that models the data.

b) Use your model from part a) to estimate on which day less than 350 mg of the substance will remain.

c) Predict the mass of substance that will remain after 20 days.

6. Investors can measure the effectiveness of an

investment by determining its doubling time; that is, the time required for the initial investment to double. Recall the compound interest formula A(t) = P(1 + r)t, where P is the initial investment, r is the annual interest rate (written as a decimal), and A(t) is the amount, after t years. a) Find the doubling time for an investment

that earns 16% annual interest. b) Find the time required for the initial

investment to double if it earns 10% annual interest.

c) Find the time required for the initial investment to quadruple if it earns 5% annual interest.

d) Find the time required for the initial investment to triple if it earns 8% annual interest.

Name: _______________________________ Date: ________________________

Calculus and Vectors 12: Teacher’s Resource Copyright ® 2008 McGraw-Hill Ryerson Limited BLM 5-5 Section 5.3 Derivatives of Exponential Functions

5.3 Derivative of Exponential Functions …BLM 5-5. . 1. Determine the derivative with respect to x

for each function. a) f(x) = 3x

b) h(x) =

1

6

!"#

$%&

x

c) P(x) = –5ex d) g(x) = πex

e)

y =3

!

"#$

%&'

x

f) y = 100x 2. Calculate the instantaneous rate of change

of the function y = 3x when x = 4. 3. Calculate the instantaneous rate of change of the function y = e7x at x = 1. 4. Determine the slope of the graph y = –2ex

at x = 1. 5. Determine the equation of the line tangent

to y = 5x at x = –2. 6. Determine the equation of the line tangent

to y = 3ex at x = 4. 7. The function N = 250e0.04t represents the

number of bacteria in a sink, where N is the number of bacteria, and t is the time, in hours. a) How many bacteria are in the sink

when t = 0? b) Determine how long it will take the

number of bacteria to i) double ii) triple iii) grow to 20 000 bacteria

c) At what rate is the number of bacteria growing after 10 h? after 20 h?

8. The value, V(t), in dollars, of a car after t years is modelled by the function V(t) = 50 000e–0.2t. a) What is the value of the car after 5 years?

after 10 years? b) What is the rate of change of the car’s

value after 5 years? after 10 years? 9. A glass of cold milk from the refrigerator is

left on a countertop on a warm summer’s day. Its temperature, y, in degrees Fahrenheit, after sitting on the counter for t minutes is given by the function

y = 72 – 30(0.98)t. a) What is the temperature of the

refrigerator? How can you tell? b) What is the temperature of the room?

How can you tell? c) Determine when the temperature of the

milk will reach 55°F. d) At what rate is the milk warming when its

temperature is 55°F? 10. The spread of measles in a school is

modelled by the function p(t) =

200

1+ e5! t

,

where t is the number of days since the measles first appeared, and p(t) is the total number of students who have caught the measles. a) Estimate the initial number of students

infected with measles. b) About how many students in total will

catch the measles? c) When will the rate of spread of the

measles be the greatest? What is this rate of spread?

Name: _______________________________ Date: ________________________

Calculus and Vectors 12: Teacher’s Resource Copyright ® 2008 McGraw-Hill Ryerson Limited BLM 5-6 Section 5.4 Differentiation Rules for Exponential Functions

5.4 Differentiation Rules for Exponential Functions …BLM 5-6. .

1. Differentiate with respect to x.

a) y = e3x

b) y = e4 – 3x c) y = e

x5

d) y = x2e2x

e) y =

ex

x

f) f(x) = (1 + 5e3x)2

g) g(x) = 5xex

h) h(x) = e5

x2

i) y = x + e1! x

2

j) y =

ex

1! e2 x

k) k(x) = etan x

l) y = e2x sin3x

m) m(x) =

e3x

1+ ex

n) n(x) = (1 + 5e–10x)4 2. Identify the coordinates of any local

maximum or minimum values of the function y = e9x – e6x – e3x.

3. A wildlife manager counted 1800 mice in

a 10-km2 area of forest. It is predicted that the mouse population in this area will grow exponentially according to the function P(t) = 1800e0.03t, where P(t) is the mouse population and t is the time, in years. a) How long will it take for the mouse

population to i) double ii) triple

b) Find the rate of change of the mouse population after i) 10 years ii) 25 years

c) At what rate is the mouse population growing at the time when its number has tripled?

4. Under certain circumstances a rumour

spreads according to the equation

p(t) =

1

1+ ae! kt

, where p(t) is the proportion

of the population that has heard the rumour at time t, measured in hours, and a and k are positive constants. a) Find the rate of spread of the rumour. b) Graph this function using graphing

technology for the case where a = 10 and k = 0.5.

c) Use the graph from part b) to estimate how long it will take for 80% of the population (p(t) = 0.8) to hear the rumour.

