1. The Natural Base, e Objective: Model exponential growth/decay.

5. The Natural Exponential Function

7. Example 1

11. Example 2

13. Try This Find the value of $500 after 4 years invested at an annual interest rate of 9% compounded monthly and continuously.

14. Try This Find the value of $500 after 4 years invested at an annual interest rate of 9% compounded monthly and continuously.

15. Try This Find the value of $500 after 4 years invested at an annual interest rate of 9% compounded monthly and continuously.

17. Example 3

20. Inverse Functions Remember, lnx and e x are inverse functions. This means that:

21. Example 4

23. Try This How long does it take for an investment to triple at an annual interest rate of 7.2% compounded continuously?

24. Try This How long does it take for an investment to triple at an annual interest rate of 7.2% compounded continuously?

25. Try This How long does it take for an investment to triple at an annual interest rate of 7.2% compounded continuously?

26. Radioactive Decay Most of the carbon found in the Earths atmosphere is the isotope cardon-12, but a small amount is the radioactive isotope carbon-14. Plants absorb carbon dioxide from the atmosphere, and animals obtain carbon from the plants they consume. When a plant or animal dies, the amount of carbon-14 it contains decays in such a way that exactly half of its initial amount is present after 5730 years. The function below models the decay of carbon-14.

27. Example 5 Suppose that archaeologists find scrolls and claim that they are 2000 years old. Tests indicate that the scrolls contain 78% of their original carbon-14.

28. Example 5 Suppose that archaeologists find scrolls and claim that they are 2000 years old. Tests indicate that the scrolls contain 78% of their original carbon-14. Could the scrolls be 2000 years old?

29. Example 5 Suppose that archaeologists find scrolls and claim that they are 2000 years old. Tests indicate that the scrolls contain 78% of their original carbon-14. Could the scrolls be 2000 years old?

30. Example 5 Suppose that archaeologists find scrolls and claim that they are 2000 years old. Tests indicate that the scrolls contain 78% of their original carbon-14. Could the scrolls be 2000 years old?

32. Homework Pages 397-399 13-59 odd 75,77,79