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

Learn about the natural exponential function, the value of e, transformations of exponential functions, and applications in calculus and finance. Explore graphs, models, and calculations related to exponential growth.

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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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