1.6 CONTINUITY OF TRIGONOMETRIC, EXPONENTIAL, AND INVERSE FUNCTIONS

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1 .6 Continuit of Trigonometric, Eponential, and Inverse Functions.6 CONTINUITY OF TRIGONOMETRIC, EXPONENTIAL, AND INVERSE FUNCTIONS In this section we will discuss the continuit properties of trigonometric functions, eponential functions, and inverses of various continuous functions. We will also discuss some important its involving such functions. c Q(cos c, sin c) P(cos, sin ) As approaches c the point P approaches the point Q. Figure.6. CONTINUITY OF TRIGONOMETRIC FUNCTIONS Recall from trigonometr that the graphs of sin and cos are drawn as continuous curves. We will not formall prove that these functions are continuous, but we can motivate this fact b letting c be a fied angle in radian measure and a variable angle in radian measure. If, as illustrated in Figure.6., the angle approaches the angle c, then the point P(cos,sin ) moves along the unit circle toward Q(cos c, sin c), and the coordinates of P approach the corresponding coordinates of Q. This implies that sin = sin c and cos = cos c () c c Thus, sin and cos are continuous at the arbitrar point c; that is, these functions are continuous everwhere. The formulas in () can be used to find its of the remaining trigonometric functions b epressing them in terms of sin and cos ; for eample, if cos c =0, then sin tan = c c cos = sin c cos c = tan c Thus, we are led to the following theorem. Theorem.6. implies that the si basic trigonometric functions are continuous on their domains. In particular, sin and cos are continuous everwhere..6. theorem If c is an number in the natural domain of the stated trigonometric function, then sin = sin c c csc = csc c c cos = cos c c sec = sec c c tan = tan c c cot = cot c c Eample Find the it ( ) cos Solution..5.5 that Since the cosine function is continuous everwhere, it follows from Theorem ( ) cos(g()) = cos g() provided g() eists. Thus, ( ) cos = cos( + ) = cos ( ) ( + ) = cos CONTINUITY OF INVERSE FUNCTIONS Since the graphs of a one-to-one function f and its inverse f are reflections of one another about the line =, it is clear geometricall that if the graph of f has no breaks or holes in it, then neither does the graph of f. This, and the fact that the range of f is the domain of f, suggests the following result, which we state without formal proof.

2 Chapter / Limits and Continuit To paraphrase Theorem.6., the inverse of a continuous function is continuous..6. theorem If f is a one-to-one function that is continuous at each point of its domain, then f is continuous at each point of its domain; that is, f is continuous at each point in the range of f. Eample [, ]. Use Theorem.6. to prove that sin is continuous on the interval Solution. Recall that sin is the inverse of the restricted sine function whose domain is the interval [ π/,π/] and whose range is the interval [, ] (Definition and Figure 0.4.3). Since sin is continuous on the interval [ π/,π/], Theorem.6. implies sin is continuous on the interval [, ]. Arguments similar to the solution of Eample show that each of the inverse trigonometric functions defined in Section 0.4 is continuous at each point of its domain. When we introduced the eponential function f() = b in Section 0.5, we assumed that its graph is a curve without breaks, gaps, or holes; that is, we assumed that the graph of = b is a continuous curve. This assumption and Theorem.6. impl the following theorem, which we state without formal proof..6.3 theorem Let b>0,b =. (a) The function b is continuous on (, + ). (b) The function log b is continuous on (0, + ). Eample 3 Where is the function f() = tan + ln 4 continuous? Solution. The fraction will be continuous at all points where the numerator and denominator are both continuous and the denominator is nonzero. Since tan is continuous everwhere and ln is continuous if >0, the numerator is continuous if >0. The denominator, being a polnomial, is continuous everwhere, so the fraction will be continuous at all points where >0and the denominator is nonzero. Thus, f is continuous on the intervals (0, ) and (, + ). OBTAINING LIMITS BY SQUEEZING In Section. we used numerical evidence to conjecture that sin = () 0 However, this it is not eas to establish with certaint. The it is an indeterminate form of tpe 0/0, and there is no simple algebraic manipulation that one can perform to obtain the it. Later in the tet we will develop general methods for finding its of indeterminate forms, but in this particular case we can use a technique called squeezing. The method of squeezing is used to prove that f() L as c b trapping or squeezing f between two functions, g and h, whose its as c are known with certaint to be L. As illustrated in Figure.6., this forces f to have a it of L as well. This is the idea behind the following theorem, which we state without proof. = h() L = g() Figure.6. c = f()

