CHAPTER 7: TRANSCENDENTAL FUNCTIONS

Transcription

1 7.0 Introduction and One to one Functions Contemporary Calculus 1 CHAPTER 7: TRANSCENDENTAL FUNCTIONS Introduction In te previous capters we saw ow to calculate and use te derivatives and integrals of many of te most important and common functions in matematics and applications. Tese common functions include polynomials, eponential functions, logaritms, te si trigonometric functions, and various combinations of tem. Tere are, owever, oter important and useful functions, and we eamine several of tem in tis capter. Te central purpose of tis capter is to etend te ideas and applications we ave already seen to additional functions, primarily te inverse trigonometric functions. Te capter begins wit a sort introduction to one to one functions. Section 7.1 is an eamination of inverse functions and some of teir properties. Section 7.2 introduces te inverses of te trigonometric functions, and Section 7.3 sows ow to calculate and use te derivatives of te inverse trigonometric functions. We ave already been using a most important pair of one to one functions, e and ln(). We ave also used te fact tat eac of tem "undoes" te effect of te oter in order to solve equations suc as 3 = 5e 2 and 8 = 2. ln(). Te functions e and ln() are inverses of eac oter, and you sould keep tese functions and teir graps in mind as we discuss general one to one and inverse functions. Section 7.4 (optional) is a calculus based presentation of te eponential and logaritm functions, and it includes verifications of a number of properties of te eponential and logaritm functions tat we ave already used, suc as te multiplication law for logaritms: ln(a. b) = ln(a) + ln(b). One to One Functions In earlier courses you saw tat some equations ave only one solution (for eample, 5 2 = 3 and 3 = 8 ), some ave two solutions ( = 7 ), and some even ave an infinite number of solutions ( sin() = 0.8 ). Te graps of te functions y = 5 2, y = 3, y = 2 + 3, and y = sin() and te solutions of te equations are sown in Fig. 1. Te functions f wose equations f() = k ave only one solution for eac value of k (eac outcome k comes from only one input ) are particularly common in applications, and tey ave a number of useful matematical properties. Te remainder of tis section focuses on tose functions and looks at some of teir properties.

2 7.0 Introduction and One to one Functions Contemporary Calculus 2 Eample 1: How many solutions does eac of te following equations ave? (a) f() = 0 for f() = ( 4) (b) g() = 3 for g given by Table 1 (c) () = 4 for given by te grap in Fig. 2. (d) f() = k for f() = e. Solution: (a) Two. (4 ) = 0 if = 0 or = 4. (b) One. g() = 3 if = 2. (c) Two. () = 4 if = 1 or = 5. (d) One, = ln(k), if k > 0. None if k 0. Practice 1: How many solutions does eac of te following equations ave? (a) f() = 4 for f() = (4 ) (b) g() = 7 for g given by Table 1 (c) () = 3 for given by te grap in Fig. 3. (d) f() = 5 for f() = ln() g() Table 1 Horizontal Line Test for One to one (Definition of One to one) A function is one to one if eac orizontal line intersects te grap of te function at most once. Equivalently, a function y = f() is one to one if two distinct values always result in two distinct y values: a b implies f(a) f(b). Tis immediately tells us tat every strictly increasing function is one to one, and every strictly decreasing function is one to one. (Wy?) For any function, if we know te input value, we can calculate te output y value, but an output may ave come from any of several different inputs. Wit a one to one function, eac output y value comes from only one input value. Eample 2: (a) Wic functions in Fig. 4 are one to one? (b) Wic functions in Table 2 are one to one? Solution: (a) In Fig. 4, functions f and are one to one. Function g is not one to one. (b) In Table 2, function is one to one. Functions f and g are not one to one. f() g() () Table 2

