1 3 Te Te Derivative 3. Limits 3. Continuity 3.3 Rates of Cange 3. Definition of te Derivative 3.5 Grapical Differentiation Capter 3 Review Etended Application: A Model for Drugs Administered Intravenously population of te United States as been increasing since 79, wen te first census was taken. Over te past few decades, te population as not only been increasing, but te level of diversity as also been increasing. Tis fact is important to scool districts, businesses, and government officials. Using eamples in te tird section of tis capter, we eplore two rates of cange related to te increase in minority population. In te first eample, we calculate an average rate of cange; in te second, we calculate te rate of cange at a particular time. Tis latter rate is an eample of a derivative, te subject of tis capter. 33
2 3 Capter 3 Te Derivative Te algebraic problems considered in earlier capters dealt wit static situations: Wat is te revenue wen items are sold? How muc interest is earned in tree years? Wat is te equilibrium price? Calculus, on te oter and, deals wit dynamic situations: At wat rate is te demand for a product canging? How fast is a car moving after ours? Wen does te growt of a population begin to slow down? Te tecniques of calculus allow us to answer tese questions, wic deal wit rates of cange. Te key idea underlying calculus is te concept of it, so we will begin by studying its. 3. Limits APPLY IT Wat appens to te demand of an essential commodity as its price continues to increase? We will find an answer to tis question in Eercise 8 using te concept of it. Te it is one of te tools tat we use to describe te beavior of a function as te values of approac, or become closer and closer to, some particular number. Eample Finding a Limit Wat appens to ƒ = wen is a number very close to (but not equal to)? Solution We can construct a table wit values getting closer and closer to and find te corresponding values of ƒ. approaces from left ƒ S d approaces from rigt ƒ() approaces ƒ() approaces Your Turn Find +. S Te table suggests tat, as gets closer and closer to from eiter side, ƒ gets closer and closer to. In fact, you can use a calculator to sow te values of ƒ can be made as close as you want to by taking values of close enoug to. Tis is not surprising, since te value of te function at = is ƒ =. We can observe tis fact by looking at te grap y =, as sown in Figure. In suc a case, we say te it of ƒ as approaces is, wic is written as ƒ =. S TRY YOUR TURN
3 3. Limits 35 y (, ) y = f() = Limit is Figure Te prase approaces from te left is written S -. Similarly, approaces from te rigt is written S +. Tese epressions are used to write one-sided its. Te it from te left (as approaces from te negative direction) is written ƒ =, S - and sown in red in Figure. Te it from te rigt (as approaces from te positive direction) is written ƒ =, S + and sown in blue in Figure. A two-sided it, suc as ƒ =, S eists only if bot one-sided its eist and are te same; tat is, if ƒ approaces te same number as approaces a given number from eiter side. caution Notice tat Sa - ƒ does not mean to take negative values of, nor does it mean to coose values of to te rigt of a and ten move in te negative direction. It means to use values less tan a 6 a tat get closer and closer to a. Te previous eample suggests te following informal definition. Teacing Tip: Tis is te first section on calculus and is very important because its are wat really distinguis calculus from algebra. Limit of a Function Let ƒ be a function and let a and L be real numbers. If. as takes values closer and closer (but not equal) to a on bot sides of a, te corresponding values of ƒ get closer and closer (and peraps equal) to L; and. te value of ƒ can be made as close to L as desired by taking values of close enoug to a; ten L is te it of ƒ as approaces a, written ƒ = L. Sa Tis definition is informal because te epressions closer and closer to and as close as desired ave not been defined. A more formal definition would be needed to prove te rules for its given later in tis section.* * Te it is te key concept from wic all te ideas of calculus flow. Calculus was independently discovered by te Englis matematician Isaac Newton (6 77) and te German matematician Gottfried Wilelm Leibniz (66 76). For te net century, supporters of eac accused te oter of plagiarism, resulting in a lack of communication between matematicians in England and on te European continent. Neiter Newton nor Leibniz developed a matematically rigorous definition of te it (and we ave no intention of doing so ere). More tan years passed before te Frenc matematician Augustin-Louis Caucy ( ) accomplised tis feat.
4 36 Capter 3 Te Derivative Te definition of a it describes wat appens to ƒ wen is near, but not equal to, te value a. It is not affected by ow (or even weter) ƒa is defined. Also te definition implies tat te function values cannot approac two different numbers, so tat if a it eists, it is unique. Tese ideas are illustrated in te following eamples. caution Note tat te it is a value of y, not. Metod Using a Table Eample Finding a Limit Find g, were g = 3 - S -. Solution Te function g is undefined wen =, since te value = makes te denominator. However, in determining te it as approaces we are concerned only wit te values of g wen is close to but not equal to. To determine if te it eists, consider te value of g at some numbers close to but not equal to, as sown in te following table. approaces from left approaces from rigt g ƒ() approaces ƒ() approaces Undefined Notice tat tis table is almost identical to te previous table, ecept tat g is undefined at =. Tis suggests tat g =, in spite of te fact tat te function g does not eist S at =. Metod Using Algebra A second approac to tis it is to analyze te function. By factoring te numerator, g simplifies to g = = -, =, provided Z. Te grap of g, as sown in Figure, is almost te same as te grap of y =, ecept tat it is undefined at = (illustrated by te ole in te grap). y y = g() Limit is Your Turn - Find S -. Figure Since we are looking at te it as approaces, we look at values of te function for close to but not equal to. Tus, te it is g = S S =. TRY YOUR TURN
