Happy Friday Eve! Do the following: Warm-up  Do on whiteboard HW #6: Clear your desks and be ready for the quiz!!!!!! Warm-up  Do on whiteboard HW #6: #1-6 (CALCULATOR ) on review study guide ( you need a calculator for these problems… can download a free graphing calculator app) UPDATE: ( next class period) 4.7-4.8 quiz ( one question from each packet)

Office Hours! After school on Thursday . Lunch everyday 1 min

Agenda Warm-Up! Review HW 4.6 Quiz Finish 4.7 Notes

Go Over HW #4 p 337 # 5-21 odd ( skip 15) , 33,41, 43, 47 Go over HW #5

Calendar Check-In

Many financial models use exponential functions. Let’s apply what we have learned to what everyone loves: $$$$$$$$$$$$$$$ !

Open Notebook Open Notes to: 4.7: Financial Models

Financial Models Table- Talk: Discuss the difference between interest compounded continuously and interest compounded monthly. If you have an investment compounded daily OR you have an investment compounded continuously, does it matter which formula that you use?

Example 4 Example 4: You invest $8,000 into an account with interest rate of 4% compounded monthly.   Write the formula that models the value of the investment after t years? What will the value of this investment be after 10 years? How long will it take for the account value to reach $20,000?

Financial Models Interest to Double or Triple Money (a) What interest rate ( compounded continuously) is required for the value of an investment to double in 10 years?  

Financial Models Interest to Double or Triple Money  (b) What interest rate ( compounded annually) is required for the value of an investment to triple in 15 years?  

Example 5- on Packet (a) What interest rate ( compounded continuously) is required for the value of an investment to double in 15 years?    (b) What interest rate ( compounded annually) is required for the value of an investment to triple in 15 years?

Financial Models Effective Rate of Interest “which is the better deal” Interest rate that is equivalent to compounding n times per year or continuously after 1 year Higher the interest the better deal  

Effective Rate of Interest Example: Determine which of the following interest rate for an investment is a better deal. 9% compounded quarterly 8.95% compounded continuously

Just like money can grow continuously, so can other natural phenomena demonstrate uninhibited growth or decay. Some examples are cell division of many living organisms which demonstrate the growth process and radioactive substances that have a specific half-life and demonstrate decay.

Open Notebook Title Notes to: 4.8: Exponential Growth/ Decay

4.8 Exponential Growth and Decay Exponential Growth/ Decay: A model that gives the amount N of radioactive materials present at a time t is given by: N(t) = final amount N0 = initial amount k= growth rate ( k>0, growth k<0, decay) t = time

Examples 1a-d: in Packet Example 1: Find Equations of Uninhibited Growth/Decay A colony of bacteria grows according to the law of uninhibited growth according to the function where N is measured in grams and t is measured in days. A ) Determine the initial amount of bacteria. b) What is the decay rate of the bacteria? c) What is the population after 5 days? d) How long will it take the population to reach 40 grams?

Examples 1e-f: in Packet Example 1: Find Equations of Uninhibited Growth/Decay A colony of bacteria grows according to the law of uninhibited growth according to the function where N is measured in grams and t is measured in days. e ) How long would it take the population to reach 13 its size? ( Just write out the equation. DO NOT solve!) f) If the colony of bacteria contains 60% of its original bacteria, how old is the colony?

Example 2- in Packet Example 2: The temperature of a cup of tea after it was brewed can be modeled by the function T (t) = 100e−0.1t + 68, where t is the number of minutes since the tea was brewed and T(t) is the temperature in degrees Fahrenheit at time t. Find and interpret T (0). Find and interpret T (10). Solve and interpret T (t) = 80. Graph y = T (t) in your calculator. What is the horizontal asymptote?

Example –In notes Doubling/Tripling Time Ex If 5,000 bacteria are present initially and the number of bacteria triples in 8 hours, how many bacteria will there be in 48 hours? How long is it until there are 100,000 bacteria?

Example 3- In Packet Example 3: Find Equations of Uninhibited Growth A colony of bacteria increases according to the law of uninhibited growth. According to the formula on the previous page, if N is the number of bacteria in the culture and t is the time in hours, then . a) If 10,000 bacteria are present initially and the number of bacteria doubles in 5 hours, how many bacteria will there be in 24 hours? b) How long is it until there are 500,000 bacteria?

4.8 Exponential Growth and Decay Half-Life FACT: Living things contain 2 kinds of carbon - carbon 12 and carbon 14. When a person dies, carbon 12 stays constant, but carbon 14 decays. In fact, carbon 14 is said to have a half-life of 5730 years. This change in the amount of carbon 14 makes it possible to calculate when the organism died.

4.8 Exponential Growth and Decay Calculating Half Life: Solve for k ( t= 5730 and N(t) is ½ N0) Solve for t using k value Ex: A bone is found to have 85% of the original amount of carbon 14. If the half-life of carbon 14 is 5730 years, how old is the bone?

Example 3- In Packet A bone is found to have 45% of the original amount of carbon 14. If the half-life of carbon 14 is 5730 years, how old is the bone?