Hyperbolic Functions

Transcription

1 88 CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions Section 58 JOHANN HEINRICH LAMBERT (78 777) The first person to publish a comprehensive stu on hperbolic functions was Johann Heinrich Lambert, a Swiss-German mathematician an colleague of Euler MathBio Hperbolic Functions Develop properties of hperbolic functions Differentiate an integrate hperbolic functions Develop properties of inverse hperbolic functions Differentiate an integrate functions involving inverse hperbolic functions Hperbolic Functions In this section ou will look briefl at a special class of eponential functions calle hperbolic functions The name hperbolic function arose from comparison of the area of a semicircular region, as shown in Figure 55, with the area of a region uner a hperbola, as shown in Figure 56 The integral for the semicircular region involves an inverse trigonometric (circular) function: arcsin 57 The integral for the hperbolic region involves an inverse hperbolic function: sinh 96 This is onl one of man was in which the hperbolic functions are similar to the trigonometric functions = + = Circle: Figure 55 Hperbola: Figure 56 FOR FURTHER INFORMATION For more information on the evelopment of hperbolic functions, see the article An Introuction to Hperbolic Functions in Elementar Calculus b Jerome Rosenthal in Mathematics Teacher MathArticle Definitions of the Hperbolic Functions sinh e e cosh e e sinh tanh cosh csch sinh, sech cosh coth tanh, 0 0 Vieo NOTE sinh is rea as the hperbolic sine of, cosh as the hperbolic cosine of, an so on

2 SECTION 58 Hperbolic Functions 89 The graphs of the si hperbolic functions an their omains an ranges are shown in Figure 57 Note that the graph of sinh can be obtaine b aition of orinates using the eponential functions f an g e e View the animations to see this f sinh e e f cosh e e Animation Animation Likewise, the graph of cosh can be obtaine b aition of orinates using the eponential functions f an h e e = cosh f() = e = sinh h() = e f() = e = tanh g() = e Domain:, Range:, Domain:, Range:, Domain:, Range:, = csch = sinh = sech = cosh = coth = tanh Domain:, 0 0, Range:, 0 0, Figure 57 Domain:, Range: 0, Domain:, 0 0, Range:,, Man of the trigonometric ientities have corresponing hperbolic ientities For instance, an cosh sinh e e e e e e sinh cosh e e e e e e e e

3 90 CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions Hperbolic Ientities cosh sinh tanh sech coth csch sinh cosh sinh sinh cosh sinh sinh cosh cosh sinh sinh sinh cosh cosh sinh cosh cosh cosh sinh sinh cosh cosh cosh sinh sinh cosh cosh cosh cosh sinh Differentiation an Integration of Hperbolic Functions Because the hperbolic functions are written in terms of e an e, ou can easil erive rules for their erivatives The following theorem lists these erivatives with the corresponing integration rules THEOREM 58 Derivatives an Integrals of Hperbolic Functions Let u be a ifferentiable function of sinh u cosh uu cosh u sinh uu tanh u sech uu coth u csch uu sech u sech u tanh uu csch u csch u coth uu cosh u u sinh u C sinh u u cosh u C sech u u tanh u C csch u u coth u C sech u tanh u u sech u C csch u coth u u csch u C Proof sinh e e e e cosh tanh sinh cosh cosh cosh sinh sinh cosh cosh sech In Eercises 98 an 0, ou are aske to prove some of the other ifferentiation rules

