The Natural Exponential Function

1. The Natural Exponential Function

2. Natural Exponential Function • Any positive number can be used as the base for an exponential function. • However, some are used more frequently than others. • We will see in the remaining sections of the chapter that the bases 2 and 10 are convenient for certain applications. • However, the most important is the number denoted by the letter e.

3. Number e • The number e is defined as the value that (1 + 1/n)n approaches as n becomes large. • In calculus, this idea is made more precise through the concept of a limit.

4. Number e • The table shows the values of the expression (1 + 1/n)nfor increasingly large values of n. • It appears that, correct to five decimal places, e ≈ 2.71828

5. Number e • The approximate value to 20 decimal places is: e≈ 2.71828182845904523536 • It can be shown that e is an irrational number. • So, we cannot write its exact value in decimal form.

6. Number e • Why use such a strange base for an exponential function? • It may seem at first that a base such as 10 is easier to work with. • However, we will see that, in certain applications, itis the best possible base.

7. Natural Exponential Function—Definition • The natural exponential functionis the exponential function f(x) = exwith base e. • It is often referred to as theexponential function.

8. Natural Exponential Function • Since 2 < e < 3, the graph of the natural exponential function lies between the graphs of y = 2xand y = 3x.

9. Natural Exponential Function • Scientific calculators have a special key for the function f(x) = ex. • We use this key in the next example.

10. E.g. 6—Evaluating the Exponential Function • Evaluate each expression correct to five decimal places. • (a) e3 • (b) 2e–0.53 • (c) e4.8

11. E.g. 6—Evaluating the Exponential Function • We use the ex key on a calculator to evaluate the exponential function. • e3≈ 20.08554 • 2e–0.53≈ 1.17721 • e4.8≈ 121.51042

12. E.g. 7—Transformations of the Exponential Function • Sketch the graph of each function. • f(x) = e–x • g(x) = 3e0.5x

13. Example (a) E.g. 7—Transformations • We start with the graph of y =exand reflect in the y-axis to obtain the graph of y =e–x.

14. Example (b) E.g. 7—Transformations • We calculate several values, plot the resulting points, and then connect the points with a smooth curve.

15. E.g. 8—An Exponential Model for the Spread of a Virus • An infectious disease begins to spread in a small city of population 10,000. • After t days, the number of persons who have succumbed to the virus is modeled by:

16. E.g. 8—An Exponential Model for the Spread of a Virus • How many infected people are there initially (at time t = 0)? • Find the number of infected people after one day, two days, and five days. • Graph the function v and describe its behavior.

17. Example (a) E.g. 8—Spread of Virus • We conclude that 8 people initially have the disease.

18. Example (b) E.g. 8—Spread of Virus • Using a calculator, we evaluate v(1), v(2), and v(5). • Then, we round off to obtain these values.

19. Example (c) E.g. 8—Spread of Virus • From the graph, we see that the number of infected people: • First, rises slowly. • Then, rises quickly between day 3 and day 8. • Then, levels off when about 2000 people are infected.

20. Logistic Curve • This graph is called a logistic curveor a logistic growth model. • Curves like it occur frequently in the study of population growth.

21. Compound Interest

22. Compound Interest • Exponential functions occur in calculating compound interest. • Suppose an amount of money P, called the principal, is invested at an interest rate i per time period. • Then, after one time period, the interest is Pi, and the amount A of money is:A = P + Pi + P(1 + i)

23. Compound Interest • If the interest is reinvested, the new principal is P(1 + i), and the amount after another time period is: A = P(1 + i)(1 + i) = P(1 + i)2 • Similarly, after a third time period, the amount is: A = P(1 + i)3

24. Compound Interest • In general, after k periods, the amount is: A = P(1 + i)k • Notice that this is an exponential function with base 1 + i.

25. Compound Interest • Now, suppose the annual interest rate is r and interest is compounded n times per year. • Then, in each time period, the interest rate is i =r/n, and there are nt time periods in t years. • This leads to the following formula for the amount after t years.

26. Compound Interest • Compound interestis calculated by the formulawhere: • A(t) = amount after t years • P = principal • t = number of years • n = number of times interest is compounded per year • r = interest rate per year

27. E.g. 9—Calculating Compound Interest • A sum of $1000 is invested at an interest rate of 12% per year. • Find the amounts in the account after 3 years if interest is compounded: • Annually • Semiannually • Quarterly • Monthly • Daily

28. E.g. 9—Calculating Compound Interest • We use the compound interest formula with: P = $1000, r = 0.12, t = 3

29. Compound Interest • We see from Example 9 that the interest paid increases as the number of compounding periods n increases. • Let’s see what happens as n increases indefinitely.

30. Compound Interest • If we let m = n/r, then

31. Compound Interest • Recall that, as m becomes large, the quantity (1 + 1/m)m approaches the number e. • Thus, the amount approaches A =Pert. • This expression gives the amount when the interest is compounded at “every instant.”

32. Continuously Compounded Interest • Continuously compounded interestis calculated by A(t) = Pert • where: • A(t) =amount after t years • P = principal • r = interest rate per year • t = number of years

33. E.g. 10—Continuously Compounded Interest • Find the amount after 3 years if $1000 is invested at an interest rate of 12% per year, compounded continuously.

34. E.g. 10—Continuously Compounded Interest • We use the formula for continuously compounded interest with: P = $1000, r = 0.12, t = 3 • Thus, A(3) = 1000e(0.12)3 = 1000e0.36 = $1433.33 • Compare this amount with the amounts in Example 9.