3 .6 Continuit of Trigonometric, Eponential, and Inverse Functions 3 The Squeezing Theorem also holds for one-sided its and its at + and. How do ou think the hpotheses would change in those cases?.6.4 theorem (The Squeezing Theorem) Let f, g, and h be functions satisfing g() f() h() for all in some open interval containing the number c, with the possible eception that the inequalities need not hold at c. Ifg and h have the same it as approaches c, sa g() = h() = L c c then f also has this it as approaches c, that is, f() = L c O = sin o sin = 0 To illustrate how the Squeezing Theorem works, we will prove the following results, which are illustrated in Figure theorem (a) sin 0 = (b) 0 cos = 0 = cos O o cos = 0 0 Figure.6.3 proof (a) In this proof we will interpret as an angle in radian measure, and we will assume to start that 0 <<π/. As illustrated in Figure.6.4, the area of a sector with central angle and radius lies between the areas of two triangles, one with area tan and the other with area sin. Since the sector has area (see marginal note), it follows that tan sin Multipling through b /(sin ) and using the fact that sin >0 for 0 <<π/, we obtain cos sin Net, taking reciprocals reverses the inequalities, so we obtain cos sin (3) which squeezes the function (sin )/ between the functions cos and. Although we derived these inequalities b assuming that 0 <<π/, the also hold for π/ <<0 [since replacing b and using the identities sin( ) = sin, and cos( ) = cos (, tan ) (cos, sin ) tan sin (, 0) Area of triangle Area of sector Area of triangle tan sin Figure.6.4

4 4 Chapter / Limits and Continuit Recall that the area A of a sector of radius r and central angle θ is A = r θ This can be derived from the relationship A πr = θ π which states that the area of the sector is to the area of the circle as the central angle of the sector is to the central angle of the circle. Area = A leaves (3) unchanged]. Finall, since the Squeezing Theorem implies that cos = and = 0 0 sin = 0 proof (b) For this proof we will use the it in part (a), the continuit of the sine function, and the trigonometric identit sin = cos. We obtain cos 0 = 0 = ( 0 [ cos sin )( 0 + cos + cos ] sin + cos sin ( + cos ) ( ) 0 = () = 0 + = 0 ) u Eample 4 Find r (a) 0 tan (b) θ 0 sin θ θ (c) 0 sin 3 sin 5 Solution (a). tan 0 ( sin = 0 ) ( = cos 0 )( ) sin = ()() = 0 cos Solution (b). The trick is to multipl and divide b, which will make the denominator the same as the argument of the sine function [just as in Theorem.6.5(a)]: sin θ sin θ = = θ 0 sin θ θ Now make the substitution = θ, and use the fact that 0 asθ 0. This ields sin θ = θ 0 sin θ θ = 0 sin = () = TECHNOLOGY MASTERY Use a graphing utilit to confirm the its in Eample 4, and if ou have a CAS, use it to obtain the its. Solution (c). sin 3 sin 3 3 sin 3 0 sin 5 = 3 = 0 sin 5 0 sin = 3 5 = 3 5 Figure.6.5 = sin Eample 5 Discuss the its ( ) (a) sin 0 ( ) (b) sin 0 Solution (a). Let us view / as an angle in radian measure. As 0 +, the angle / approaches +, so the values of sin(/) keep oscillating between and without approaching a it. Similarl, as 0, the angle / approaches, so again the values of sin(/) keep oscillating between and without approaching a it. These conclusions are consistent with the graph shown in Figure.6.5. Note that the oscillations become more and more rapid as 0 because / increases (or decreases) more and more rapidl as approaches 0.