4 SECTION 58 Hperbolic Functions 9 EXAMPLE Differentiation of Hperbolic Functions a b sinh cosh c sinh cosh cosh sinh sinh cosh Tr It Eploration A sinh lncosh tanh cosh Eploration B Vieo f() = ( ) cosh sinh (0, ) (, sinh ) f0 < 0, so 0, is a relative maimum f > 0, so, sinh is a relative minimum Figure 58 EXAMPLE Fining Relative Etrema Fin the relative etrema of f cosh sinh Solution Begin b setting the first erivative of f equal to 0 f sinh cosh cosh 0 sinh 0 So, the critical numbers are an 0 Using the Secon Derivative Test, ou can verif that the point 0, iels a relative maimum an the point, sinh iels a relative minimum, as shown in Figure 58 Tr using a graphing utilit to confirm this result If our graphing utilit oes not have hperbolic functions, ou can use eponential functions as follows f e e e e e e e e e e e e e Tr It Eploration A When a uniform fleible cable, such as a telephone wire, is suspene from two points, it takes the shape of a catenar, as iscusse in Eample EXAMPLE Hanging Power Cables Power cables are suspene between two towers, forming the catenar shown in Figure 59 The equation for this catenar is = a cosh a a a cosh a The istance between the two towers is b Fin the slope of the catenar at the point where the cable meets the right-han tower Solution Differentiating prouces b b a a sinh a sinh a Catenar Figure 59 At the point b, a coshba, the slope (from the left) is given b m sinh b a Tr It Eploration A Open Eploration FOR FURTHER INFORMATION In Eample, the cable is a catenar between two supports at the same height To learn about the shape of a cable hanging between supports of ifferent heights, see the article Reeamining the Catenar b Paul Cella in The College Mathematics Journal

5 9 CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions EXAMPLE Integrating a Hperbolic Function Fin cosh sinh Solution cosh sinh sinh cosh sinh C sinh 6 C u sinh Tr It Eploration A Inverse Hperbolic Functions Unlike trigonometric functions, hperbolic functions are not perioic In fact, b looking back at Figure 57, ou can see that four of the si hperbolic functions are actuall one-to-one (the hperbolic sine, tangent, cosecant, an cotangent) So, ou can appl Theorem 57 to conclue that these four functions have inverse functions The other two (the hperbolic cosine an secant) are one-to-one if their omains are restricte to the positive real numbers, an for this restricte omain the also have inverse functions Because the hperbolic functions are efine in terms of eponential functions, it is not surprising to fin that the inverse hperbolic functions can be written in terms of logarithmic functions, as shown in Theorem 59 THEOREM 59 Function sinh ln cosh ln tanh coth sech ln csch ln ln ln Inverse Hperbolic Functions Domain,,,,, 0,, 0 0, Proof The proof of this theorem is a straightforwar application of the properties of the eponential an logarithmic functions For eample, if f sinh e e an g ln ou can show that fg an g f, which implies that g is the inverse function of f

6 SECTION 58 Hperbolic Functions 9 = = Graphs of the hperbolic tangent function an the inverse hperbolic tangent function Figure 50 TECHNOLOGY You can use a graphing utilit to confirm graphicall the results of Theorem 59 For instance, graph the following functions tanh Hperbolic tangent e e Definition of hperbolic tangent e e tanh Inverse hperbolic tangent ln Definition of inverse hperbolic tangent The resulting ispla is shown in Figure 50 As ou watch the graphs being trace out, notice that an Also notice that the graph of is the reflection of the graph of in the line The graphs of the inverse hperbolic functions are shown in Figure 5 = sinh = cosh = tanh Domain:, Range:, Domain:, Range: 0, Domain:, Range:, = csch = sech = coth Domain:, 0 0, Range:, 0 0, Figure 5 Domain: 0, Range: 0, Domain:,, Range:, 0 0, The inverse hperbolic secant can be use to efine a curve calle a tractri or pursuit curve, as iscusse in Eample 5

