Exponential Functions: Differentiation and Integration. The Natural Exponential Function

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1 46_54.q //4 :59 PM Page 5 5 CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions Section 5.4 f () = e f() = ln The inverse function of the natural logarithmic function is the natural eponential function. Figure 5.9 Eponential Functions: Differentiation an Integration Develop properties of the natural eponential function. Differentiate natural eponential functions. Integrate natural eponential functions. The Natural Eponential Function The function f ln is increasing on its entire omain, an therefore it has an inverse function f. The omain of f is the set of all reals, an the range is the set of positive reals, as shown in Figure 5.9. So, for an real number, f f ln f. If happens to be rational, then lne ln e. is an real number. is a rational number. Because the natural logarithmic function is one-to-one, ou can conclue that f an e agree for rational values of. The following efinition etens the meaning of e to inclue all real values of. Definition of the Natural Eponential Function The inverse function of the natural logarithmic function f ln is calle the natural eponential function an is enote b That is, f e. e if an onl if ln. THE NUMBER e The smbol e was first use b mathematician Leonhar Euler to represent the base of natural logarithms in a letter to another mathematician, Christian Golbach, in 7. The inverse relationship between the natural logarithmic function an the natural eponential function can be summarize as follows. lne EXAMPLE an e ln Solving Eponential Equations Inverse relationship Solve 7 e. Solution You can convert from eponential form to logarithmic form b taking the natural logarithim of each sie of the equation. 7 e Write original equation. ln 7 lne Take natural logarithm of each sie. ln 7 Appl inverse propert. ln 7 Solve for..946 Use a calculator. Check this solution in the original equation.

2 46_54.q //4 :59 PM Page 5 SECTION 5.4 Eponential Functions: Differentiation an Integration 5 EXAMPLE Solving a Logarithmic Equation Solve ln 5. Solution To convert from logarithmic form to eponential form, ou can eponentiate each sie of the logarithmic equation. ln 5 Write original equation. e ln e 5 Eponentiate each sie. e 5 Appl inverse propert. e 5 Solve for Use a calculator. The familiar rules for operating with rational eponents can be etene to the natural eponential function, as shown in the following theorem. THEOREM 5. Operations with Eponential Functions Let a an b be an real numbers.. e a e b e ab e a. e b eab (, e ) (, e ) (, e) = e (, ) Proof To prove Propert, ou can write lne a e b lne a lne b a b lne ab. Because the natural logarithmic function is one-to-one, ou can conclue that e a e b e ab. The proof of the secon propert is left to ou (see Eercise 9). The natural eponential function is increasing, an its graph is concave upwar. Figure 5. In Section 5., ou learne that an inverse function f shares man properties with f. So, the natural eponential function inherits the following properties from the natural logarithmic function (see Figure 5.). Properties of the Natural Eponential Function. The omain of f e is,, an the range is,.. The function f e is continuous, increasing, an one-to-one on its entire omain.. The graph of f e is concave upwar on its entire omain. 4. lim an lim e e

3 46_54.q //4 :59 PM Page 5 5 CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions Derivatives of Eponential Functions One of the most intriguing (an useful) characteristics of the natural eponential function is that it is its own erivative. In other wors, it is a solution to the ifferential equation This result is state in the net theorem.. FOR FURTHER INFORMATION To fin out about erivatives of eponential functions of orer /, see the article A Chil s Garen of Fractional Derivatives b Marcia Kleinz an Thomas J. Osler in The College Mathematics Journal. To view this article, go to the website THEOREM 5. Derivative of the Natural Eponential Function Let u be a ifferentiable function of... e e eu e u u Proof To prove Propert, use the fact that ln e, an ifferentiate each sie of the equation. ln e ln e e e e e The erivative of follows from the Chain Rule. e u Definition of eponential function Differentiate each sie with respect to. NOTE You can interpret this theorem geometricall b saing that the slope of the graph of f e at an point, e is equal to the -coorinate of the point. EXAMPLE Differentiating Eponential Functions a. b. e e u u e e e u u e e u u f() = e (, e ) Relative minimum The erivative of f changes from negative to positive at. Figure 5. EXAMPLE 4 Locating Relative Etrema Fin the relative etrema of f e. Solution The erivative of f is given b f e e Prouct Rule e. Because e is never, the erivative is onl when. Moreover, b the First Derivative Test, ou can etermine that this correspons to a relative minimum, as shown in Figure 5.. Because the erivative f e is efine for all, there are no other critical points.