3 7.0 Introduction and One to one Functions Contemporary Calculus 3 f() g() () Eample 3: Practice 2: (a) Wic functions in Fig. 5 are one to one? (b) Wic functions in Table 3 are one to one? Let f() = (Fig. 6). Find te values of so tat (a) f() = 9 and (b) f() = a. (c) Solve f(y) = for y. ( f(y) = 2y + 1 ) Table 3 Solution: (a) 9 = f() = so 8 = 2 and = 8/2 = 4. (b) a = so 2 = a 1 and = (a 1)/2. (c) = f(y) = 2y + 1 so 2y = 1 and y = ( 1)/2. Practice 3: Let f() = 3 5. Find values of so (a) f() = 7 and (b) f() = a. (c) Solve f(y) = for y. Practice 4: Sow tat eponential growt, f() = e 3, and eponential decay, g() = e 2, are one to one. PROBLEMS g() In problems 1 4, state weter te given functions are one to one. 1. f() = 3 5, y = 3, g() given by Table 4, and () given by te grap in Fig Table 4 Fig. 7 g() Table 5 Fig f() = /4, y = 2 + 3, g() given by Table 5, and () given by te grap in Fig f() = sin(), y = e 2, g() given by Table 6, and () given by te grap in Fig. 9. g() Table 6 Fig f() = 17, y = 3 1, g() given by Table 7, and () given by te grap in Fig Are Social Security numbers one to one? Telepone numbers? 6. Wat would it mean if te scores on a calculus test were one to one? 7. Wat is te legal/social term for one to one in marriage? g() Table 7 Fig. 10

4 7.0 Introduction and One to one Functions Contemporary Calculus 4 8. Te function given below represents "y is married to." (a) Is tis f a function? (b) Is f one to one? (d) Is A breaking te law? A B C D y P Q P R 9. How many places can a one to one function touc te ais? (c) Is P breaking te law? 10. Can a continuous one to one function ave te values given below Eplain wy it is possible or wy it is not possible? f() Te grap of f() = 2. INT() for 2 3 is given in Fig. 11. (a) Is f a one to one function? (b) Is f an increasing function? a decreasing function? 12. Is every linear function f() = a + b one to one? 13. Sow tat te function f() = ln() is one to one for > Sow tat te function f() = e is one to one. In problems 15 18, rules are given for encoding a 6 letter alpabet. For eac problem: (a) Is te encoding rule a function? (b) Is te encoding rule one to one? (c) Encode te word "bad." (d) Write a table for decoding te encoded letters and use it to decode your answer to part (c). (e) Grap te encoding rule and te decoding rule. (Fig. 12 sows te graps for te code in problem 15). How are te encoding and decoding graps related? 15. original letter a b c d e f 16. original letter a b c d e f encoded letter d c f e b a encoded letter b d f b a c 17. original letter a b c d e f encoded letter d f e a c b How does your decoding rule compare wit te encoding rule? Wat appens if you encode a word and ten encode te encoded word: for eample, Encode( Encode("bad") ) =? 18. original letter a b c d e f ncoded letter e a f c b d Wat appens if you apply tis coding rule tree times: Encode( Encode( Encode("bad") ) ) =?

5 7.0 Introduction and One to one Functions Contemporary Calculus 5 Section 7.0 PRACTICE Answers Practice 1: (a) One. Solve (4 ) = 4 to get = 2. (b) At least two, = 1 and = 5. (c) At least one, 4. (d) Eactly one. Solve 5 = ln() to get = e Practice 2: (a) Only g is 1 1. (b) From te values in te table, f anf g are one to one. (However, if f and g are continuous on [0,5], ten neiter of tem is one to one. Wy?) Practice 3: (a) 3 5 = 7 so = 4. (b) 3 5 = a so = a (c) f() = 3 5 so f(y) = 3y 5. f(y) = means tat 3y 5 = and y = Practice 4: f() = e 3. f '() = 3. e 3 > 0 so f is increasing, one to one, and as an inverse. g() = e 2. g '() = 2. e 2 < 0 so g is decreasing, one to one, and as an inverse.