5 3. Limits 37 Metod 3 Graping Calculator We can use te TRACE feature on a graping calculator to determine te it. Figure 3 sows te grap of te function in Eample drawn wit a TI-8 Plus C. Notice tat te function as a small gap at te point,, wic agrees wit our previous observation tat te function is undefined at =, were te it is. (Due to te itations of te graping calculator, tis gap may vanis wen te viewing window is canged very sligtly.) Te result after pressing te TRACE key is sown in Figure. Te cursor is already located at = ; if it were not, we could use te rigt or left arrow key to move te cursor tere. Te calculator does not give a y-value because te function is undefined at =. Moving te cursor back a step gives =.98885, y = Moving te cursor forward two steps gives =.555, y = It seems tat as approaces, y approaces, or at least someting close to. Zooming in on te point, (suc as using te window 3.9,. by 33.9,.) allows te it to be estimated more accurately and elps ensure tat te grap as no unepected beavior very close to =. 8 g() 3 8 Y (X 3 X ) (X ) 3 Figure 3 X Y 3 Figure Te Table feature of a graping calculator can also be used to investigate values of te function for values of close to. See Eample. y = () y Limit is (, ) Note tat a it can be found in tree ways:. algebraically;. using a grap (eiter drawn by and or wit a graping calculator); and 3. using a table (eiter written out by and or wit a graping calculator). Wic metod you coose depends on te compleity of te function and te accuracy required by te application. Algebraic simplification gives te eact answer, but it can be difficult or even impossible to use in some situations. Calculating a table of numbers or tracing te grap may be easier wen te function is complicated, but be careful, because te results could be inaccurate, inconclusive, or misleading. A graping calculator does not tell us wat appens between or beyond te points tat are plotted. 3 3 Figure 5 Your Turn 3 Find S3 ƒ if ƒ = e - if Z 3 if = 3. Eample 3 Finding a Limit Determine S for te function defined by = e,, if Z, if =. Solution A function defined by two or more cases is called a piecewise function. Te domain of is all real numbers, and its grap is sown in Figure 5. Notice tat =, but = wen Z. To determine te it as approaces, we are concerned only wit te values of wen is close but not equal to. Once again, = S S =. TRY YOUR TURN 3
6 38 Capter 3 Te Derivative Eample Finding a Limit Find S - ƒ, were ƒ = Solution Te grap of te function is sown in Figure 6. A table wit te values of ƒ as gets closer and closer to - is given below. approaces - from left approaces - from rigt ƒ.5.5.5,.5-9, Teacing Tip: Altoug te it does not eist wen ƒ = or wen Sa ƒ = -, te preferred answer is to Sa write te it equal to or - because tis answer provides us wit more information about ow te function is beaving near = a. 3 + y = + y y = 3 = Figure 6 Your Turn - Find. S Bot te grap and te table suggest tat as approaces - from te left, ƒ becomes larger and larger witout bound. Tis appens because as approaces -, te denominator approaces, wile te numerator approaces -, and - divided by a smaller and smaller number becomes larger and larger. Wen tis occurs, we say tat te it as approaces - from te left is infinity, and we write ƒ =. S- - Because is not a real number, te it in tis case does not eist. In te same way, te beavior of te function as approaces - from te rigt is indicated by writing ƒ = -, S- + since ƒ becomes more and more negative witout bound. Since tere is no real number tat ƒ approaces as approaces - (from eiter side), nor does ƒ approac eiter or -, we simply say 3 + S - + does not eist. TRY YOUR TURN Note In general, if bot te it from te left and from te rigt approac, so tat ƒ =, te it would not eist because is not a real number. It is customary, Sa owever, to give as te answer, since it describes ow te function is beaving near = a. Likewise, if Sa ƒ = -, we give - as te answer.
7 3. Limits 39 Metod Algebraic Approac Eample 5 Find S. Solution Finding a Limit Te function ƒ = / is not defined wen =. Wen 7, te definition of absolute value says tat =, so ƒ = / = / =. Wen 6, ten = - and ƒ = / = -/ = -. Terefore, ƒ = and ƒ =. S + S - Since te its from te left and from te rigt are different, te it does not eist. Metod Graping Calculator Approac A calculator grap of ƒ is sown in Figure 7. As approaces from te rigt, is always positive and te corresponding value of ƒ is, so ƒ =. S + But as approaces from te left, is always negative and te corresponding value of ƒ is -, so ƒ =. S - As in te algebraic approac, te its from te left and from te rigt are different, so te it does not eist. f() Figure 7 Te discussion up to tis point can be summarized as follows. Eistence of Limits Te it of ƒ as approaces a may not eist.. If ƒ becomes infinitely large in magnitude (positive or negative) as approaces te number a from eiter side, we write ƒ = or ƒ = -. In eiter Sa Sa case, te it does not eist.. If ƒ becomes infinitely large in magnitude (positive) as approaces a from one side and infinitely large in magnitude (negative) as approaces a from te oter side, ten ƒ does not eist. Sa 3. If ƒ = L and ƒ = M, and L Z M, ten ƒ does not eist. Sa - Sa + Sa
8 Capter 3 Te Derivative Figure 8 illustrates tese tree facts. f() = 3., S even toug f() =. f() (, 3) 3 f() does not eist. S (, ) 6 y = f() f() = S 6 (6, ) (, ) f() =, even toug S f() is not defined. 5 6 f() does S 6 not eist f() =, so te S 3 it does not eist Figure 8 Rules for Limits As sown by te preceding eamples, tables and graps can be used to find its. However, it is usually more efficient to use te rules for its given below. (Proofs of tese rules require a formal definition of it, wic we ave not given.) Rules for Limits Let a, A, and B be real numbers, and let ƒ and g be functions suc tat ƒ = A and g = B. Sa Sa. If k is a constant, ten Sa k = k and Sa [k # ƒ] = k # Sa ƒ = k # A.. 3ƒ ± g = ƒ ± g = A ± B Sa Sa Sa (Te it of a sum or difference is te sum or difference of te its.) 3. 3ƒ # g = 3 ƒ # 3 g = A # B Sa Sa Sa (Te it of a product is te product of te its.) ƒ ƒ. Sa g = Sa g = Sa A B if B Z (Te it of a quotient is te quotient of te its, provided te it of te denominator is not zero.) 5. If p is a polynomial, ten Sa p = pa. 6. For any real number k, Sa 3 ƒ k = 3 Sa ƒ k = A k, provided tis it eists.* 7. ƒ = g if ƒ = g for all Z a. Sa Sa 8. For any real number b 7, b ƒ = b 3 ƒ Sa = b A. Sa 9. For any real number b suc tat 6 b 6 or 6 b, 3log b ƒ = log b 3 ƒ = log b A if A 7. Sa Sa *Tis it does not eist, for eample, wen A 6 and k = /, or wen A = and k.
