1

Sections 4.1 Exponential Functions

Find the following:

91

2 −81 −3

4 𝑑−1

5 ∙ 𝑑 1

3 ( 𝑥1

3𝑥2)

4

3 (1252)−

2

3

Solve for x:

Ex/ 164 x Ex/ 24 x Ex/ 933 x Ex/ 4

14 x Ex/

27

19 x

Sketch a quick graph using the asymptote and your calculator. 12)( xxf 1

( ) 3 2x

f x

HA: y= HA: y=

2

Present value Future value 1

P B 1

nt

nt

r

n

r

n

Future value=Present value•

B=P•

rt

rt

e

e

-

-

Present value=Future value•

P=B•

rt

rt

e

e

Future value=Present value 1

B=P 1

nt

nt

r

n

r

n

Compounded Continuously

Present value=P= Amount invested in the account initially, Principle.

Future value=B= Future amount in the account after t years.

r= Annual interest rate

n= Number of times the account is compounded, interest given, per year.

t= Number of years.

1. How much will an account with $5000 compounded monthly with a rate of 5% have after 5 years?

2. What is the present value of an account with a future value of $4000 compounded quarterly with a rate of

8.5% after 3 years?

3. How much will an account with $40 compounded continuously with a rate of 28% have after 30 years?

4. What is the present value of an account with a future value of $4000 compounded continuously with a rate

of 8.5% have after 3 years?

3

ANNUAL PERCENTAGE YIELD

𝐴𝑃𝑌 = (1 +𝑟

𝑛)

𝑛− 1 for compounded n times a year

𝐴𝑃𝑌 = 𝑒𝑟 − 1 for compounded continuously

1. You have a choice of three accounts. One account is compounded continuously at 12%, one account is

compounded monthly at 12.05%, and the last account is a simple interest rate of 15%. Which account has the

highest APY?

2. A certain bank offers an interest rate of 6% per year compounded annually. A competing bank compounds

its interest continuously. What interest rate should the competing bank off so that the effective interest rates

of the two banks will be equal?

Other exponential functions:

A manufacturer estimates that when x units of a particular commodity are produced, the market price p

(dollars per unit) is given by the demand function

𝑝 = 7 + 50𝑒−𝑥

200⁄

a) What market price corresponds to the production of x=0 units.

b) How much revenue is obtained when 200 units are produced?

4

Logartithms

Formulas

1. xyb log yb x

2. yxxy bbb logloglog

3. yxy

xbbb logloglog

4. crc b

r

b loglog

5. xb x

b log , 1log bb ,

xe x ln

6. xbxb

log

7. 01log b

8. errororb 0log , b>0

9. b

a

b

aab

ln

ln

log

loglog

Equation type 1

31log2 x

Use formula 1

123 x

Solve for x

x

x

7

18

x

x

x

x

x

1002

1002

21000

210

32log

3

48

480

3240

432

42

54log

54loglog

2

2

5

2

22

xorx

xx

xx

xx

xx

xx

xx

-8 gives an error when

checked, so 4 is the only

answer.

X=4

Equation type 2

xx log2log1log

Combine log’s

xx log12log

This means the insides are =

x

x

xx

xx

2

2

22

12

-2 gives an error message when

checked, so

No Solution

x

x

xx

xx

xx

x

x

x

x

xx

4

28

335

131

51

31

5

3log1

5log

3log1log5log

22

222

Equation type 3

1153 x

Need to bring the variable down using

formula 4

4966.

5log3

11log

5log3

11log

5log3

5log3

11log5log3

11log5log 3

x

x

x

x

x

m

m

mm

mm

mm

mm

mm

2571.13

4054651.375278.5

3862944.1375278.57917595.1

3862944.17917595.1)3(

4ln6ln)3(

4ln6ln

46

3.

3

q

30lnln

30

006.

006.

x

x

e

e

Using formula 5

866.566

006.

30ln

006.

006.

30ln006.

x

x

x

Section 4.2

xyb log yb x

Find without using a calculator:

125log5 9log3 27log3 25

1log5

log 2= ln 2=

Find with a calculator:

log 25 = ln 3 = log (2

5+ 3.22) =

log 0 = ln −2 = log −3 =

5

log3 5log 2logx x 5 5 5

2log log 1 2log 1

3x x x 3ln lnx x

2

5 3log

1

x

x

log3

x 2

ln3

xxe

Find the domain of :

Change of Base.

b

a

b

aab

ln

ln

log

loglog

log4 25 = log710 = log3 (2

5+ 3.22) =

yxxy bbb logloglog yxy

xbbb logloglog crc b

r

b loglog

Write as the sum and difference of logarithms

Write as a single logarithm

6

5log 1 2x

Solving equations.

log and ln.

a) b) 364log x c) 2ln x b d) )5ln3(ln2ln x

Try: 2log4 9 x

With exponentials.