5 .6 Continuit of Trigonometric, Eponential, and Inverse Functions 5 Confirm (4) b considering the cases >0 and <0 separatel. = Solution (b). Since it follows that if =0, then sin sin ( ) ( ) (4) Since 0as 0, the inequalities in (4) and the Squeezing Theorem impl that ( ) sin = 0 0 This is consistent with the graph shown in Figure.6.6. = sin = Figure.6.6 REMARK It follows from part (b) of this eample that the function { sin( /), =0 f() = 0, = 0 is continuous at = 0, since the value of the function and the value of the it are the same at 0. This shows that the behavior of a function can be ver comple in the vicinit of = c, even though the function is continuous at c. QUICK CHECK EXERCISES.6 (See page 8 for answers.). In each part, is the given function continuous on the interval [0,π/)? (a) sin (b) cos (c) tan (d) csc. Evaluate sin (a) 0 cos (b) Suppose a function f has the propert that for all real numbers 3 f() 3 + From this we can conclude that f() as. 4. In each part, give the largest interval on which the function is continuous. (a) e (b) ln (c) sin (d) tan EXERCISE SET.6 Graphing Utilit 8 Find the discontinuities, if an. ( ). f() = sin( ). f() = cos π 3. f() = cot 4. f() = sec 5. f() = csc 6. f() = + sin 7. f() = 8. f() = + tan sin 9 4 Determine where f is continuous. 9. f() = sin 0. f() = cos (ln ). f() = ln(tan ) 9 3. f() = sin (/) ( ) sin. f() = ep 4. f() = ln ln( + 3)

6 6 Chapter / Limits and Continuit 5 6 In each part, use Theorem.5.6(b) to show that the function is continuous everwhere. 5. (a) sin( ) (b) sin (c) cos 3 ( + ) 6. (a) 3 + sin (b) sin(sin ) (c) cos 5 cos Find the its. ( ) 7. cos + ( ) 9. + sin ( ) π 8. sin + 3 ( ) + 0. ln +. 0 e sin. + cos( tan ) sin 3θ 3. sin θ 5. θ 0 + θ tan sin 3 sin sin 3. 0 t 33. t 0 cos t 35. cos θ ( ) 37. sin 0 + θ 0 θ cos 3 cos tan 3 + sin h 0 sin h h sin θ 6. sin sin 8 sin sin h 3. h 0 cos h cos ( π ) cos 3h 36. h 0 cos 5h 3 sin (a) Complete the table and make a guess about the it indicated. (b) Find the eact value of the it. sin( 5) 4. f() = 5 ; f() 5 f() f() = sin( ) ; + f() f(). Table E-4 Table E True False Determine whether the statement is true or false. Eplain our answer. 43. Suppose that for all real numbers, a function f satisfies Then f() = 5. f() For 0 <<π/, the graph of = sin lies below the graph of = and above the graph of = cos. 45. If an invertible function f is continuous everwhere, then its inverse f is also continuous everwhere. 46. Suppose that M is a positive number and that for all real numbers, a function f satisfies Then M f() M f() f() = 0 and = FOCUS ON CONCEPTS 47. In an attempt to verif that 0 (sin )/ =, a student constructs the accompaning table. (a) What mistake did the student make? (b) What is the eact value of the it illustrated b this table? sin / Table E In the circle in the accompaning figure, a central angle of measure θ radians subtends a chord of length c(θ) and a circular arc of length s(θ). Based on our intuition, what would ou conjecture is the value of θ 0 + c(θ)/s(θ)? Verif our conjecture b computing the it. u c(u) s(u) Figure E Find a nonzero value for the constant k that makes tan k, < 0 f() = 3 + k, 0 continuous at = Is sin f() =, =0, = 0 continuous at = 0? Eplain.