9 3. Limits Tis list may seem imposing, but tese it rules, once understood, agree wit common sense. For eample, Rule 3 says tat if ƒ becomes close to A as approaces a, and if g becomes close to B, ten ƒ # g sould become close to A # B, wic seems plausible. Your Turn 5 Find S 3 ƒ + g. Eample 6 Rules for Limits Suppose S ƒ = 3 and S g =. Use te it rules to find te following its. (a) S 3ƒ + 5g Solution 3 ƒ (b) S ln g Solution 3 ƒ + 5g = ƒ + 5g Rule S S S = S ƒ + 5 S g Rule = = 3 3ƒ 3ƒ S ln g = S ln g Rule S = 3 ƒ S ln3 g S = 3 ln Rule 6 and Rule TRY YOUR TURN 5 Eample 7 Find S Solution - - S3 + Finding a Limit = = - - S3 + Rule S3 - - S3 + S3 = = 5 Rule 6 a = a / Rule 5 = 5 As Eamples 6 and 7 suggest, te rules for its actually mean tat many its can be found simply by evaluation. Tis process is valid for polynomials, rational functions, eponential functions, logaritmic functions, and roots and powers, as long as tis does
10 Capter 3 Te Derivative not involve an illegal operation, suc as division by or taking te logaritm of a negative number. Division by presents particular problems tat can often be solved by algebraic simplification, as te following eample sows. Your Turn 6 Find - -. S Eample 8 Finding a Limit Find. S - Solution Rule cannot be used ere, since - =. S Te numerator also approaces as approaces, and / is meaningless. For Z, we can, owever, simplify te function by rewriting te fraction as Now Rule 7 can be used = S = + 3. = S + 3 = + 3 = 5 TRY YOUR TURN 6 Note Matematicians often refer to a it tat gives /, as in Eample 8, as an indeterminate form. Tis means tat wen te numerator and denominator are polynomials, tey must ave a common factor, wic is wy we factored te numerator in Eample 8. Metod Rationalizing te Numerator Eample 9 Finding a Limit - Find S -. Solution As S, te numerator approaces and te denominator also approaces, giving te meaningless epression /. In an epression suc as tis involving square roots, rater tan trying to factor, you may find it simpler to use algebra to rationalize te numerator by multiplying bot te numerator and te denominator by +. Tis gives - # = = - + = + if Z. Now use te rules for its. - S - = S + = + = + = a b a + b = a b Your Turn 7 - Find S -. Metod Factoring Alternatively, we can take advantage of te fact tat - = - = + - because of te factoring a - b = a + ba - b. Ten - S - = - S + - = S + = + = + = ṪRY YOUR TURN 7
11 3. Limits 3 caution Simply because te epression in a it is approacing /, as in Eamples 8 and 9, does not mean tat te it is or tat te it does not eist. For suc a it, try to simplify te epression using te following principle: To calculate te it of ƒ/g as approaces a, were ƒa = ga =, you sould attempt to factor a from bot te numerator and te denominator. Metod Algebraic Approac Eample Finding a Limit - + Find. S - 3 Solution Again, Rule cannot be used, since - 3 =. If Z, te function can be S rewritten as Ten = - + S - 3 by Rule 7. None of te rules can be used to find S = -. -, = S - but as approaces, te denominator approaces wile te numerator stays at, making te result larger and larger in magnitude. If 7, bot te numerator and denominator are positive, so / - =. If 6, te denominator is negative, S + so / - = -. Terefore, S S - 3 = does not eist. S - Metod Graping Calculator Approac Using te TABLE feature on a TI-8 Plus C, we can produce te table of numbers sown in Figure 9, were Y represents te function y = / -. Figure sows a graping calculator view of te function on 3, by 3-,. Te beavior of te function indicates a vertical asymptote at =, wit te it approacing - from te left and from te rigt, so - + S - 3 = does not eist. S - Bot te table and te grap can be easily generated using a spreadseet. Consult te Graping Calculator and Ecel Spreadseet Manual, available wit tis tet, for details. X X Y Figure 9 Figure y
12 Capter 3 Te Derivative Note Anoter way to understand te beavior of te function in te previous eample near = is to recall from Section.3 on Polynomial and Rational Functions tat a rational function often as a vertical asymptote at a value of were te denominator is, altoug it may not if te numerator tere is also. In tis eample, we see after simplifying tat te function as a vertical asymptote at = because tat would make te denominator of / - equal to, wile te numerator is. Limits at Infinity Sometimes it is useful to eamine te beavior of te values of ƒ as gets larger and larger (or more and more negative). Te prase approaces infinity, written S, epresses te fact tat becomes larger witout bound. Similarly, te prase approaces negative infinity (symbolically, S - ) means tat becomes more and more negative witout bound (suc as -, -, -,, etc.). Te net eample illustrates a it at infinity. Eample Oygen Concentration Suppose a small pond normally contains units of dissolved oygen in a fied volume of water. Suppose also tat at time t = a quantity of organic waste is introduced into te pond, wit te oygen concentration t weeks later given by ƒt = t - 5t +. t + As time goes on, wat will be te ultimate concentration of oygen? Will it return to units? Solution After weeks, te pond contains ƒ = # - 5 # + + = 3 5 = 6 units of oygen, and after weeks, it contains ƒ = # - 5 # units. Coosing several values of t and finding te corresponding values of ƒt, or using a graping calculator or computer, leads to te table and grap in Figure. Te grap suggests tat, as time goes on, te oygen level gets closer and closer to te original units. If so, te line y = is a orizontal asymptote. Te table suggests tat ƒt =. ts Tus, te oygen concentration will approac, but it will never be eactly. f(t) t ƒt ,.9985, t 5t + f(t) = t Figure t
13 3. Limits 5 As we saw in te previous eample, its at infinity or negative infinity, if tey eist, correspond to orizontal asymptotes of te grap of te function. In te previous capter, we saw one way to find orizontal asymptotes. We will now sow a more precise way, based upon some simple its at infinity. Te graps of ƒ = / (in red) and g = / (in blue) sown in Figure, as well as te table tere, indicate tat / =, / =, S S - / =, and / =, suggesting te following rule. S S - y g() = f() = Figure Limits at Infinity For any positive real number n, S H n = and S H n =.* For Review In Section.3, we saw a way to find orizontal asymptotes by considering te beavior of te function as (or t) gets large. For large t, t - 5t + t, because te t-term and te constant term are small compared wit te t -term wen t is large. Similarly, t + t. Tus, for large t, ƒt = t - 5t + t + t =. Tus te function ƒ t as a orizontal asymptote at y =. Te rules for its given earlier remain uncanged wen a is replaced wit or -. To evaluate te it at infinity of a rational function, divide te numerator and denominator by te largest power of te variable tat appears in te denominator, t ere, and ten use tese results. In te previous eample, we find tat t - 5t + ts t + = ts t t - 5t + t t t t + t - 5 # + # t t = ts +. t *If is negative, n does not eist for certain values of n, so te second it is undefined for tose values of n.
