Derivatives of Exponentials

Transcription

1 mat 0 more on derivatives: day 0 Derivatives of Eponentials Recall tat DEFINITION... An eponential function as te form f () =a, were te base is a real number a > 0. Te domain of an eponential function is te set of all real numbers (, ). Notice tat unlike power functions of te form n were te base varies and te eponent is constant, in an eponential function a te base a is fied and it is te eponent varies. We ave mentioned previously tat eponential functions are continuous. Tis means tat! a = a. Take a moment rigt now to answer te following questions. YOU TRY IT.. Fill in te table below for te values of y = a. Ten draw te graps. How would you describe te relationsip between corresponding pairs of graps? How are teir slopes (derivatives) related? answer to you try it.. You sould find tat a and a are reflections of eac oter in te y- ais. Tis means tat te slopes at corresponding points ave te same magnitude but are oppositely signed (one is positive, te oter negative). As calculus students, te first question we sould now ave regarding any function is: Is f () differentiable... and if so, wat is f 0 ()? So our problem is to determine D (a ). Let s use te definition of te derivative Lecture0.te Version: Mitcell-06/09/0.4:0:

2 and see were tis takes us. So let f () =a, were a > 0. Ten f 0 f ( + ) f () a () = = + a a = a a a = () = a. (.7) a Notice tat te epression = a 0 is really just te slope of a at = 0. Tere is no additional algebra tat we can perform to simplify tis it. It s value is not obvious since bot te numerator and denominator are bot going to 0 as goes to 0. Let s eamine a table of values to estimate te it, if it eists. In te last step we ave used tat a is constant because it is tat is going to 0. So we may pull te term a outside te it. YOU TRY IT.4. Let a =. Fill in te rest of te values in tis table and ten estimate te value of answer to you try it.4. You sould find tat YOU TRY IT.5. Remember tat in equation (.7), we found tat D (a )=a, if te it eists. So letting a = and using (approimate) value of tat you just found above, determine te formula for D ( ) answer to you try it.5. You sould find tat D( ) (0.69). Te eercise above sows tat we will know te derivative of a once we are able to determine. Te following problem asks you to estimate tis it for several different values of a. You will need a calculator. Try tis now. YOU TRY IT.6. For various values of a, we can estimate by using by using = In oter words, using a calculator we obtain a good estimate of te it using a Fill in your estimates in te table below. Use at least tree decimal places a answer to you try it.6.!0 a

3 mat 0 more on derivatives: day Stop and Tink: If you look at te answer to You Try It.6, you will see tat te its for a = and a = ave te same magnitude but are oppositely signed. Te result is similar for a = and a =. Tis makes sense and follows from two observations tat we made earlier. Te first observation was made in te answer to You Try It. were we noted tat a and a are reflections of eac oter in te y- ais, wic means tat te slopes at corresponding points on tese curves ave te same magnitude but are oppositely signed. Te second observation tat te it epression is really just te slope of a at = 0. Any point wit = 0 lies on te y-ais and so is reflected to itself under reflection in te y-ais. So te its wen a = and a = represent te slopes of te curves y = and y = at = 0, tat is, at te point (0, ) tat lies on every te grap of every eponential function. Tese slopes sould ave te same magnitude but different signs. So wat about te Derivative of a? Remember tat in equation (.7), we found tat D (a )=a, if te it eists. Using te table of its in You Try It.6, tis means tat we ave D (a )=a!0 D ( ) 0.69( ) D ( ).0986( ) D (.5 ) 0.96( ) D ( ) 0.69( ) D ( ).0986( ) Sould we memorize tese? Impossible! You would never be able to do it, especially if we added some additional functions suc as 4,5, and so on. Life would be simple if we could locate a base a suc tat =. For suc a value of a, we would ave D (a )=a = a = a. Tis would be te easiest derivative formula to remember: Te derivative of te function would be te function we started wit! We would not ave to remember anyting for tis derivative formula. YOU TRY IT.7. OK, try to locate a value of a so tat =. Estimate tis it using = From te table above, sould you start wit a >? Between.5 and? Less tan? Your a!0 Your a answer to you try it.7. Here are te values of a tat I tried. You sould end up wit a Te eact base tat we seek is given a special designation: Lecture0.te Version: Mitcell-06/09/0.4:0:

4 e DEFINITION..4. e is te real number suc tat =. Its approimate value is e Furter, e is a transcendental, irrational number (like p.) From our earlier work, tis means THEOREM..5. Te function e is te natural eponential function. It is differentiable (and ence continuous) wit d d (e )=e. Te number e may seem unnatural at first, but it is actually te most natural coice for a base of an eponential function because its derivative is itself. In oter words, at any point along te curve y = e, te slope of te function is te same as its y-value. We can use our new derivative rule in combination wit our earlier derivative properties. EXAMPLE..6. Determine te derivatives of te following functions. (a) f () =6e + p 5 (b) g(s) = es SOLUTION. In bot cases it is convenient to rewrite te functions before taking te derivatives. (a) Use te sum and constant multiple rules along wit power rule and eponential rule. D 6e + p = D 6e + 5/ = 6e 55 5 /. (b) Similarly, D s e s 7s 7s e = D s es 7 s e = es + s 7. EXAMPLE..7. Find te equation of te tangent line to f () =e e + at =. SOLUTION. We need a point and a slope to determine te equation of te tangent. Since f () =e e + = e +, te point is (, f ()) = (, e + ). Te slope is m = df d = = e e e = e. = So te equation of te tangent line to f () = e e + at = is y y = m( ) or y (e + ) =e( ) or y = e +. Higer-Order Derivatives Wen a function f () is differentiable, we obtain a new function f 0 (). An precisely because f 0 () is also a function, we can ask weter f 0 () itself is differentiable. Tat is, does D ( f 0 ()) eist? If it does, ten te derivative of te derivative of f is called te second derivative of f () and is denoted by f 00 (). In oter words, f 00 () = d d [ f 0 ()] = d apple d d d ( f ()). In a similar way, if te derivative of te second derivative of f () eists, we call it te tird derivative of f () and denote it by f 000 (). We can continue in tis manner as long as te derivatives can be calculated.

5 mat 0 more on derivatives: day 4 EXAMPLE..8. Find te first four derivatives of f () = p + 7. SOLUTION. Use te general power rule and te general sum and and constant multiple rules to determine te derivatives. f 0 () = / f 00 () = + 6 f 000 () = / f 0(4) () = / 4 / Note: To avoid using large numbers of 0 symbols, for n > we use f (n) () to denote te n-t derivative of f, if it eists. EXAMPLE..9. Find te first tree derivatives of f () =6e +. Wat can you say about f (n) () for n >? SOLUTION. Use te general power rule, eponential rule, and te general sum and and constant multiple rules to determine te derivatives. f 0 () =6e + 4 f 00 () =6e + 4 f 000 () =6e If n >, ten f (n) () =6e. We ave several interpretations for te (first) derivative of a function. Te most general interpretation of f 0 () is as te rate of cange in te quantity f per unit cange in. Wen we interpret tis geometrically using te grap of f, ten f 0 () represents te slope of te grap at eac. We will see later in te term tat f 00 () can be interpreted as te concavity or bend of te grap. Because te derivative of f in general represents te rate of cange in te quantity f, wen f () represents te position of an object along a straigt line, ten f 0 () represents te velocity of tat same object. Taking te derivative of f 0 (), we get te rate of cange in te velocity of te object. Having driven a car, you know tat te rate of cange in velocity is acceleration. Tus, for motion problems f 00 () represents acceleration. In tis same contet, te tird derivative of te position function is called te jerk of an object. In pysics, jerk, also known as jolt, surge, or lurc. (I tink te motion sickness you sometimes feel in elevators (or rollercoasters?) is a biological sensation of jerk.) Oter notation If y = f (), ten f 00 () can be indicated in several ways. Te most common are f 00 () = f () () =y 00 = d f d = d d ( f ()) = d y d. Lecture0.te Version: Mitcell-06/09/0.4:0: