1. Writing and applying exponential functions
Ch 5- day 3
Writing and applying exponential functions
Leave the class being able to:
1: choose the appropriate form of equation for the given context
2. Write an equation from context
3. Use your equation to answer questions.

2. 5.3 Exponential Functions
Convert the 2nd form to the 1st form.

3. 5.3 Exponential Functions

4. 5.3 Exponential Functions
How long will it take for a savings account of $1000 to grow to $2000 if it earns a 9% annual rate of interest?
72 9 =8 𝑦𝑒𝑎𝑟𝑠

5. 5.3 practice
Class exercises:
P all

6. Section 5.4 Two new forms of the exponential function:
One dealing with compound interest
A 2nd also dealing with compound interest and population models

7. 5.4 The Function e^x Complete the table. Lim 10 2.5937 100 2.7048 1000
2.7169
10,000
2.7181
100,000
2.7183
Lim

8. Leonhard Euler http://www.storyofmathematics.com/18th_euler.html

9. Some fun facts about Leonard Euler
he spent most of his academic life in Russia and Germany, especially in the burgeoning St. Petersburg of Peter the Great and Catherine the Great
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 of one mathematical paper every week 
 he had a photographic memory 

11. 5.4 The Function e^x

12. Compound interest
12% annual interested, compounded twice per year.

13. Interest Period % growth each period Amount Annually 12% 6% Quarterly
Growth factor
during period
Amount
Annually
12%
Semiannually 6%
Quarterly 3%
Monthly 1%
Daily
(365 days)
(12/365)%
k times per year
(12/k)% e

14. Exponential equation form #4
Compound interest, when the interest is compounded LESS THAN daily.
𝐴 𝑡 = 𝐴 𝑜 (1+ 𝑟 𝑛 ) 𝑛𝑡
Where r= the annual rate, as a decimal.
Where n= the number of times per year the interest is compounded.

15. Compound interest example

16. What if we continue to compound more frequently:
The answers level off at …
No matter how much more frequently you compound interest, you will not earn any more money- as you notice the front decimal places are no longer changing.
This value that the calculations approach but don’t exceed is called a limit.
Calculate the expression for:
K=500
K=1000
K=5000
K=10,000
And
K=1,000,000≈ twice per minute
Lim
Hence we have a new equation to model compound interest
When we compound daily or more:
𝑃 𝑡 = 𝑃 𝑜 𝑒 𝑟𝑡

17. Exponential equation form #5
Use this form of the equation for most population problems as well.

18. Effective annual yield
= actual % growth after 1 year
This will be more than the given annual rate, if interest is compounded more than once per year
𝑎𝑚𝑜𝑢𝑛𝑡 𝑎𝑓𝑡𝑒𝑟 1 𝑦𝑒𝑎𝑟−𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 ∗100

19. 5.2 Exponential Functions
Homework: (due Wednesday)
5.4: pg 189
3-17odd, +18