14 6 Capter 3 Te Derivative Now apply te it rules and te fact tat ts /t n =. ts a - 5 # t + # t b ts a + t b = = - 5 # + # ts ts t ts t + ts ts t - 5a b + a ts H t ts H t b + ts H t = - 5 # + # + Rules and Rule =. Limits at infinity Eample Limits at Infinity Find eac it. (a) S Solution We can use te rule S / n = to find tis it by first dividing te numerator and denominator by, as follows S 3 - = S # = S 3 - = = # 3 + (b) S 3 - = + # 3 S - = = = Here, te igest power of in te denominator is 3, wic is used to divide eac term in te numerator and denominator (c) S - 3 = S - 3 Te igest power of in te denominator is (to te first power). Tere is a iger power of in te numerator, but we don t divide by tis. Notice tat te denominator approaces, wile te numerator becomes infinitely large, so 3 + S - 3 =. (d) S = S
15 3. Limits 7 Your Turn 8 Find S Te igest power of in te denominator is. Te denominator approaces 3, wile te numerator becomes a negative number tat is larger and larger in magnitude, so S = -. TRY YOUR TURN 8 Te metod used in Eample is a useful way to rewrite epressions wit fractions so tat te rules for its at infinity can be used. Teacing Tip: Students will ave te best understanding of its if tey ave studied tem grapically (as in Eercises 5 ), numerically (as in Eercises 5 ), and analytically (as in Eercises 3 5). Finding Limits at Infinity If ƒ = p/q, for polynomials p and q, q Z, can be found as follows.. Divide p and q by te igest power of in q. S-. Use te rules for its, including te rules for its at infinity, S H n = and S H to find te it of te result from Step. n =, For an alternate approac to finding its at infinity, see Eercise 83. ƒ and S ƒ 3. Warm-up Eercises Factor eac of te following epressions. (Sec. R.) W Simplify eac of te following epressions. (Sec. R.3) W / + W W / Eercises In Eercises, coose te best answer for eac it.. If ƒ = 5 and ƒ = 6, ten ƒ (c) S - + S S (a) is 5. (b) is 6. (c) does not eist.. If S - (a) is -. (d) is infinite. ƒ = ƒ = -, but ƒ =, ten ƒ (a) + S S (b) does not eist. (c) is infinite. (d) is. 3. If S - S ƒ = ƒ = 6, but ƒ does not eist, ten + S ƒ (b) (a) does not eist. (b) is 6. (c) is -. (d) is.. If S - (a) is. ƒ = - and ƒ = -, ten ƒ (b) + S S (b) is -. (c) does not eist. (d) is.
16 8 Capter 3 Te Derivative Decide weter eac it eists. If a it eists, estimate its value. 5. (a) S3 ƒ 3 f() 3 F() 3 (b) S ƒ 6. (a) S F (b) S - F. (a) a = (b) a = (i) (ii) (iii) (iv) (i) (ii) (iii) (iv) f() 3 Decide weter eac it eists. If a it eists, find its value.. S ƒ 3 f() (a) S ƒ (b) S ƒ Does not eist f() 3. g (does not eist) S - g() (a) S3 g (b) S5 g Does not eist g() Eplain wy F in Eercise 6 eists, but ƒ in S S- Eercise 9 does not.. In Eercise, wy does ƒ =, even toug ƒ =? S 5. Use te table of values to estimate S ƒ. In Eercises 9 and, use te grap to find (i) ƒ, Sa (ii) ƒ, (iii) ƒ, and (iv) ƒa if it eists. Sa + Sa 9. (a) a = - (b) a = - f() (a) (i) -; (ii) -/ (iii) Does not eist; (iv) Does not eist (b) (i) -/ (ii) -/ (iii) -/ (iv) -/ ƒ Complete te tables and use te results to find te indicated its. 6. If ƒ = - + 7, find S ƒ ƒ 5. 5.
17 3. Limits 9 7. If k = 3 - -, find k. - S k 8. If ƒ = , find ƒ. - + S ƒ 9. If = -, find. Does not eist - S If ƒ = - 3, find ƒ. Does not eist - 3 S ƒ Let ƒ = 9 and g = 7. Use te it rules to find S S eac it.. 3ƒ - g S g # ƒ 3 S ƒ 3. S g /3. 3 ƒ S 5. ƒ S g 3 S 7. ƒ S ƒ S ƒ + g 9. S g /33. 5g + S - ƒ -37/8 Use te properties of its to elp decide weter eac it eists. If a it eists, find its value. 3. S S S S / / S S 3 3. S S /3. S S /. S36-36 /. S /7. S /5 -/ + + / -/938. / S /6. S S 3 / S S S S (does not eist) (does not eist) S S (does not eist) - (does not eist) 53. Let ƒ = e 3 + if Z - 5 if = -. Find ƒ. S - 5. Let g = e Let ƒ = + 3 if = - if Z -. if 6 3 if 3 5. if 7 5 Find g. - S - (a) Find ƒ. S3 (b) Find ƒ. S Let g = - 7 if 6 if 3. if 7 3 Does not eist (a) Find g. (b) Find g. 7 S Does not eist S3 57. Does a value of k eist suc tat te following it eists? 3 + k - S If so, find te value of k and te corresponding it. If not, eplain wy not. -5, Repeat te instructions of Eercise 57 for te following it. + k - 9 S , 9/ In Eercises 59 6, calculate te it in te specified eercise, using a table suc as in Eercises 5. Verify your answer by using a graping calculator to zoom in on te point on te grap. 59. Eercise Eercise 3-6. Eercise Eercise Let F = +. 3 (a) Find F. Does not eist S - (b) Find te vertical asymptote of te grap of F. = - (c) Compare your answers for parts (a) and (b). Wat can you conclude? If = a is a vertical asymptote for te grap of ƒ, ten ƒ does not eist. -6 Sa 6. Let G = -. (a) Find G. - (does not eist) S (b) Find te vertical asymptote of te grap of G. = (c) Compare your answers for parts (a) and (b). Are tey related? How? If = a is a vertical asymptote, ten ƒ does not eist Sa
18 5 Capter 3 Te Derivative 65. Describe ow te beavior of te grap in Figure near = can be predicted by te simplified epression for te function y = / A friend wo is confused about its wonders wy you investigate te value of a function closer and closer to a point, instead of just finding te value of a function at te point. How would you respond? 67. Use a grap of ƒ = e to answer te following questions. (a) Find S - e. (b) Were does te function e ave a orizontal asymptote? y = 68. Use a graping calculator to answer te following questions. (a) From a grap of y = e -, wat do you tink is te value of e -? Support tis by evaluating te function for S several large values of. (b) Repeat part (a), tis time using te grap of y = e -. (c) Based on your results from parts (a) and (b), wat do you tink is te value of n e -, were n is a positive S integer? Support tis by eperimenting wit oter positive integers n. 69. Use a grap of ƒ = ln to answer te following questions. (a) Find ln. - (does not eist) S + (b) Were does te function ln ave a vertical asymptote? = 7. Use a graping calculator to answer te following questions. (a) From a grap of y = ln, wat do you tink is te value of ln? Support tis by evaluating te function for S + several small values of. (b) Repeat part (a), tis time using te grap of y = ln. (c) Based on your results from parts (a) and (b), wat do you tink is te value of ln n, were n is a positive integer? S + Support tis by eperimenting wit oter positive integers n. 7. Eplain in