7 .6 Continuit of Trigonometric, Eponential, and Inverse Functions 7 5. In parts (a) (c), find the it b making the indicated substitution. (a) (b) sin + ; (c) π π sin ; t = ( cos t = π ) ; t = cos(π/) [ 5. Find. Hint: Let t = π π ]. sin(π) tan 53. Find. 54. Find π/4 π/ Find π/4 cos sin. π/4 56. Suppose that f is an invertible function, f(0) = 0, f is continuous at 0, and 0 (f()/) eists. Given that L = 0 (f()/), show 0 f () = L [Hint: Appl Theorem.5.5 to the composition h g, where { f() /, =0 h() = L, = 0 and g() = f ().] Appl the result of Eercise 56, if needed, to find the its. tan sin sin 5 FOCUS ON CONCEPTS 60. sin ( ) 6. Use the Squeezing Theorem to show that 50π cos 0 = 0 and illustrate the principle involved b using a graphing utilit to graph the equations =, =, and = cos(50π/) on the same screen in the window [, ] [, ]. 6. Use the Squeezing Theorem to show that ( ) 50π 0 sin 3 = 0 and illustrate the principle involved b using a graphing utilit to graph the equations =, =, and = sin(50π/ 3 ) on the same screen in the window [ 0.5, 0.5] [ 0.5, 0.5]. 63. In Eample 5 we used the Squeezing Theorem to prove that ( ) sin = 0 0 Wh couldn t we have obtained the same result b writing ( ) sin = 0 0 = 0 0 sin 0 sin ( ( ) ) = 0? 64. Sketch the graphs of the curves =, = cos, and = f(), where f is a function that satisfies the inequalities f() cos for all in the interval ( π/,π/). What can ou sa about the it of f() as 0? Eplain. 65. Sketch the graphs of the curves = /, = /, and = f(), where f is a function that satisfies the inequalities f() for all in the interval [, + ). What can ou sa about the it of f() as +? Eplain our reasoning. 66. Draw pictures analogous to Figure.6. that illustrate the Squeezing Theorem for its of the forms + f() and f(). 67. (a) Use the Intermediate-Value Theorem to show that the equation = cos has at least one solution in the interval [0,π/]. (b) Show graphicall that there is eactl one solution in the interval. (c) Approimate the solution to three decimal places. 68. (a) Use the Intermediate-Value Theorem to show that the equation + sin = has at least one solution in the interval [0,π/6]. (b) Show graphicall that there is eactl one solution in the interval. (c) Approimate the solution to three decimal places. 69. In the stud of falling objects near the surface of the Earth, the acceleration g due to gravit is commonl taken to be a constant 9.8 m/s. However, the elliptical shape of the Earth and other factors cause variations in this value that depend on latitude. The following formula, known as the World Geodetic Sstem 984 (WGS 84) Ellipsoidal Gravit Formula, is used to predict the value of g at a latitude of φ degrees (either north or south of the equator): g = sin φ sin φ m /s (cont.)

8 8 Chapter / Limits and Continuit (a) Use a graphing utilit to graph the curve = g(φ) for 0 φ 90. What do the values of g at φ = 0 and at φ = 90 tell ou about the WGS 84 ellipsoid model for the Earth? (b) Show that g = 9.8 m/s somewhere between latitudes of 38 and Writing In our own words, eplain the practical value of the Squeezing Theorem. 7. Writing A careful eamination of the proof of Theorem.6.5 raises the issue of whether the proof might actuall be a circular argument! Read the article A Circular Argument b Fred Richman in the March 993 issue of The College Mathematics Journal, and write a short report on the author s principal points. QUICK CHECK ANSWERS.6. (a) es (b) es (c) es (d) no. (a) (b) ; 0 4. (a) (, + ) (b) (0, + ) (c) [, ] (d) (, + ) CHAPTER REVIEW EXERCISES Graphing Utilit C CAS. For the function f graphed in the accompaning figure, find the it if it eists. (a) f() (b) f() (c) f() 3 (d) f() (e) f() (f ) f() 4 + (g) f() (h) f() (i) f() Figure E-. In each part, complete the table and make a conjecture about the value of the it indicated. Confirm our conjecture b finding the it analticall. (a) f() = 4 ; f().0000 f() tan 4 (b) f() = ; f() f() 3. (a) Approimate the value for the it 3 0 to three decimal places b constructing an appropriate table of values. (b) Confirm our approimation using graphical evidence. C 4. Approimate both b looking at a graph and b calculating values for some appropriate choices of. Compare our answer with the value produced b a CAS. 5 0 Find the its ( ) (3 + 7)( 3 9) In each part, find the horizontal asmptotes, if an. (a) = 7 4 (b) = (c) = In each part, find a f(), if it eists, where a is replaced b 0, 5 +, 5, 5, 5,, and +. (a) f() = 5 (b) f() = { ( 5)/ 5, =5 0, = Find the its. sin 3 sin tan 3 0 cos 3 sin(k) 5., k =0 0 ( ) cos θ 6. tan 7. e tan t 8. ln(sin θ) ln(tan θ) t π/ + θ 0 +

