Compound Interest. Lecture 34 Section 4.1. Robb T. Koether. Hampden-Sydney College. Tue, Mar 28, 2017

Transcription

1 Compound Interest Lecture 34 Section 4.1 Robb T. Koether Hampden-Sydney College Tue, Mar 28, 2017 Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8

2 Reminder Reminder Test #3 is this Friday, March 31. Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8

3 Reminder Reminder Test #3 is this Friday, March 31. It will cover Chapter 3. Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8

4 Reminder Reminder Test #3 is this Friday, March 31. It will cover Chapter 3. Be there. Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8

5 Objectives Objectives Learn about the natural base. Apply exponential functions to compound interest. Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8

6 The Natural Base e The Natural Base e Let f (x) = b x for some base b > 0. What is f (x)? Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8

7 The Natural Base e The Natural Base e If we apply the definition, we get f (x) = Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8

8 The Natural Base e The Natural Base e If we apply the definition, we get f (x) = lim h 0 f (x + h) f (x) h Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8

9 The Natural Base e The Natural Base e If we apply the definition, we get f f (x + h) f (x) (x) = lim b x+h b x = lim Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8

10 The Natural Base e The Natural Base e If we apply the definition, we get f f (x + h) f (x) (x) = lim b x+h b x = lim b x b h b x = lim Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8

11 The Natural Base e The Natural Base e If we apply the definition, we get f f (x + h) f (x) (x) = lim b x+h b x = lim b x b h b x = lim b x (b h 1) = lim Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8

12 The Natural Base e The Natural Base e If we apply the definition, we get f f (x + h) f (x) (x) = lim b x+h b x = lim b x b h b x = lim b x (b h 1) = lim = b x ( lim h 0 b h 1 h ). Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8

13 The Natural Base e The Natural Base e b h 1 Note that lim is a number whose value depends solely on b. Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8

14 The Natural Base e The Natural Base e b h 1 Note that lim is a number whose value depends solely on b. For the right choice of b, that value will be 1. Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8

15 The Natural Base e The Natural Base e b h 1 Note that lim is a number whose value depends solely on b. For the right choice of b, that value will be 1. How do we know that? Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8

16 The Natural Base e The Natural Base e b h 1 Note that lim is a number whose value depends solely on b. For the right choice of b, that value will be 1. How do we know that? That choice is approximately Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8

17 The Natural Base e The Natural Base e b h 1 Note that lim is a number whose value depends solely on b. For the right choice of b, that value will be 1. How do we know that? That choice is approximately We call that number e, the natural base. Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8

18 Compound Interest Compound Interest Let P be the present value of an investment, Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8

19 Compound Interest Compound Interest Let P be the present value of an investment, r be the annual interest rate, Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8

20 Compound Interest Compound Interest Let P be the present value of an investment, r be the annual interest rate, k be the number of compounding periods per year, Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8

21 Compound Interest Compound Interest Let P be the present value of an investment, r be the annual interest rate, k be the number of compounding periods per year, t be the duration (in years) of the investment, and Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8

22 Compound Interest Compound Interest Let P be the present value of an investment, r be the annual interest rate, k be the number of compounding periods per year, t be the duration (in years) of the investment, and B(t) be the future value after t years. Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8

23 Compound Interest Compound Interest Let Then P be the present value of an investment, r be the annual interest rate, k be the number of compounding periods per year, t be the duration (in years) of the investment, and B(t) be the future value after t years. B(t) = P ( 1 + r k ) kt. Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8

24 Continuous Compounding Continuous Compounding If the interest is compounded continuously, then the future value is B(t) = Pe rt. Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, / 8