your own words wy te rules for its at infinity sould be true. 7. Eplain in your own words wat Rule for its means. Find eac of te following its (a) by investigating values of te function near te -value were te it is taken, and (b) using a graping calculator to view te function near tat value of S - 5 / S S / or /376. S + / /3 + 5/ /3 + 5/ S 5 S - 5 (a) 8 (a) Eplain wy te following rules can be used to find 3 p/q: S (a) If te degree of p is less tan te degree of q, te it is. (b) If te degree of p is equal to te degree of q, te it is A/B, were A and B are te leading coefficients of p and q, respectively. (c) If te degree of p is greater tan te degree of q, te it is or -. Applications Business and Economics 8. apply IT Consumer Demand Wen te price of an essential commodity (suc as gasoline) rises rapidly, consumption drops slowly at first. If te price continues to rise, owever, a tipping point may be reaced, at wic consumption takes a sudden substantial drop. Suppose te accompanying grap sows te consumption of gasoline, Gt, in millions of gallons, in a certain area. We assume tat te price is rising rapidly. Here t is time in monts after te price began rising. Use te grap to find te following. G(t) (6,.5) (, 3) (6, ) 8 6 t (a) Gt 3 million gallons (b) Gt Does not eist ts ts6 (c) G6 million gallons (d) Te tipping point (in monts) 6 monts 85. Sales Ta Officials in California tend to raise te sales ta in years in wic te state faces a budget deficit and ten cut te ta wen te state as a surplus. Te grap below sows te California state sales ta in recent years. Let T represent te sales ta per dollar spent in year. Find te following. Source: California State. T() 8 Use a graping calculator to grap te function. (a) Determine te it from te grap. (b) Eplain ow your answer could be determined from te epression for ƒ. Ta (in cents) S S - 3 (a) (a).578. S - (a) S (a) Year
19 3. Limits 5 + (a) S9 (c) Cost (in cents) T 7.5 cents (b) T 7.5 cents - S3 (d) T Does not eist T 7.5 cents + S3 (e) T3 7.5 cents C(t) Year S3 86. Postage Te grap below sows ow te postage required to mail a letter in te United States as canged in recent years. Let Ct be te cost to mail a letter in te year t. Find te following. Source: United States Postal Service. (a) (c) Ct 6 cents (b) Ct 9 cents - + ts ts Ct Does not eist (d) C 9 cents ts Average Cost Te cost (in dollars) for manufacturing a particular DVD is $6; te average cost approaces $6 as te number of DVDs becomes very large. C = 5, + 6, were is te number of DVDs produced. Recall from te previous capter tat te average cost per DVD, denoted by C, is found by dividing C by. Find and interpret C. S 88. Average Cost In Capter, we saw tat te cost to fly miles on American Airlines could be approimated by te equation.7; te average cost approaces $.7 per mile as te number of miles becomes very large. C = Recall from te previous capter tat te average cost per mile, denoted by C, is found by dividing C by. Find and interpret S C. Source: American Airlines. 89. Employee Productivity A company training program as determined tat, on te average, a new employee produces Ps items per day after s days of on-te-job training, were Find and interpret ss Ps. Ps = 63s s Preferred Stock In business finance, an annuity is a series of equal payments received at equal intervals for a finite period of time. Te present value of an n-period annuity takes te form - + i-n P = R c d, i were R is te amount of te periodic payment and i is te fied interest rate per period. Many corporations raise money by issuing preferred stock. Holders of te preferred stock, called + indicates more callenging problem items; te number of items a new employee produces gets closer and closer to 63 as te number of days of training increases. + a perpetuity, receive payments tat take te form of an annuity in tat te amount of te payment never canges. However, normally te payments for preferred stock do not end but teoretically continue forever. Find te it of tis present value equation as n approaces infinity to derive a formula for te present value of a sare of preferred stock paying a periodic dividend R. Source: Robert D. Campbell. R/i 9. Growing Annuities For some annuities encountered in business finance, called growing annuities, te amount of te periodic payment is not constant but grows at a constant periodic rate. Leases wit escalation clauses can be eamples of growing annuities. Te present value of a growing annuity takes te form were P = R i - g c - a + g n + i b d, R = amount of te net annuity payment, g = epected constant annuity growt rate, i = required periodic return at te time te annuity is evaluated, n = number of periodic payments. A corporation s common stock may be tougt of as a claim on a growing annuity were te annuity is te company s annual dividend. However, in te case of common stock, tese payments ave no contractual end but teoretically continue forever. Compute te it of te epression above as n approaces infinity to derive te Gordon Sapiro Dividend Model popularly used to estimate te value of common stock. Make te reasonable assumption tat i 7 g. (Hint: Wat appens to a n as n S if 6 a 6?) Source: Robert D. Campbell. R/i - g Life Sciences 9. Alligator Teet Researcers ave developed a matematical model tat can be used to estimate te number of teet Nt at time t (days of incubation) for Alligator mississippiensis, were Nt = 7.8e -8.96e-.685t. Source: Journal of Teoretical Biology. (a) Find N65, te number of teet of an alligator tat atced after 65 days. 65 teet (b) Find Nt and use tis value as an estimate of te ts number of teet of a newborn alligator. (Hint: See Eercise 67.) Does tis estimate differ significantly from te estimate of part (a)? 7 teet 93. Sediment To develop strategies to manage water quality in polluted lakes, biologists must determine te depts of sediments and te rate of sedimentation. It as been determined tat te dept of sediment Dt (in centimeters) wit respect to time (in years before 99) for Lake Coeur d Alene, Idao, can be estimated by te equation Dt = 55 - e -.33t. Source: Matematics Teacer. (a) Find D and interpret. 36. cm; te dept of te sediment layer deposited below te bottom of te lake in 97 is 36. cm. (b) Find Dt and interpret. 55 cm; te dept of te sedi- ts ment approaces 55 cm going back in time.