9 ( ) ( 0. + a ) b, a,b > If $000 is invested in an account that pas 7% interest compounded n times each ear, then in 0 ears there will be 000( /n) 0n dollars in the account. How much mone will be in the account in 0 ears if the interest is compounded quarterl (n = 4)? Monthl (n = )? Dail (n = 365)? Determine the amount of mone that will be in the account in 0 ears if the interest is compounded continuousl, that is, as n +.. (a) Write a paragraph or two that describes how the it of a function can fail to eist at = a, and accompan our description with some specific eamples. (b) Write a paragraph or two that describes how the it of a function can fail to eist as + or, and accompan our description with some specific eamples. (c) Write a paragraph or two that describes how a function can fail to be continuous at = a, and accompan our description with some specific eamples. 3. (a) Find a formula for a rational function that has a vertical asmptote at = and a horizontal asmptote at =. (b) Check our work b using a graphing utilit to graph the function. 4. Paraphrase the ɛ-δ definition for a f() = L in terms of a graphing utilit viewing window centered at the point (a, L). 5. Suppose that f() is a function and that for an given ɛ>0, the condition 0 < < 3 ɛ guarantees that 4 f() 5 <ɛ. (a) What it is described b this statement? (b) Find a value of δ such that 0 < < δguarantees that 8f() 40 < The it sin = 0 ensures that there is a number δ such that sin < 0.00 if 0 < <δ. Estimate the largest such δ. 7. In each part, a positive number ɛ and the it L of a function f at a are given. Find a number δ such that f() L <ɛ if 0 < a <δ. (a) (4 7) = ; ɛ = 0.0 (b) 3/ = 6; ɛ = 0.05 (c) 4 = 6; ɛ = 0.00 Chapter Review Eercises 9 8. Use Definition.4. to prove the stated its are correct. 4 9 (a) (4 7) = (b) 3/ 3 = 6 9. Suppose that f is continuous at 0 and that f( 0 )>0. Give either an ɛ-δ proof or a convincing verbal argument to show that there must be an open interval containing 0 on which f() > (a) Let f() = sin sin Approimate f()b graphing f and calculating values for some appropriate choices of. (b) Use the identit sin α sin β = sin α β cos α + β to find the eact value of f(). 3. Find values of, if an, at which the given function is not continuous. (a) f() = (c) f() = Determine where f is continuous. (a) f() = 3 (c) f() = e ln 33. Suppose that f() = (b) f() = 3 (b) f() = cos ( { 4 + 3, + 9, > Is f continuous everwhere? Justif our conclusion. 34. One dictionar describes a continuous function as one whose value at each point is closel approached b its values at neighboring points. (a) How would ou eplain the meaning of the terms neighboring points and closel approached to a nonmathematician? (b) Write a paragraph that eplains wh the dictionar definition is consistent with Definition Show that the conclusion of the Intermediate-Value Theorem ma be false if f is not continuous on the interval [a,b]. 36. Suppose that f is continuous on the interval [0, ], that f(0) =, and that f has no zeros in the interval. Prove that f() > 0 for all in [0, ]. 37. Show that the equation = 0 has at least two real solutions in the interval [ 6, ]. )