7 9 CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions EXAMPLE 5 A Tractri (0, ) A person is holing a rope that is tie to a boat, as shown in Figure 5 As the person walks along the ock, the boat travels along a tractri, given b the equation Person a sech a a (, ) 0 0 where a is the length of the rope If a 0 feet, fin the istance the person must walk to bring the boat 5 feet from the ock 0 0 = 0 sech 0 0 A person must walk 7 feet to bring the boat 5 feet from the ock Figure 5 Solution In Figure 5, notice that the istance the person has walke is given b 0 0 sech When 5, this istance is 0 sech 0 0 sech 5 0 ln 0 0 ln 5 7 feet Tr It Eploration A Differentiation an Integration of Inverse Hperbolic Functions The erivatives of the inverse hperbolic functions, which resemble the erivatives of the inverse trigonometric functions, are liste in Theorem 50 with the corresponing integration formulas (in logarithmic form) You can verif each of these formulas b appling the logarithmic efinitions of the inverse hperbolic functions (See Eercises 99 0) THEOREM 50 Let u be a ifferentiable function of sinh u tanh u u u a u a ln a u a u C Differentiation an Integration Involving Inverse Hperbolic Functions u u u sech u u u u u C u ± a lnu u ± a csch u u ua ± u a ln a a ± u u C u u cosh u u coth u u u u u

8 SECTION 58 Hperbolic Functions 95 EXAMPLE 6 More About a Tractri For the tractri given in Eample 5, show that the boat is alwas pointing towar the person Solution For a point, on a tractri, the slope of the graph gives the irection of the boat, as shown in Figure 5 0 sech However, from Figure 5, ou can see that the slope of the line segment connecting the point 0, with the point, is also m 0 So, the boat is alwas pointing towar the person (It is because of this propert that a tractri is calle a pursuit curve) Tr It EXAMPLE 7 Eploration A Integration Using Inverse Hperbolic Functions Fin 9 Solution Let a an u 9 9 Tr It ln 9 Eploration A C u ua u a ln a a u u C EXAMPLE 8 Integration Using Inverse Hperbolic Functions Fin 5 Solution Let a 5 an u 5 5 Tr It 5 ln 5 5 C 5 ln 5 5 C Eploration A u a u a ln a u a u C

9 96 CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions Eercises for Section 58 The smbol Click on Click on inicates an eercise in which ou are instructe to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarge cop of the graph In Eercises 6, evaluate the function If the value is not a rational number, give the answer to three-ecimal-place accurac (a) sinh (a) cosh 0 (b) tanh (b) sech (a) cschln (a) sinh 0 (b) cothln 5 (b) tanh 0 5 (a) cosh 6 (a) csch (b) sech (b) coth 7 cosh sinh 8 5 (0, ) e sinh (0, ) In Eercises 7, verif the ientit 7 8 tanh sech cosh In Eercises an, use the value of the given hperbolic function to fin the values of the other hperbolic functions at In Eercises 5, fin the erivative of the function ln tanh cosh 9 sinh sinh cosh cosh sinh 0 sinh sinh cosh sinh sinh sinh cosh cosh cosh cosh sinh tanh 5 sech 6 coth 7 f lnsinh 8 g lncosh 9 0 cosh sinh In Eercises 9, fin an relative etrema of the function Use a graphing utilit to confirm our result 9 0 f sin sinh cos cosh, f sinh cosh g sech h tanh In Eercises an, show that the function satisfies the ifferential equation Function a sinh a cosh Linear an Quaratic Approimations In Eercises 5 an 6, use a computer algebra sstem to fin the linear approimation P fa fa a an the quaratic approimation Differential Equation 0 0 P fa fa a fa a of the function f at a Use a graphing utilit to graph the function an its linear an quaratic approimations h sinh ht t coth t 5 f tanh, a 0 6 f cosh, a 0 ft arctansinh t g sech In Eercises 5 8, fin an equation of the tangent line to the graph of the function at the given point 5 sinh 6 cosh (, 0) (, ) Catenar In Eercises 7 an 8, a moel for a power cable suspene between two towers is given (a) Graph the moel, (b) fin the heights of the cable at the towers an at the mipoint between the towers, an (c) fin the slope of the moel at the point where the cable meets the right-han tower cosh, cosh, In Eercises 9 50, fin the integral cosh 9 sinh 0 cosh sinh sinh sinh