5

2 3 1 7x 3 5xe

Try:

a) 218 1.5 x b) x335647.234 c)

24 xe d) xe 358

7

Applications.

1. How long will it take a bank 2. How long will it take a bank 3. An exponential decay

account with $5000 compounded account with $4000 compounded is given by the formula

monthly at 5% to reach 410,000? continuously at 8% take to double? teA 24 . Find the half

life.

4. Carbon 14 dating. A monkey from a refuse deposit near the Strait of Magellan had 40 grams of the

carbon14 when it died. If the formula for amount for carbon 14 left after t years is given by

tetA 0001216.040)( ,

a) What is the initial amount of carbon 14? b) What was the amount of Carbon 14 left after 1000

years?

c) If ¼ of the amount of carbon 14 was found in the Monkey, then how old is the monkey?

5. The basic formula for a decay model for carbon 14 is teAtA 000124.0

0)( where t is in years.

If a random object has 65 % of its carbon-14 left in 2004, then find when the object died or made from dead

stuff.

8

Sections 4.3 #1-53 eoo.65, 69, 73, 75

Derivative of Exponential Functions

𝑑

𝑑𝑥(𝑒𝑥) = 𝑒𝑥

𝑎) 𝑦 = 𝑒𝑥 b) 𝑦 = 2𝑒𝑥 c) 𝑦 = −3𝑒𝑥

Find the derivative of (chain rule)

𝑑

𝑑𝑥(𝑒𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡) = 𝑒𝑥 ∙

𝑑

𝑑𝑥(𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡)

𝑎) 𝑦 = 𝑒2𝑥 b) 𝑦 = 5𝑒4𝑥2 c) 𝑦 = −3𝑒−7𝑥 − 2𝑒5𝑥

Derivative for non-natural exponential

𝑑

𝑑𝑥(𝑏𝑎𝑠𝑒𝑥) = 𝑏𝑎𝑠𝑒𝑥 ∙ ln (𝑏𝑎𝑠𝑒)

𝑎) 𝑦 = 2𝑥 b) 𝑦 = 2(4)𝑥 c) 𝑦 = −3(5)𝑥

Try: Find the derivative basic natural exponential

a) 325)( xexf b)

27

7

1)( xexf

9

Using the product rule. Using the power rule.

xxexf

1

)( 21

2)(

xexf

Derivative of Logarithmic Functions

𝑑

𝑑𝑥(𝑙𝑛𝑥) =

1

𝑥

𝑎) 𝑦 = ln 𝑥 b) 𝑦 = 2 ln 𝑥 c) 𝑦 = −3 ln 𝑥

Chain rule

𝑑

𝑑𝑥(ln (𝑖𝑛𝑠𝑖𝑑𝑒)) =

1

𝑖𝑛𝑠𝑖𝑑𝑒∙

𝑑

𝑑𝑥(𝑖𝑛𝑠𝑖𝑑𝑒)

𝑎) 𝑦 = ln 2𝑥 b) 𝑦 = −4 ln 3𝑥3 c) 𝑦 = 5 ln𝑥

7

Derivative for non-natural logarithm

𝑑

𝑑𝑥(𝑙𝑜𝑔𝑏𝑥) =

1

𝑥𝑙𝑛𝑏

𝑑

𝑑𝑥(log 𝑏(𝑖𝑛𝑠𝑖𝑑𝑒)) =

1

𝑖𝑛𝑠𝑖𝑑𝑒∙𝑙𝑛𝑏∙

𝑑

𝑑𝑥(𝑖𝑛𝑠𝑖𝑑𝑒)

𝑎) 𝑦 = log3 𝑥 b) 𝑦 = 4log 3𝑥 c) 𝑦 = 15 log5(3𝑥2)

10

Try: Find the derivative of

𝑎) 𝑦 = ln 𝑥2 b) 𝑦 = −4 ln(−5𝑥3) c) 𝑦 = (5𝑥 + 7 log9 𝑥)3

Rewriting the natural logarithm.

d) 323ln xy d)

12ln

2

x

xy

Group work:

Form 8 groups and I will assign one of the following problems for your group to perform on the board.

1) 𝑓(𝑥) = ln (𝑥+1

𝑥−1) 2) 𝑓(𝑥) = 𝑒𝑥𝑙𝑛𝑥

11

3) 𝑓(𝑥) =𝑒𝑥+𝑒−𝑥

2 4) 𝑓(𝑥) =

𝑒𝑥+𝑒−𝑥

𝑒𝑥−𝑒−𝑥

5) y= ln(𝑥2 − 1)3 6) 𝑓(𝑥) = ln(𝑢 + √𝑢2 + 1)

7) Determine the equation of the tangent line to the function 2x

xf x

e

at the point

2

11, .

e

8) Find dy

dximplicitly: 𝑥3𝑒−𝑥 + 2𝑦 = 𝑦2

12

Price elasticity of demand, E(p) is:

𝑬(𝒑) = −𝒑

𝒒∙ 𝒒′, where p is the demand price and q is the demanded quantity.