4 46_54.q //4 :59 PM Page 5 SECTION 5.4 Eponential Functions: Differentiation an Integration 5 EXAMPLE 5 The Stanar Normal Probabilit Densit Function Show that the stanar normal probabilit ensit function f e has points of inflection when ±. Two points of inflection... f() = e / π The bell-shape curve given b a stanar normal probabilit ensit function Figure 5. Solution To locate possible points of inflection, fin the -values for which the secon erivative is. f f f e e e e e Write original function. First erivative Prouct Rule Secon erivative So, f when ±, an ou can appl the techniques of Chapter to conclue that these values iel the two points of inflection shown in Figure 5.. NOTE The general form of a normal probabilit ensit function (whose mean is ) is given b f e where is the stanar eviation ( is the lowercase Greek letter sigma). This bell-shape curve has points of inflection when ±. EXAMPLE 6 Shares Trae Shares trae (in millions) 4, 5,, 5,, 5,, 5, = 6,66e.9t t = Year ( 99) t The number of shares trae (in millions) on the New York Stock Echange from 99 through can be moele b 6,66e.9t where t represents the ear, with t corresponing to 99. At what rate was the number of shares trae changing in 998? (Source: New York Stock Echange, Inc.) Solution The erivative of the given moel is.96,66e.9t 697e.9t. B evaluating the erivative when t 8, ou can conclue that the rate of change in 998 was about,9 million shares per ear. Figure 5. The graph of this moel is shown in Figure 5..

5 46_54.q //4 :59 PM Page CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions Integrals of Eponential Functions Each ifferentiation formula in Theorem 5. has a corresponing integration formula. THEOREM 5. Integration Rules for Eponential Functions Let u be a ifferentiable function of.. e e. C e u u e u C EXAMPLE 7 Integrating Eponential Functions Fin e. Solution If ou let u, then u. e e eu u Multipl an ivie b. Substitute: u. eu C Appl Eponential Rule. e C Back-substitute. NOTE In Eample 7, the missing constant factor was introuce to create u. However, remember that ou cannot introuce a missing variable factor in the integran. For instance, e e. EXAMPLE 8 Integrating Eponential Functions Fin 5e. Solution If ou let u, then u or u. 5e 5e 5e u u Regroup integran. Substitute: u. 5 e u u Constant Multiple Rule 5 eu C Appl Eponential Rule. 5 e C Back-substitute.

6 46_54.q //4 :59 PM Page 55 SECTION 5.4 Eponential Functions: Differentiation an Integration 55 EXAMPLE 9 Integrating Eponential Functions a. e e e C u e u u b. e sin ecos u u ecos sin e cos C u cos EXAMPLE Fining Areas Boune b Eponential Functions Evaluate each efinite integral. a. e b. c. e e e cose Solution a. e e See Figure 5.4(a). e e.6 e b. See Figure 5.4(b). e ln e ln e ln.6 c. e cose sine See Figure 5.4(c). sin sine.48 = e e = + e = e cos(e ) (a) (b) (c) Figure 5.4

7 46_54.q //4 :59 PM Page CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions Eercises for Section 5.4 In Eercises 4, solve for accurate to three ecimal places.. e ln 4. e ln. e 4. 4e e e e 8. e ln. ln. ln. ln 4. ln 4. ln In Eercises 5 8, sketch the graph of the function. 9. Use a graphing utilit to graph f e an the given function in the same viewing winow. How are the two graphs relate? (a) g e (b) h e (c) q e. Use a graphing utilit to graph the function. Use the graph to etermine an asmptotes of the function. (a) f 8 e.5 8 (b) g e.5 e 5. e e 8. e In Eercises 4, match the equation with the correct graph. Assume that a an C are positive real numbers. [The graphs are labele (a), (b), (c), an ().] See for worke-out solutions to o-numbere eercises. In Eercises 5 8, illustrate that the functions are inverses of each other b graphing both functions on the same set of coorinate aes. 5. f e 6. f e 9. Graphical Analsis Use a graphing utilit to graph an in the same viewing winow. What is the relationship between f an g as?. Conjecture Use the result of Eercise 9 to make a conjecture about the value of as. In Eercises an, compare the given number with the number e. Is the number less than or greater than e?. (See Eercise.),,,,. g ln 7. f e 8. f e g ln f.5 r g ln g ln g e (a) (b) In Eercises an 4, fin an equation of the tangent line to the graph of the function at the point,.. (a) e (b) e (c) () (, ) (, ) 4. (a) e (b) e. Ce a.. C e a 4. Ce a C e a (, ) (, )