20 5 Capter 3 Te Derivative 9. Drug Concentration Te concentration of a drug in a patient s bloodstream ours after it was injected is given by Find and interpret S A. Social Sciences A =.7 +. ; te concentration of te drug in te bloodstream approaces as te number of ours after injection increases. 95. Legislative Voting Members of a legislature often must vote repeatedly on te same bill. As time goes on, members may cange teir votes. Suppose tat p is te probability tat an individual legislator favors an issue before te first roll call vote, and suppose tat p is te probability of a cange in position from one vote to te net. Ten te probability tat te legislator will vote yes on te nt roll call is given by p n = + ap - b - pn. For eample, te cance of a yes on te tird roll call vote is p 3 = + ap - b - p3. Source: Matematics in te Beavioral and Social Sciences. Suppose tat tere is a cance of p =.7 tat Congressman Stepens will favor te budget appropriation bill before te first roll call, but only a probability of p =. tat e will cange is mind on te subsequent vote. Find and interpret te following. (a) p.57 (b) p.56 (c) p 8.53 (d) p n ns YOUR TURN ANSWERS Does not eist / 8. /3 95. (d).5; te numbers in (a), (b), and (c) give te probability tat te legislator will vote yes on te second, fourt, and eigt votes. In (d), as te number of roll calls increases, te probability of a yes vote approaces.5 but is never less tan APPLY IT Continuity How does te average cost per day of a rental car cange wit te number of days te car is rented? We will answer tis question in Eercise 38. In 9, Congress passed legislation raising te federal minimum wage for te tird time in tree years. Figure 3 below sows ow tat wage as varied since it was instituted in 938. We will denote tis function by ƒt, were t is te year. Source: U.S. Department of Labor Minimum wage Year Figure 3
21 3. Continuity 53 Teacing Tip: Eplain to students tat, intuitively, a function is continuous at a point if you can draw te grap of te function around tat point witout lifting your pencil from te paper. Notice from te grap tat ƒt =.75 and tat ƒt = 5.5, so tat - + ts997 ts997 ƒt does not eist. Notice also tat ƒ997 = 5.5. A point suc as tis, were a ts997 function as a sudden sarp break, is a point were te function is discontinuous. In tis case, te discontinuity is caused by te jump in te minimum wage from $.75 per our to $5.5 per our in 997. Intuitively speaking, a function is continuous at a point if you can draw te grap of te function in te vicinity of tat point witout lifting your pencil from te paper. As we already mentioned, tis would not be possible in Figure 3 if it were drawn correctly; tere would be a break in te grap at t = 997, for eample. Conversely, a function is discontinuous at any -value were te pencil must be lifted from te paper in order to draw te grap on bot sides of te point. A more precise definition is as follows. Continuity at = c A function ƒ is continuous at = c if te following tree conditions are satisfied:. ƒc is defined,. ƒ eists, and Sc 3. ƒ = ƒc. Sc If ƒ is not continuous at c, it is discontinuous tere. Te following eample sows ow to ceck a function for continuity at a specific point. We use a tree-step test, and if any step of te test fails, te function is not continuous at tat point. Eample Continuity Determine if eac function is continuous at te indicated -value. (a) ƒ in Figure at = 3 Solution Step Does te function eist at = 3? Te open circle on te grap of Figure at te point were = 3 means tat ƒ does not eist at = 3. Since te function does not pass te first test, it is discontinuous at = 3, and tere is no need to proceed to Step. f() () 3 Figure Figure 5 (b) in Figure 5 at = Solution Step Does te function eist at =? According to te grap in Figure 5, eists and is equal to -.
