Write a function to represent the amount after t years for each situation.

1. 100 grams of a compound with a half-life of 5000 years

2. 12 bacteria that quadruple themselves every 2 years

3. A new car worth $35,000 that depreciates 15% per year

4. A $75,000 student loan with a 6% annual interest rate

Warm Up

𝑨 (𝒕 )=𝟏𝟎𝟎 (𝟏𝟐 )

𝒕𝟓𝟎𝟎𝟎

𝑨 (𝒕 )=𝟏𝟐 (𝟒)𝒕𝟐

𝑨 (𝒕 )=𝟑𝟓 ,𝟎𝟎𝟎 (𝟏−𝟎 .𝟏𝟓)𝒕=𝟑𝟓𝟎𝟎𝟎(𝟎 .𝟖𝟓)𝒕

𝑨 (𝒕 )=𝟕𝟓 ,𝟎𝟎𝟎 (𝟏+𝟎 .𝟎𝟔)𝒕=𝟕𝟓 ,𝟎𝟎𝟎 (𝟏 .𝟎𝟔)𝒕

a= starting value b= multiplier

Exponential growth and decay- given a rate r is the rate, as a decimal r is positive for growth, negative for decay t is positive for the future, negative for the past or Exponential growth and decay- given an outcome and the time to achieve it b is the outcome, in other words, “what happened”examples: reduced by ½, doubled, multiplied by 4 k is how long it takes to do that

Exponential Equation Reminders

Homework Questions?

5.3 Exponential Functions

Homework: Page 1831-23 odd, 27, 28

5.4 The Function : To define and apply the natural exponential function.Objective

Lim

e 11n

n n

10 2.5937

100 2.7048

1000 2.7169

10,000 2.7181

100,000 2.7183

n 11n

n

2.718..

1As increases, 1

gets closer to .

n

nn

Complete the table.

http://www.storyofmathematics.com/18th_euler.html

Leonhard Euler

Leonhard Euler (1707-1783)

He spent most of his academic life in Russia and Germany.

He had a long life and thirteen children. His collected works comprise nearly 900

books and, in the year 1775, he is said to have produced on average one mathematical paper every week. 

He had a photographic memory. 

Facts About Leonard Euler

5.4 The Function The function is called the natural exponential function.xeThe number 2.718 is extremely important in advanced mathematics.e It appears in unexpected places including statistics and physics.

The equations of each of these graphs contain .xe

Compound Interest

2

4

Suppose that you invest dollars at 12% compounded semiannually.Each half year, your money grows by 6%.

At the end of the year you have (1.06) 1.1236 dollars.

Compounded quarterly (1.03) 1.1255

P

P P

P P

dollarsMonthly, Daily, etc...

Invest $1.00 and see what happens when we compound more often.

Annually 12%

Semiannually 12%2=6%

Quarterly 12%4=3%

Monthly 12%12=1%

Daily(365 days)

(12/365)%

k times per year

(12/k)%

12% Annual Growth

Compounded:% growth each period

Growth factorduring period

Amount

0.121+1

0.121+2

0.121+4

0.121+120.121+365

0.121+k

11.12 1.12

21.06 1.1236

41.03 1.1255

0.121 1.1275365

365

121.01 1.1268

0.121k

k

12% Annual Growth

Compound Interest Equation

= initial investmentr = rate as a decimalk = number of times compounded per yeart = timeIf you invest $500 at a 10% annual interest rate that is compounded monthly, how much money will you have after 6 years?

𝑃 (6 )=500(1+.1012 )

6 ∙ 12

=$ 908.80

0.12

0.12

0.12

/ 0.12

12

0.12 0.12

.1275

1 1

.75% annu

11 1 1

al yield1

k k

n

k k n

e

1.12, 1.1236, 1.1255, 1.1268, 1.1275,...It looks like we are approaching a fixed value.Find it!

1 0.12 k

k

Lim

e 11n

n n

0.12k n

0( ) rtP t P e

The exponential formula when compounding is continuous (occurs all of the time).

= initial investmentr = rate as a decimalt = time

If you invest $500 at a 10% annual interest rate that is compounded continuously, how much money will you have after 6 years?

𝑃 (6 )=500 (𝑒 ).1 ∙6=$911.06

Actual % growth after 1 year This will be more than the given annual

rate, if interest is compounded more than once per year.

Effective Annual Yield

After a year during which interest is compounded quarterly, an investment of $750 is worth $790. What is the effective annual yield?

790−750750 ∗100=5.33 %

5.4 The Function e^xSuppose you invest $1.00 at 6% annual interest.Calculate the amount that you would have after one year if the interest is compounded quarterly.

40.061 1

4$1.06A t

5.4 The Function e^xSuppose you invest $100.00 at 6% annual interest.Calculate the amount that you would have after one year if the interest is compounded quarterly.

40.06100 1

4$106.14A t

Page 189 # 1-11 odd, 13-16 all

Homework