8 46_54.q //4 :59 PM Page 57 SECTION 5.4 Eponential Functions: Differentiation an Integration 57 In Eercises 5 48, fin the erivative. 5. f e 6. e 7. e 8. e 9. gt e t e t 4. gt e t 4. ln e e e 45. e 46. ln 47. F sin cos cos e t t 48. In Eercises 49 56, fin an equation of the tangent line to the graph of the function at the given point. 5. lne,, 4 5. In Eercises 57 an 58, use implicit ifferentiation to fin /. 57. e 58. e In Eercises 59 an 6, fin an equation of the tangent line to the graph of the function at the given point. 59. e e,, 6. ln e,, In Eercises 6 an 6, fin the secon erivative of the function. 6. f e 6. g e ln In Eercises 6 an 64, show that the function solution of the ifferential equation e cos sin e cos 4 sin 5 f is a In Eercises 65 7, fin the etrema an the points of inflection (if an eist) of the function. Use a graphing utilit to graph the function an confirm our results f e e f e e 67. g e 68. g ln e e e e ln e e F lnt t 49. f e,, 5. e, ln e e, 5. e e e,, e e e, f e ln,,, 56. f e ln,,,, e 69. f e 7. f e 7. gt te t 7. f e 4 7. Area Fin the area of the largest rectangle that can be inscribe uner the curve e in the first an secon quarants. 74. Area Perform the following steps to fin the maimum area of the rectangle shown in the figure. (a) Solve for c in the equation f c f c. (b) Use the result in part (a) to write the area A as a function of. Hint: A fc (c) Use a graphing utilit to graph the area function. Use the graph to approimate the imensions of the rectangle of maimum area. Determine the maimum area. () Use a graphing utilit to graph the epression for c foun in part (a). Use the graph to approimate an Use this result to escribe the changes in imensions an position of the rectangle for < <. 75. Verif that the function lim c 4 c L ae b, f() = e c + lim c. a >, b >, L > increases at a maimum rate when L Writing Consier the function f e. (a) Use a graphing utilit to graph f. (b) Write a short paragraph eplaining wh the graph has a horizontal asmptote at an wh the function has a nonremovable iscontinuit at. 77. Fin a point on the graph of the function f e such that the tangent line to the graph at that point passes through the origin. Use a graphing utilit to graph f an the tangent line in the same viewing winow. 78. Fin the point on the graph of e where the normal line to the curve passes through the origin. (Use Newton s Metho or the zero or root feature of a graphing utilit.) 79. Depreciation The value V of an item t ears after it is purchase is V 5,e.686t, t. (a) Use a graphing utilit to graph the function. (b) Fin the rate of change of V with respect to t when t an t 5. (c) Use a graphing utilit to graph the tangent line to the function when t an t 5.