22 5 Capter 3 Te Derivative Step Does te it eist at =? As approaces from te left, is -. As approaces from te rigt, owever, is. In oter words, wile = -, - S =. + S Since no single number is approaced by te values of as approaces, te it does not eist. Since te function does not pass te S second test, it is discontinuous at =, and tere is no need to proceed to Step 3. (c) g in Figure 6 at = Solution Step Is te function defined at =? In Figure 6, te eavy dot above sows tat g is defined. In fact, g =. Step Does te it eist at =? Te grap sows tat S S g = -, and g = Terefore, te it eists at = and g = -. S Step 3 Does g = S g? Using te results of Step and Step, we see tat g Z S g. Since te function does not pass te tird test, it is discontinuous at =. f() g() Figure 6 Figure 7 (d) ƒ in Figure 7 at = -. Solution Step Does te function eist at = -? Te function ƒ graped in Figure 7 is not defined at = -. Since te function does not pass te first test, it is discontinuous at = -. (Function ƒ is continuous at any value of greater tan, owever.) Notice tat te function in part (a) of Eample could be made continuous simply by defining ƒ3 =. Similarly, te function in part (c) could be made continuous by
23 3. Continuity 55 y Figure 8 redefining g = -. In suc cases, wen te function can be made continuous at a specific point simply by defining or redefining it at tat point, te function is said to ave a removable discontinuity. A function is said to be continuous on an open interval if it is continuous at every -value in te interval. Continuity on a closed interval is sligtly more complicated because we must decide wat to do wit te endpoints. We will say tat a function ƒ is continuous from te rigt at = c if ƒ = ƒc. A function ƒ is continuous from te left at = c Sc + if ƒ = ƒc. Wit tese ideas, we can now define continuity on a closed interval. Sc - Continuity on a Closed Interval A function is continuous on a closed interval 3a, b if. it is continuous on te open interval a, b,. it is continuous from te rigt at = a, and 3. it is continuous from te left at = b. For eample, te function ƒ = -, sown in Figure 8, is continuous on te closed interval 3-,. By defining continuity on a closed interval in tis way, we need not worry about te fact tat - does not eist to te left of = - or to te rigt of =. Te table below lists some key functions and tells were eac is continuous. Continuous Functions Type of Function Were It Is Continuous Grapic Eample Polynomial Function y = a n n + a n - n - + g + a + a, were a n, a n -,..., a, a are real numbers, not all For all y Rational Function y = p, were p and q q are polynomials, wit q Z For all were q Z y Root Function y = a + b, were a and b are real numbers, wit a Z and a + b Ú For all were a + b Ú y (continued)
24 56 Capter 3 Te Derivative Continuous Functions (cont.) Type of Function Were It Is Continuous Grapic Eample Eponential Function y = a were a 7 For all y Logaritmic Function y = log a were a 7, a Z For all 7 y Continuous functions are nice to work wit because finding Sc ƒ is simple if ƒ is continuous: just evaluate ƒc. Wen a function is given by a grap, any discontinuities are clearly visible. Wen a function is given by a formula, it is usually continuous at all -values ecept tose were te function is undefined or possibly were tere is a cange in te defining formula for te function, as sown in te following eamples. Your Turn Find all values = a were te function is discontinuous. ƒ = Eample Continuity Find all values = a were te function is discontinuous. (a) ƒ = Solution Tis rational function is discontinuous werever te denominator is zero. Tere is a discontinuity wen a = 7/. (b) g = e - 3 Solution Tis eponential function is continuous for all. TRY YOUR TURN Eample 3 Continuity Find all values of were te following piecewise function is discontinuous. ƒ = c if 6 if 3. if 7 3 Solution Since eac piece of tis function is a polynomial, te only -values were ƒ migt be discontinuous ere are and 3. We investigate at = first. From te left, were -values are less tan, S ƒ = = + =. S
25 3. Continuity 57 From te rigt, were -values are greater tan, ƒ = = =. + S S + Furtermore, ƒ = =, so ƒ = ƒ =. Tus ƒ is continuous at S =, since ƒ = ƒ. S Now let us investigate = 3. From te left, From te rigt, ƒ = = =. - S3 S3 - Your Turn Find all values of were te piecewise function is discontinuous. 5 - ƒ = c + 6 if 6 if 3 if 7 3 Because S3 - S3 ƒ = = 5-3 =. S3 ƒ Z ƒ, te it ƒ does not eist, so ƒ is discontinuous at + S3 S3 = 3, regardless of te value of ƒ3. Te grap of ƒ can be drawn by considering eac of te tree parts separately. In te first part, te line y = + is drawn including only te section of te line to te left of =. Te oter two parts are drawn similarly, as illustrated in Figure 9. We can see by te grap tat te function is continuous at = and discontinuous at = 3, wic confirms our solution above. TRY YOUR TURN y 3 3 Figure 9 Tecnology Note Some graping calculators ave te ability to draw piecewise functions. On te TI-8 Plus C, letting Y = (X + )(X 6 ) + (X - 3X + )( X)(X 3) + (5 - X)(X 7 3) produces te grap sown in Figure. 6 Figure
26 58 Capter 3 Te Derivative C() Eample Cost Analysis Figure A trailer rental firm carges a flat $8 to rent a itc. Te trailer itself is rented for $ per day or fraction of a day. Let C represent te cost of renting a itc and trailer for days. (a) Grap C. Solution Te carge for one day is $8 for te itc and $ for te trailer, or $3. In fact, if 6, ten C = 3. To rent te trailer for more tan one day, but not more tan two days, te carge is 8 + # = 5 dollars. For any value of satisfying 6, te cost is C = 5. Also, if 6 3, ten C = 7. Tese results lead to te grap in Figure. (b) Find any values of were C is discontinuous. Solution As te grap suggests, C is discontinuous at =,, 3,, and all oter positive integers. One application of continuity is te Intermediate Value Teorem, wic says tat if a function is continuous on a closed interval [a, b], te function takes on every value between ƒa and ƒb. For eample, if ƒ = -3 and ƒ = 5, ten ƒ must take on every value between -3 and 5 as varies over te interval [, ]. In particular (in tis case), tere must be a value of in te interval, suc tat ƒ =. If ƒ were discontinuous, owever, tis conclusion would not necessarily be true. Tis is important because, if we are searcing for a solution to ƒ = in [, ], we would like to know tat a solution eists. 3. Warm-up Eercises Find eac of te following its. (Sec. 3.) W. S W. S if * 3 Let ƒ = if 3 " " 5. Find eac of te following its. (Sec. 3.) + if + 5 W3. ƒ W. S3 ƒ Does not eistw5. S5 ƒ 5 S6 3. Eercises In Eercises 6, find all values = a were te function is discontinuous. For eac point of discontinuity, give (a) ƒa if it eists, (b) ƒ, (c) ƒ, (d) ƒ, and (e) identify Sa - + Sa Sa wic conditions for continuity are not met. Be sure to note wen te it doesn t eist.. f() a = -: (a) ƒ- does not eist. (b) / (c) / (d) / (e) ƒ- does not eist.. f() a = -: (a) (b) (c) (d) Does not eist. (e) Limit does not eist.