5. A bacteria colony’s population is

modelled by the function P(t) = 40e0.4t, where P is the number of bacteria after t days. a) What is the bacterial population after

2 days? b) How long will it take for the population

to reach 8 times its initial level? c) Rewrite this function as an exponential

function using a base of 10. d) Determine the bacterial population after

10 days, using your equation from part c).

e) Use the function from the initial question to find the bacterial population in part d). What is the difference in the values?

6. A pond has an algae population of

1500. After 10 min, the population is 3000. The population can be modelled by P = P0(at), where P is the population after t hours, and P0 is the initial population. a) Determine the values of P0 and a. b) Find the algae population after

6 min. c) Find the rate of change of the algae

population after i) 1 h ii) 2.5 h

Name: _______________________________ Date: ________________________

Calculus and Vectors 12: Teacher’s Resource Copyright ® 2008 McGraw-Hill Ryerson Limited BLM 5-7 Section 5.5 Making Connections: Exponential Models

5.5 Making Connections: Exponential Models …BLM 5-7. .

1. A bacterial culture has an initial population

of 2000 bacteria. The predicted population of the culture after 3 h is 10 000 bacteria. a) Assume that the bacteria population

grows exponentially. Find an exponential function that models the population growth.

b) Find the number of bacteria after 2 h. c) Find the rate of growth of the number

of bacteria after 2 h. d) When will the bacteria population

reach 18 000? 2. A 300 mg sample of polonium-210

(Po-210) decays to 150 mg after 138 days. a) Write a function that gives the mass of

Po-210 remaining as a function of time, in terms of its half-life.

b) How long will it take a 300 mg sample of Po-210 to decay to a mass of 200 mg?

c) What is the sample’s rate of decay when the mass remaining is 200 mg?

3. As lava from a volcano cools, it flows

more slowly. The distance, s(t), in kilometres, from the crater to the leading edge of lava during a particular volcanic eruption is modelled by the equation s(t) = 12(2 – 0.8t), where t is the time, in hours. a) Determine the derivative of s(t). b) How fast is the lava travelling down

the mountain after i) 1 hour? ii) 4 hours?

4. A company decides the price of a new product should be determined by the formula

p(t) = 100e0.1t, where p(t) is the price, in dollars, and t is the time, in months. It was determined that when !p (t) = 1.02 , the price formula p(t) should be revised. What is the value of t when the price formula should be revised?

5. An automatic door has been programmed so that the angle, in degrees, that the door is open is determined by the equation a(t) = 180t(2)–t, where t is the time in seconds, measured from when a button is pressed. a) Find an equation that models the rate at

which the angle is changing with respect to time.

b) How quickly is the door closing after 5 s? 6. The mass of a sample of strontium-90 (Sr-90), in milligrams, as a function of time

is given by the equation M (t) = M

0e

! ln 2( )t

29 , where M0 is the mass of the sample, and M(t) is the mass of strontium remaining at time t, in years. a) Find the mass of a 24 mg sample of Sr-90 remaining after 40 years. b) At what rate is the 24 mg sample of Sr-90 decaying after 40 years? c) How long does it take for 24 mg of Sr-90

to decay to 5 mg? 7. The frequencies of the first six A-notes on

a piano are shown in the table, where f represent the frequency of the note, in hertz, and N represent the A-note. N = 1 corresponds to A1, N = 2 corresponds to A2, etc.

N f (Hz) 1 27.5 2 55 3 110 4 220 5 440 6 880

a) Use graphing technology to create a scatter plot with N on the horizontal axis and log2 f on the vertical axis.

b) What type of curve would best fit the data? Use the appropriate regression to determine an equation relating log2 f to N.

c) Write an equation relation f to N in exponential form.

d) Determine the frequency of note A7.

Name: _______________________________ Date: ________________________

Calculus and Vectors 12: Teacher’s Resource Copyright ® 2008 McGraw-Hill Ryerson Limited BLM 5-9 Chapter 5 Review

Chapter 5 Review …BLM 5-9. .

5.1 Rates of Change and the Number e 1. Without using technology, sketch the

following graphs on the same set of axes. Then, sketch the graph of the derivative of each function. a) y = ex b) y = 4x

c) y = 1x d) y =

1

2

!"#

$%&

x

2. What is the domain and range of each

function? a) y = 100x b) y = 0.1x

5.2 The Natural Logarithm 3. Evaluate, correct to three decimal places.

a) e–5 b) ln(8.7) c) ln(eπ) d) eln(4.57)

4. Solve for x, correct to three decimal places.

a) ln x =

1

3 b) e

ex

= 17

c) 26x – 1 = 28 d) x = 15e

!1

5 5.3 Derivatives of Exponential Functions 5. Differentiate each function with respect to x.

a) y = !5ex b) y = 7x

6. Find the equation of the line tangent to the

curve y =

1

2

!"#

$%&

x

at (–3, 8).