𝑬(𝒑) = −% 𝒓𝒂𝒕𝒆 𝒐𝒇 𝒄𝒉𝒂𝒏𝒈𝒆 𝒐𝒇 𝒒𝒖𝒂𝒏𝒕𝒊𝒕𝒚, 𝒒

% 𝒓𝒂𝒕𝒆 𝒐𝒇 𝒄𝒉𝒂𝒏𝒈𝒆 𝒐𝒇 𝒑𝒓𝒊𝒄𝒆, 𝒑

𝑬

𝟏=

𝑬% 𝒅𝒆𝒄𝒓𝒆𝒂𝒔𝒆 𝒊𝒏 𝒒𝒖𝒂𝒏𝒕𝒊𝒕𝒚 𝒒

𝟏% 𝒊𝒏𝒄𝒓𝒆𝒂𝒔𝒆 𝒊𝒏 𝒑𝒓𝒊𝒄𝒆 𝒑

Elasticity 1% increase

in price Implies

Following A

Price

Increase

Following A

Price

Decrease

Inelastic E<1 Demand

decreases less than 1%

R' is + Revenue increases. Revenue decreases.

Unit elasticity E=1

Demand decreases 1%

R'=0 Revenue

unchanged.

Revenue

unchanged.

Elastic E>1 Demand

decreases more than 1%

R' is - Revenue decreases. Revenue increases

1) The demand function, 𝑞 = 𝐷(𝑝) = 10000𝑒−0.025𝑝, for a particular commodity where p is the price per unit

and q is the number of units sold at that price.

a) Find the elasticity of demand and determine the values of p for which the demand is elastic, inelastic, and of

unit elasticity.

b) If the price is increased by 2% from $15, what is the approximate effect on demand?

c) Find the revenue R(p) obtained by selling q units at the unit price p. For what value of p is revenue

maximized?

13

2) The function 𝐶(𝑡) = 456 + 1234𝑡𝑒−0.137𝑡 as a model for the number of cases of AIDS reported t years after

the base year of 1990.

a) What was the number of cases in 1990.

b) In what year will the largest number of cases be reported?

3) Find the largest and smallest values of the given function over the prescribed closed, bounded interval.

𝑓(𝑥) =ln (𝑥 + 1)

𝑥 + 1 𝑓𝑜𝑟 0 ≤ 𝑥 ≤ 2

14

Section 4.4 Additional Applications

Exponential functions – the basic ones.

1) Once the initial publicity surrounding the release of a new book is over, sales of the hardcover edition tend

to decrease exponentially. At the time publicity was discontinued, 2009, a certain book was experiencing sales

of 25,000 copies per year. One year later, 2010, sales of the book had dropped to 10,000 copies per year. What

will the sales be after two more years? Will the number of sales continue to decrease? Graph the data.

(t, A), need to find the constants P and k.

( , ) ( , )

𝐴 = 𝑃𝑒𝑘𝑡 𝐴 = 𝑃𝑒𝑘𝑡

2) The population of India increased exponentially from 650 million in 1987 to 790 in 1993.

a) Find the exponential model that fits the data and graph the model. Use ktA Pe

b) What is the population of India in 2005? Is the population increasing or decreasing?

c) If the population continues to grow at that rate, in what year will the population reach 1.2 billion?

15

Logistic curves and other exponential curves

1) It is estimated that t years from now, the population of a certain country will be 𝑃(𝑡) =20

2+3𝑒−0.06𝑡

a) Graph the function.

b) What is the current population?

c) What will happen to the population in the long run?

2) When a certain industrial machine has become t years old, its resale value will be

𝑉(𝑡) = 48000𝑒−𝑡

5⁄ + 400 dollars.

a) Graph the function.

b) What is the value of the machine when it is new?

c) What will happen to the value in the long run?

16

Try:

1) Public health records indicate that t weeks after the outbreak of a certain strain of influenza, approximately

𝒇(𝒕) =𝟐

𝟏+𝟑𝒆−𝟎.𝟖𝒕 thousand people had contracted the disease. Sketch a graph. If the trend continues,

approximately how many people in all will contract the disease?

2) A small business assumes that the demand function for one of its new products can be modeled by ,kxp Ce

When $45,p 1000x units, and when $40,p 1200x units.

(a) Find the exponential model that fits the data and graph the model. Use kxp Ce

(b) Find the value of x and p that will maximize the revenue for this product.