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.

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

5.3 Exponential Functions

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

5.3 practice Class exercises: P182 1-10 all

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

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

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

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 

5.4 The Function e^x

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

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

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.

Compound interest example

What if we continue to compound more frequently: The answers level off at 1.12749… 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

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

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

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