9 46_54.q //4 :59 PM Page CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions 8. Harmonic Motion The isplacement from equilibrium of a mass oscillating on the en of a spring suspene from a ceiling is.56e.t cos 4.9t where is the isplacement in feet an t is the time in secons. Use a graphing utilit to graph the isplacement function on the interval,. Fin a value of t past which the isplacement is less than inches from equilibrium. 8. Moeling Data A meteorologist measures the atmospheric pressure P (in kilograms per square meter) at altitue h (in kilometers). The ata are shown below. h P (a) Use a graphing utilit to plot the points h, ln P. Use the regression capabilities of the graphing utilit to fin a linear moel for the revise ata points. (b) The line in part (a) has the form Write the equation in eponential form. (c) Use a graphing utilit to plot the original ata an graph the eponential moel in part (b). () Fin the rate of change of the pressure when h 5 an h Moeling Data The table lists the approimate value V of a mi-size sean for the ears 997 through. The variable t represents the time in ears, with t 7 corresponing to 997. t V t V 5 5, ln P ah b $7,4 $4,59 $,845 $,995 $9 $895 $685 (a) Use a computer algebra sstem to fin linear an quaratic moels for the ata. Plot the ata an graph the moels. (b) What oes the slope represent in the linear moel in part (a)? (c) Use a computer algebra sstem to fit an eponential moel to the ata. () Determine the horizontal asmptote of the eponential moel foun in part (c). Interpret its meaning in the contet of the problem. (e) Fin the rate of ecrease in the value of the sean when t 8 an t using the eponential moel. Linear an Quaratic Approimations In Eercises 8 an 84, use a graphing utilit to graph the function. Then graph P f f an P f f f in the same viewing winow. Compare the values of f, P, an P an their first erivatives at. 8. f e 84. f e In Eercises 85 98, fin the inefinite integral e 4 e e e e e e e e e e e e e e e e e e e 5 e e e e e e e 97. e 98. tane lne In Eercises 99 6, evaluate the efinite integral. Use a graphing utilit to verif our result. 99. e.. e. e e sin cos 6. 4 e Differential Equations In Eercises 7 an 8, solve the ifferential equation e e ea Differential Equations In Eercises 9 an, fin the particular solution that satisfies the initial conitions. 9. f e e,. f sin e, f, f f 4, f e e e sec sec tan

10 46_54.q //4 4: PM Page 59 SECTION 5.4 Eponential Functions: Differentiation an Integration 59 Slope Fiels In Eercises an, a ifferential equation, a point, an a slope fiel are given. (a) Sketch two approimate solutions of the ifferential equation on the slope fiel, one of which passes through the given point. (b) Use integration to fin the particular solution of the ifferential equation an use a graphing utilit to graph the solution. Compare the result with the sketches in part (a). To print an enlarge cop of the graph, go to the website e, e. Area In Eercises 6, fin the area of the region boune b the graphs of the equations. Use a graphing utilit to graph the region an verif our result.. e,,, 5 4. e,, a, b 5. e 4,,, 6 6. e,,, Numerical Integration In Eercises 7 an 8, approimate the integral using the Mipoint Rule, the Trapezoial Rule, an Simpson s Rule with n. Use a graphing utilit to verif our results e 8. 5 e 9. Probabilit A car batter has an average lifetime of 48 months with a stanar eviation of 6 months. The batter lives are normall istribute. The probabilit that a given batter will last between 48 months an 6 months is e.9t48 t. 48 Use the integration capabilities of a graphing utilit to approimate the integral. Interpret the resulting probabilit.. Probabilit The meian waiting time (in minutes) for people waiting for service in a convenience store is given b the solution of the equation.e.t t. Solve the equation Given e for, it follows that e t t t. Perform this integration to erive the inequalit e for.. Moeling Data A valve on a storage tank is opene for 4 hours to release a chemical in a manufacturing process. The flow rate R (in liters per hour) at time t (in hours) is given in the table. t R (a) Use the regression capabilities of a graphing utilit to fin a linear moel for the points t, ln R. Write the resulting equation of the form ln R at b in eponential form. (b) Use a graphing utilit to plot the ata an graph the eponential moel. (c) Use the efinite integral to approimate the number of liters of chemical release uring the 4 hours. 7. Fin, to three ecimal places, the value of such that e. (Use Newton s Metho or the zero or root feature of a graphing utilit.) 8. Fin the value of a such that the area boune b e, the 8 -ais, a, an a is 9. Prove that Writing About Concepts. In our own wors, state the properties of the natural eponential function. 4. Describe the relationship between the graphs of f ln an g e. 5. Is there a function f such that f f? If so, ientif it. 6. Without integrating, state the integration formula ou can use to integrate each of the following. e (a) (b) e e e a e b eab.. Let f ln. (a) Graph f on, an show that f is strictl ecreasing on e,. (b) Show that if e A < B, then A B > B A. (c) Use part (b) to show that e >. e.