27 3. Continuity f() a = : (a) (b) - (c) - (d) - (e) ƒ does not equal te it. 5. k = e - 6. j = e / a 6, it does not eist. a =, it does not eist. 7. r = ln ` - ` 8. j = ln ` ` 7. a =, - (it does not eist); a =, (it does not eist). 8. a = -, - (it does not eist); a = 3, (it does not eist). In Eercises 9, (a) grap te given function, (b) find all values of were te function is discontinuous, and (c) find te it from te left and from te rigt at any values of found in part (b).. f() f() 5 6. f() a = -: (a) (b) - (c) - (d) - (e) ƒ- does not equal te it; a = 3: (a) (b) - (c) - (d) - (e) ƒ3 does not equal te it. a = -5: (a) ƒ-5 does not eist (b) (does not eist) (c) - (does not eist) (d) Limit does not eist. (e) ƒ-5 does not eist and te it does not eist; a = : (a) ƒ does not eist. (b) (c) (d) (e) ƒ does not eist. a = : (a) ƒ does not eist. (b) - (does not eist) (c) - (does not eist) (d) - (does not eist) (e) ƒ does not eist and te it does not eist; a = : (a) ƒ does not eist. (b) - (c) - (d) - (e) ƒ does not eist. 9. ƒ = c ƒ = c -. g = c +. g = c = b = b if 6 if if 7 if 6 if if 7 if 6 - if - 3 if 7 3 if 6 if 5 if 7 5 if if 7 if if 7 (a) * (b) (c), 5 (a) * (b) (c), (a) * (b) (c), 3 (a) * (b) 5 (c), 5 (a) * (b) None (a) * (b) (c) -, In Eercises 5 8, find te value of te constant k tat makes te function continuous. 5. ƒ = b k + k 6. g = b 3 + k k - 5 if if 7 / g = c - 3 k - if 3 if if Z 3 if = 3 Find all values = a were te function is discontinuous. For eac value of, give te it of te function as approaces a. Be sure to note wen te it doesn t eist. 7. ƒ = ƒ = 3. p = a =, it does not eist; a =, it does not eist. a = -, it does not eist. a = -, it does not eist; a = -/, it does not eist. 9. ƒ = -. ƒ = a =, it. p = is a = -5, it is Nowere. q = Nowere. r = a = 5, it does not eist if Z - 8. = c k if = - 9. Eplain in your own words wat te Intermediate Value Teorem says and wy it seems plausible. 3. Eplain wy can be evaluated by substituting S =. In Eercises 3 3, (a) use a graping calculator to tell were te rational function P/Q is discontinuous, and (b) verify your answer from part (a) by using te graping calculator to plot Q and determine were Q =. You will need to coose te viewing window carefully ƒ = * indicates answer is in te Additional Instructor Answers at end of te book. + indicates more callenging problem. Discontinuous at =.
28 6 Capter 3 Te Derivative ƒ = Discontinuous at = Let g =. Determine all values of at wic g is discontinuous, and for eac of tese values of, define g in suc a manner so as to remove te discontinuity, if possible. Coose one of te following. Source: Society of Actuaries. (a) (a) g is discontinuous only at - and. Define g- = - 6 to make g continuous at -. g cannot be defined to make g continuous at. (b) g is discontinuous only at - and. Define g- = - 6 to make g continuous at -. Define g = 6 to make g continuous at. (c) g is discontinuous only at - and. g- cannot be defined to make g continuous at -. g() cannot be defined to make g continuous at. (d) g is discontinuous only at. Define g = 6 to make g continuous at. (e) g is discontinuous only at. g cannot be defined to make g continuous at. 3. Tell at wat values of te function ƒ in Figure 8 from te previous section is discontinuous. Eplain wy it is discontinuous at eac of tese values. Discontinuous at = -6, -, and 3 because te it does not eist tere. Also, te function is not defined at = -6 and = 3. Discontinuous at = because te function is not defined tere. Discontinuous at = because ƒ Z ƒ. Applications Business and Economics 35. Production Te grap sows te profit from te daily production of tousand kilograms of an industrial cemical. Use te grap to find te following its. (a) P $5 S6 (c) P $ S + (b) P $5 S - (d) P Does not eist S (e) Were is te function discontinuous? Wat migt account for suc a discontinuity? Discontinuous at = ; a cange in sifts (f) Use te grap to estimate te number of units of te cemical tat must be produced before te second sift is as profitable as te first. 5 Profit (in dollars) S P() 5 5 First sift (, 5) (, ) Second sift Number of units (tousands of kilograms) 36. Cost Analysis Te cost to transport a mobile ome depends on te distance,, in miles tat te ome is moved. Let C represent te cost to move a mobile ome miles. One firm carges as follows. Cost per Mile Distance in Miles $. 6 5 $ $.5 6 Find te cost to move a mobile ome te following distances. (a) 3 miles (b) 5 miles (c) miles $5 $6 $63 (d) miles (e) 5 miles $ $5 (f) Were is C discontinuous? At 5 and miles 37. Cost Analysis A company carges $.5 per lb for a certain fertilizer on all orders lb or less, and $ per lb for orders over lb. Let F represent te cost for buying lb of te fertilizer. Find te cost of buying te following. (a) 8 lb $ (b) 5 lb $5 (c) lb $5 (d) Were is F discontinuous? At = 38. apply IT Car Rental Recently, a car rental firm carged $36 per day or portion of a day to rent a car for a period of to 5 days. Days 6 and 7 were ten free, wile te carge for days 8 troug was again $36 per day. Let At represent te average cost to rent te car for t days, were 6 t. Find te average cost of a rental for te following number of days. (a) (b) 5 (c) 6 (d) 7 (e) 8 $36 $36 $3 $5.7 $7 (f) Find At. 36 (g) Find At. 36 S5 - + S5 () Were is A discontinuous on te given interval? t =,, 3,, 7, 8, 9,, 39. Postage In, it cost $.98 to send a large envelope witin te United States for te first ounce, and $. for eac additional ounce, or fraction tereof, up to 3 ounces. Let C be te cost to mail ounces. Find te following. Source: U.S. Postal Service. (a) C Does not (b) S3 - S3 +C (c) C eist $. $.6 S3 (d) C3 $. (e) S8.5-C $.66 (f) S8.5 +C (g) C () C8.5 $.66 $.66 $.66 S8.5 (i) Find all values on te interval,3 were te function C is discontinuous.,, 3,..., Life Sciences. Pregnancy A woman s weigt naturally increases during te course of a pregnancy. Wen se delivers, er weigt immediately decreases by te approimate weigt of te cild. Suppose tat a -lb woman gains 7 lb during pregnancy, delivers a 7-lb baby, and ten, troug diet and eercise, loses te remaining weigt during te net weeks. (a) Grap te weigt gain and loss during te pregnancy and te weeks following te birt of te baby. Assume tat te pregnancy lasts weeks, tat delivery occurs immediately after tis time interval, and tat te weigt gain/loss before and after birt is linear. * (b) Is tis a continuous function? If not, ten find te value(s) of t were te function is discontinuous. No; weeks