10 SECTION 58 Hperbolic Functions 97 cosh sech sinh 5 csch 6 sech tanh csch coth cosh sinh 9 50 In Eercises 5 56, evaluate the integral ln 5 tanh 5 cosh ln e cosh 0 In Eercises 57 6, fin the erivative of the function 57 cosh 58 tanh sinh tan sech cos, tanh sin csch 0 < < 6 6 sinh tanh ln Writing About Concepts 65 Discuss several was in which the hperbolic functions are similar to the trigonometric functions 66 Sketch the graph of each hperbolic function Then ientif the omain an range of each function Limits In Eercises 67 7, fin the limit 67 lim 68 lim 69 lim 70 lim 7 lim sinh 0 7 lim coth 0 In Eercises 7 80, fin the inefinite integral using the formulas of Theorem e In Eercises 8 8, solve the ifferential equation Area In Eercises 85 88, fin the area of the region 85 sech 86 tanh In Eercises 89 an 90, evaluate the integral in terms of (a) natural logarithms an (b) inverse hperbolic functions Chemical Reactions Chemicals A an B combine in a -to- ratio to form a compoun The amount of compoun being prouce at an time t is proportional to the unchange amounts of A an B remaining in the solution So, if kilograms of A is mie with kilograms of B, ou have t k k 6 One kilogram of the compoun is forme after 0 minutes Fin the amount forme after 0 minutes b solving the equation k 6 t 8 8 6

11 98 CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions 9 Vertical Motion An object is roppe from a height of 00 feet (a) Fin the velocit of the object as a function of time (neglect air resistance on the object) (b) Use the result in part (a) to fin the position function (c) If the air resistance is proportional to the square of the velocit, then vt kv, where feet per secon per secon is the acceleration ue to gravit an k is a constant Show that the velocit v as a function of time is Tractri b performing the following integration an simplifing the result () Use the result in part (c) to fin lim vt an give its t interpretation (e) Integrate the velocit function in part (c) an fin the position s of the object as a function of t Use a graphing utilit to graph the position function when k 00 an the position function in part (b) in the same viewing winow Estimate the aitional time require for the object to reach groun level when air resistance is not neglecte (f) Give a written escription of what ou believe woul happen if k were increase Then test our assertion with a particular value of k 9 Fin vt tanhk t k v kv t In Eercises 9 an 9, use the equation of the tractri a sech a a, a > 0 9 Let L be the tangent line to the tractri at the point P If L intersects the -ais at the point Q, show that the istance between P an Q is a 95 Prove that 96 Show that arctansinh arcsintanh 97 Let > 0 an b > 0 Show that In Eercises 98 0, verif the ifferentiation formula 98 cosh sinh 99 sech 00 0 sinh cosh 0 tanh ln, sech sech tanh < < b e t t b Putnam Eam Challenge sinh b 0 From the verte 0, c of the catenar c cosh c a line L is rawn perpenicular to the tangent to the catenar at a point P Prove that the length of L intercepte b the aes is equal to the orinate of the point P 0 Prove or isprove that there is at least one straight line normal to the graph of cosh at a point a, cosh a an also normal to the graph of sinh at a point c, sinh c [At a point on a graph, the normal line is the perpenicular to the tangent at that point Also, cosh e e an sinh e e ] These problems were compose b the Committee on the Putnam Prize Competition The Mathematical Association of America All rights reserve

