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College Algebra Sixth Edition James Stewart  Lothar Redlin  Saleem Watson

College Algebra Sixth Edition James Stewart  Lothar Redlin  Saleem Watson. Exponential and Logarithmic Functions. 4. The Natural Exponential Function. 4.2. Natural Exponential Function. Any positive number can be used as the base for an exponential function.

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College Algebra Sixth Edition James Stewart  Lothar Redlin  Saleem Watson

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  1. College Algebra Sixth Edition James StewartLothar RedlinSaleem Watson

  2. Exponential and Logarithmic Functions 4

  3. The Natural Exponential Function 4.2

  4. Natural Exponential Function • Any positive number can be used as the base for an exponential function. • In this section we study the special base e,which is convenient for applications involving calculus.

  5. The Number e

  6. 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.

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

  8. 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.

  9. The Natural Exponential Function

  10. 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.

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

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

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

  14. E.g. 1—Evaluating the Exponential Function • Evaluate each expression correct to five decimal places. • (a) e3 • (b) 2e–0.53 • (c) e4.8

  15. E.g. 1—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

  16. E.g. 2—Transformations of the Exponential Function • Sketch the graph of each function. • f(x) = e–x • g(x) = 3e0.5x

  17. Example (a) E.g. 2—Transformations • We start with the graph of y =exand reflect in the y-axis to obtain the graph of y =e–x.

  18. Example (b) E.g. 2—Transformations • We calculate several values, plot the resulting points, and then connect the points with a smooth curve.

  19. E.g. 3—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 the function :

  20. E.g. 3—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.

  21. Example (a) E.g. 3—Spread of Virus • We conclude that 8 people initially have the disease.

  22. Example (b) E.g. 3—Spread of Virus • Using a calculator, we evaluate v(1), v(2), and v(5). • Then, we round off to obtain these values.

  23. Example (c) E.g. 3—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.

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

  25. ContinuouslyCompounded Interest

  26. Compound Interest • we saw that the interest paid increases as the number of compounding periods n increases. • Let’s see what happens as n increases indefinitely.

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

  28. 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.”

  29. 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

  30. E.g. 4—Calculating Continuously Compounded Interest • Find the amount after 3 years if $1000 is invested at an interest rate of 12% per year, compounded continuously.

  31. E.g. 4—Calculating 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

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