7. A bacteria culture has an initial

population of 5000 bacteria. The number of bacteria doubles every 2 h. a) Write an exponential function that

models the population growth of the bacteria as a function of time.

b) What is the population of the culture after 12 h?

c) How long will it take for the population to grow to 1 000 000 bacteria?

d) Take the derivative of the exponential function you created in part a). What does this new equation represent?

5.4 Differentiation Rules for Exponential Functions 8. Determine the derivative of each function.

a) y = x3 + 2ex b) f(x) = e

4 x2!7 x+11

c) g(x) = e–2xx2

d) h(x) = x ! ex

9. The absorption into the bloodstream of an

800 mg dose of medication administered orally is modelled by the function

A(t) = 800e

!t

4 , where A(t) represents the mass of medication that has yet to be absorbed, in milligrams, and t is the time, in hours. a) What mass of medication is yet to be

absorbed after 6 h? b) How long will it take for half of the

medication to be absorbed? c) At what rate is the medication being

absorbed when half of the medication has been absorbed?

5.5 Making Connections: Exponential Models 10. To determine whether a person has a

thyroid deficiency, a radioactive iodine isotope with a half-life of 8 days is injected into the bloodstream. The amount, I(t), of iodine present in a healthy patient’s body after t days can be modelled with the

function I(t) =

I0

1

2

!"#

$%&

t

8

, where I0 is the

initial dose. How long will it take for 87.5% of a dose of iodine to decay?

11. Medical researchers are testing thermal

recovery of human skin. For one group of patients, the temperature of the forehead, T, in degrees Celsius, decreased according to the function T (t) = 35 + 2.1e

!0.17 t , where t is the time, in minutes. The researchers are interested in the value of t when

!T (t) = "0.028 . Determine this value of t.

Name: _______________________________ Date: ________________________

Calculus and Vectors 12: Teacher’s Resource Copyright ® 2008 McGraw-Hill Ryerson Limited BLM 5-10 Chapter 5 Test

Chapter 5 Test …BLM 5-10. .

1. Differentiate each function. a) y = e6x b) y = e

x2+ 3

c) y = 2e1+ 4 x

2

d) y = xe4x e) y = x

!3

ex

2!1

f) y =

e3x

2x ! 5

g) y = e5x + e–5x

h) y =

1

2 + e! x

2. A virus is spreading according to the

function p(t) = 125(2)

t

4 , where p(t) is the number of people infected after t days. a) How many people had the virus initially? b) How many people will be infected after

4 weeks? c) How fast will the virus be spreading

after 4 weeks? d) How long will it take until

22 500 people are infected? 3. In a science laboratory, water is brought to

a boil and then removed from heat. The temperature of the water is modelled by the function T = 80e–0.57t + 20, where T is the temperature, in degrees Celsius, and t is the time, in minutes. a) Determine the rate at which the

temperature of the water is decreasing after 5 min.

b) Determine how long it will take the temperature of the water to reach 21ºC.

4. The value of a Guaranteed Investment

Certificate (GIC) is given by the function V = 500(1.03)t, where V is the value, in dollars, after t years. a) How long will it take the GIC to double in

value? b) Determine the derivative of V with respect

to t. c) How fast is the value of the GIC changing

at the end of 15 years?

5. The amount of a medication in the bloodstream can be modelled with the function C(t) = C0te–0.5t, where C(t) represents the amount of the medication in the bloodstream, in milligrams, and t represents the time, in hours, since the medication was taken. a) Determine the amount of medication

remaining in the bloodstream after 8 h, as a percent.

b) At what rate is the amount of medication decreasing after 1.5 h?

6. In a particular circuit, the current, I in

amperes, can be found using the formula I = 0.6(1 – e–0.1t), where t is the time, in seconds. a) What value does the current approach,

as t increases without bound? b) Find the rate of change of the current

after i) 40 s ii) 60 s

7. In a chemistry lab, there is initially

500 mg of an unknown substance. The table shows the results of measuring the mass of the substance remaining in 1 h intervals.

Time (h) Mass (mg) 0 1390.3 1 1223.5 2 1076.6 3 947.5 4 833.8 5 733.7 6 645.7 7 568.2 8 500.0

a) Determine a suitable exponential equation to model the remaining mass of unknown substance after t hours.

b) Estimate the amount of unknown substance remaining after 12 h.

c) What is the rate of decay of the substance after i) 6 h? ii) 12 h?

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..

(page 2)

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..

(page 3)

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