12 REVIEW EXERCISES 99 Review Eercises for Chapter 5 The smbol Click on Click on inicates an eercise in which ou are instructe to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarge cop of the graph In Eercises an, sketch the graph of the function b han Ientif an asmptotes of the graph f ln f ln In Eercises an, use the properties of logarithms to epan the logarithmic function ln 5 ln In Eercises 5 an 6, write the epression as the logarithm of a single quantit 5 ln ln ln 6 ln ln ln 5 In Eercises 7 an 8, solve the equation for 7 ln 8 ln ln 0 0 sec f f In Eercises 5 0, (a) fin the inverse function of f, (b) use a graphing utilit to graph f an f in the same viewing winow, an (c) verif that f f an 5 f 6 f f 8 f 9 f 0 f 5, 0 In Eercises, fin f a for the function f an the given real number a f, a f, f tan,, f ln, a 0 a tan 0 a In Eercises 9, fin the erivative of the function 9 g ln 0 b a b a lna b a b a b a ln In Eercises 5 an 6, fin an equation of the tangent line to the graph of the function at the given point 5 ln 6 (, ) In Eercises 7, fin or evaluate the integral sin ln 9 0 cos ln h ln f ln f ln ln e (, ln ) In Eercises 5 an 6, (a) fin the inverse function of f, (b) use a graphing utilit to graph f an f in the same viewing winow, an (c) verif that f f an f f 5 f ln 6 f e In Eercises 7 an 8, graph the function without the ai of a graphing utilit 7 e 8 e In Eercises 9, fin the erivative of the function 9 gt t e t 0 g ln e e e hz e z g e t e In Eercises 5 an 6, fin an equation of the tangent line to the graph of the function at the given point 5 f lne,, 6 f esin, In Eercises 7 an 8, use implicit ifferentiation to fin / 7 ln 0 8 cos e In Eercises 9 56, fin or evaluate the integral e 9 e e e e e e e e e 0,

13 00 CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions 5 5 e e e e 57 Show that e a cos b sin satisfies the ifferential equation 58 Depreciation The value V of an item t ears after it is purchase is V 8000e 06t, 0 t 5 (a) Use a graphing utilit to graph the function (b) Fin the rates of change of V with respect to t when t an t (c) Use a graphing utilit to graph the tangent lines to the function when t an t 0 0 In Eercises 59 an 60, fin the area of the region boune b the graphs of the equations 59 e, 0, 0, 60 e, 0, 0, In Eercises 6 6, sketch the graph of the function b han log 6 log In Eercises 65 70, fin the erivative of the function 65 f 66 f e g log 70 h log 5 In Eercises 7 an 7, fin the inefinite integral t t t 7 Climb Rate The time t (in minutes) for a small plane to climb to an altitue of h feet is 8,000 t 50 log 0 8,000 h 0 e e where 8,000 feet is the plane s absolute ceiling (a) Determine the omain of the function appropriate for the contet of the problem (b) Use a graphing utilit to graph the time function an ientif an asmptotes (c) Fin the time when the altitue is increasing at the greatest rate 7 Compoun Interest (a) How large a eposit, at 7% interest compoune continuousl, must be mae to obtain a balance of $0,000 in 5 ears? (b) A eposit earns interest at a rate of r percent compoune continuousl an oubles in value in 0 ears Fin r In Eercises 75 an 76, sketch the graph of the function 75 f arctan 76 h arcsin In Eercises 77 an 78, evaluate the epression without using a calculator (Hint: Make a sketch of a right triangle) 77 (a) sinarcsin 78 (a) tanarccot (b) cosarcsin (b) cosarcsec 5 In Eercises 79 8, fin the erivative of the function 79 tanarcsin 80 arctan 8 arcsec 8 arctan e 8 arcsin arcsin 8 arcsec, In Eercises 85 90, fin the inefinite integral e 5 87 e 88 6 arcsin 89 arctan 90 In Eercises 9 an 9, fin the area of the region Harmonic Motion A weight of mass m is attache to a spring an oscillates with simple harmonic motion B Hooke s Law, ou can etermine that A k m t where A is the maimum isplacement, t is the time, an k is a constant Fin as a function of t, given that 0 when t 0 In Eercises 9 an 95, fin the erivative of the function 9 cosh 95 tanh In Eercises 96 an 97, fin the inefinite integral sech < <

