Section 6.7 – Financial Models

1. Section 6.7 – Financial Models Simple Interest Formula Compound Interest Formula Continuous Compounding Interest Formula

2. Section 6.7 – Financial Models Example - Simple Interest What is the future value of a $34,100 principle invested at 4% for 3 years Examples - Compound Interest The amount of $12,700 is invested at 8.8% compounded semiannually for 1 year. What is the future value? $21,000 is invested at 13.6% compounded quarterly for 4 years. What is the return value?

3. Section 6.7 – Financial Models Examples - Compound Interest How much money will you have if you invest $4000 in a bank for sixty years at an annual interest rate of 9%, compounded monthly? Example - Continuous Compounding Interest If you invest $500 at an annual interest rate of 10% compounded continuously, calculate the final amount you will have in the account after five years.

4. Section 6.7 – Financial Models Effective Interest Rate – the actual annual interest rate that takes into account the effects of compounding. Compounding n times per year: Continuous compounding: Which is better, to receive 9.5% (annual rate) continuously compounded or 10% (annual rate) compounded 4 times per year? Continuous compounding Compounding 4 times per year

5. Section 6.7 – Financial Models Present Value – the initial principal invested at a specific rate and time that will grow to a predetermined value. Compounding n times per year: Continuous compounding: How much money do you have to put in the bank at 12% annual interest for five years (a) compounded 6 times per year and (b) compounded continuously to end up with $2,000? Compounding 6 times per year Continuous compounding P P

6. Section 6.7 – Financial Models Example What rate of interest (a) compounded monthly and (b) continuous compounding is required to triple an investment in five years?

7. Section 6.8 – Exponential Growth/Decay Models; Newton's Law of Cooling and Logistic Growth/Decay Models Uninhibited Exponential Growth Uninhibited Exponential Decay

8. Section 6.8 – Exponential Growth/Decay Models; Newton's Law of Cooling and Logistic Growth/Decay Models Examples The population of the United States was approximately 227 million in 1980 and 282 million in 2000. Estimate the population in the years 2010 and 2020. Find k 2010 2020

9. Section 6.8 – Exponential Growth/Decay Models; Newton's Law of Cooling and Logistic Growth/Decay Models Examples A radioactive material has a half-life of 700 years. If there were ten grams initially, how much would remain after 300 years? When will the material weigh 7.5 grams? Find k 300 years 7.5 grams or or

10. Section 6.8 – Exponential Growth/Decay Models; Newton's Law of Cooling and Logistic Growth/Decay Models Newton’s Law of Cooling

11. Section 6.8 – Exponential Growth/Decay Models; Newton's Law of Cooling and Logistic Growth/Decay Models Newton’s Law of Cooling Example A pizza pan is removed at 3:00 PM from an oven whose temperature is fixed at 450 Finto a room that is a constant 70 F. After 5 minutes, the pizza pan is at 300 F.At what time is the temperature of the pan 135 F? Find k t @ 135 F

12. Section 6.8 – Exponential Growth/Decay Models; Newton's Law of Cooling and Logistic Growth/Decay Models Logistic Growth/Decay

13. Section 6.8 – Exponential Growth/Decay Models; Newton's Law of Cooling and Logistic Growth/Decay Models Logistic Growth/Decay Example The logistic growth model relates the proportion of U.S. households that own a cell phone to the year. Let represent 2000, represent 2001, and so on. (a) What proportion of households owned a cell phone in 2000, (b) what proportion of households owned a cell phone in 2005, and (c) when will 85% of the households own a cell phone? P(t) = 85% 2005 2000