1. Exponential & Logarithmic Models MATH 109 - Precalculus S. Rook

2. Overview Section 3.5 in the textbook: – Compound interest – Exponential growth & decay 2

3. Compound Interest

4. 4 We put a principal P (starting amount) into an account with a constant growth rate r (expressed as a %) for t years – The amount gained while in the account is known as interest Each year, interest is added to the account n periods per year – Some common values for n include annually (n = 1), quarterly (n = 4), monthly (n = 12), and daily (n = 365)

5. 5 Compound Interest (Continued) Compound Interest Formula: where A is the amount when principal P has been invested at a rate r for t years with n compounding periods each year The product nt is the total number of compounding periods – e.g. If the amount were to be invested for 5 years with interest compounded biannually, there would be 5 ∙ 2 = 10 occasions where interest would be added to the account

6. 6 Compounding Continuously Compounding Continuously: when the number of compounding periods increases without bound w here A is the amount when investing principal P at a rate for t years and e is Euler’s constant – Only need to know the formula, not how to derive it

7. 7 Compound Interest (Example) Ex 1: $7500 is deposited into an account with a 6.5 % interest rate for 4 years. Find the amount that results when the interest is compounded: a) Monthly b) Three times a year

8. 8 Compound Interest (Example) Ex 2: $10,000 is deposited into an account with an interest rate of 4.05 % that is compounded biannually. Approximately how long would it take for the $10,000 to double?

9. 9 Compound Interest & Continuous Compounding (Example) Ex 3: What interest rate would be required for $1500 to accrue to $2500 in 8 years if the account is compounded: a)continuously b)quarterly

10. Exponential Growth & Decay

11. 11 Exponential Growth & Decay Both exponential growth and decay follow the formula N(t) = N 0 e kt – If k > 0, the formula models exponential growth – If k < 0, the formula models exponential decay Given the points (0, N 0 ) and (t, N t ), we can solve for the constant k – i.e. If we know the initial quantity and the quantity after some time t

12. 12 Half-life & Carbon Dating Half-life: the amount of time it takes for half of a substance to dissolve – Don’t need to memorize half-lives for substances Carbon Dating: the process of using the amount of carbon-14 left in an organism to determine its age Model is P(t) = 0.5 t/5730 where P(t) is the percent of carbon-14 remaining after t years Again, don’t need to memorize

13. 13 Exponential Growth (Example) Ex 4: Suppose a population has 25,000 members in 2000 and 40,000 members in 2005 a) Find the exponential growth equation that models this situation b) Find the population that the equation predicts in 2010 to the nearest thousand

14. 14 Half-life (Example) Ex 5: Polonium has a half-life of 138 days. Use this information in order to: a) Construct the exponential decay equation for the amount of Polonium that remains after t days b) Use the equation to determine the fraction of Polonium that remains after 500 days

15. 15 Carbon Dating (Example) Ex 6: Approximately how old is a bone that contains 70% Carbon-14?

16. Summary After studying these slides, you should be able to: – Solve problems involving: Compound and continuously compounded interest Exponential growth and decay Additional Practice – See the list of suggested problems for 3.5 Next lesson – Radian & Degree Measure (Section 4.1) 16