14 PS Problem Solving 0 PS Problem Solving The smbol Click on Click on inicates an eercise in which ou are instructe to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarge cop of the graph Fin the value of a that maimizes the angle shown in the figure What is the approimate measure of this angle? 6 θ 0 a 0 Recall that the graph of a function f is smmetric with respect to the origin if whenever, is a point on the graph,, is also a point on the graph The graph of the function f is smmetric with respect to the point a, b if, whenever a, b is a point on the graph, a, b is also a point on the graph, as shown in the figure Let f sinln (a) Determine the omain of the function f (b) Fin two values of satisfing f (c) Fin two values of satisfing f () What is the range of the function f? (e) Calculate f an use calculus to fin the maimum value of f on the interval, 0 (f) Use a graphing utilit to graph f in the viewing winow 0, 5, an estimate lim f, if it eists (g) Determine lim 0 0 f analticall, if it eists 5 Graph the eponential function a for a 05,, an 0 Which of these curves intersects the line? Determine all positive numbers a for which the curve a intersects the line 6 (a) Let Pcos t, sin t be a point on the unit circle in the first quarant (see figure) Show that t is equal to twice the area of the shae circular sector AOP (a +, b + ) (a, b) (a, b ) O t P A(, 0) (a) Sketch the graph of sin on the interval 0, Write a short paragraph eplaining how the smmetr of the graph with respect to the point 0, allows ou to conclue that 0 (b) Sketch the graph of sin on the interval 0, Use the smmetr of the graph with respect to the point, to evaluate the integral 0 sin 0 sin (c) Sketch the graph of arccos on the interval, Use the smmetr of the graph to evaluate the integral arccos () Evaluate the integral 0 tan ln (a) Use a graphing utilit to graph f on the interval, (b) Use the graph to estimate lim f 0 (c) Use the efinition of erivative to prove our answer to part (b) (b) Let Pcosh t, sinh t be a point on the unit hperbola in the first quarant (see figure) Show that t is equal to twice the area of the shae region AOP Begin b showing that the area of the shae region AOP is given b the formula At cosh t cosh t sinh t O t A(, 0) P

15 0 CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions 7 Consier the three regions A, B, an C etermine b the graph of f arcsin, as shown in the figure (a) Calculate the areas of regions A an B (b) Use our answers in part (a) to evaluate the integral arcsin (c) Use our answers in part (a) to evaluate the integral ln () Use our answers in part (a) to evaluate the integral arctan 8 Let L be the tangent line to the graph of the function ln at the point a, b Show that the istance between b an c is alwas equal to b c π π 6 9 Let L be the tangent line to the graph of the function e at the point a, b Show that the istance between a an c is alwas equal to b c L A B a a C L 0 Use integration b substitution to fin the area uner the curve between an Use integration b substitution to fin the area uner the curve sin cos between 0 an (a) Use a graphing utilit to compare the graph of the function e with the graphs of each of the given functions (i) (ii)!!! (iii)!!! (b) Ientif the pattern of successive polnomials in part (a) an eten the pattern one more term an compare the graph of the resulting polnomial function with the graph of e (c) What o ou think this pattern implies? A $0,000 home mortgage for 5 ears at 9 % has a monthl pament of $9859 Part of the monthl pament goes for the interest charge on the unpai balance an the remainer of the pament is use to reuce the principal The amount that goes for interest is u M M Pr r t an the amount that goes towar reuction of the principal is v M Pr r t In these formulas, P is the amount of the mortgage, r is the interest rate, M is the monthl pament, an t is the time in ears (a) Use a graphing utilit to graph each function in the same viewing winow (The viewing winow shoul show all 5 ears of mortgage paments) (b) In the earl ears of the mortgage, the larger part of the monthl pament goes for what purpose? Approimate the time when the monthl pament is evenl ivie between interest an principal reuction (c) Use the graphs in part (a) to make a conjecture about the relationship between the slopes of the tangent lines to the two curves for a specifie value of t Give an analtical argument to verif our conjecture Fin u5 an v5 () Repeat parts (a) an (b) for a repament perio of 0 ears M $856 What can ou conclue?