1.5 Functions and Their Rates of Change

Transcription

1 66_cpp-75.qd /6/8 4:8 PM Page CHAPTER Introduction to Functions and Graps.5 Functions and Teir Rates of Cange Identif were a function is increasing or decreasing Use interval notation Use and interpret average rate of cange Calculate te difference quotient Introduction Sales of rock music ave not remained constant during te past two decades. In 99, rock music accounted for 6% of all U.S. music sales. Tis percentage decreased to a low of 4% in and ten increased to 4% in 6. (Source: Recording Industr Association of America.) A linear function cannot be used to describe tese data because te grap of a (nonconstant) linear function eiter alwas rises or alwas falls. Te concepts of increasing and decreasing are important to functions and teir rates of cange. Increasing and Decreasing Functions Figure.8 sows sales of rock music modeled b a continuous nonlinear function ƒ. For eample, ƒ(99) = 6 indicates tat rock music accounted for 6% of all music sales in 99. Rock music sales decreased from 99 to and ten increased from to 6. Matematicall, we sa tat function ƒ decreases for 99 and increases for 6. 5 Sales (percentage) 4 (99, 6) (6, 4) = f() (, 4) Year Figure.8 U.S. Rock Music Sales NOTE Te inequalit a b means tat Ú a and b. Te concepts of increasing and decreasing are defined as follows. Increasing and Decreasing Functions Suppose tat a function ƒ is defined over an interval I on te number line. If and are in I, (a) ƒ increases on I if, wenever 6, ƒ( ) 6 ƒ( ); (b) ƒ decreases on I if, wenever 6, ƒ( ) 7 ƒ( ). Figures.84 and.85 illustrate tese concepts at te top of te net page. Te concepts of increasing and decreasing relate to weter te grap of a function rises or falls. If we could walk from left to rigt along te grap of an increasing function, it would be upill. For a decreasing function, we would walk downill. We speak of a function ƒ increasing or decreasing over an interval of its domain. For eample, in Figure.86 te function is decreasing (te grap falls) wen - and increasing (te grap rises) wen.

2 66_cpp-75.qd /6/8 4:8 PM Page 57.5 Functions and Teir Rates of Cange 57 f ( ) f ( ) f( ) f( ) Wen <, f( ) < f( ), f is increasing. Figure.84 Wen <, f( ) > f( ), f is decreasing. Figure.85 Figure.86 EXAMPLE Recognizing increasing and decreasing graps Te graps of tree functions are sown in Figure.87. Determine intervals were eac function is increasing or decreasing. f() = f() = f() = (a) Figure.87 (b) (c) (a) Moving from left to rigt, te grap of ƒ() = is decreasing for and increasing for Ú. (b) Moving from left to rigt, te grap of ƒ() = is increasing for all real numbers. Note tat te -values alwas increase as te -values increase. (c) Te grap of ƒ() = is increasing for Ú. Now Tr Eercises 7, 8, and 9 NOTE Wen stating were a function is increasing and were it is decreasing, it is important to give -intervals and not -intervals Figure.88 7 Interval Notation To describe intervals were functions are increasing or decreasing, a number line grap is sometimes used. Te set { 7 }, wic includes all real numbers greater tan, is graped in Figure.88. Note tat a parentesis at = indicates tat te endpoint is not included. Te set { - 4} is sown in Figure.89 on te net page, and te set E F is sown in Figure.9 on te net page. Note tat brackets, eiter [ or ], are used wen endpoints are included.

3 66_cpp-75.qd /6/8 4:8 PM Page CHAPTER Introduction to Functions and Graps Figure Figure A convenient notation for number line graps is called interval notation. Instead of drawing te entire number line, as in Figure.89, we can epress te set as [-, 4]. Because te set includes te endpoints - and 4, te interval is a closed interval and brackets are used. A set tat included all real numbers satisfing - 7 would be epressed as te open interval A- 7, - B Parenteses indicate tat te endpoints are not included in te set. An eample of a alf-open interval is [, 4), wic represents te interval 6 4. Table.5 provides some eamples of interval notation. Te smbol q refers to infinit; it does not represent a real number. Te notation (, q) means { 7 }, or simpl 7. Since tis interval as no maimum -value, q is used in te position of te rigt endpoint. A similar interpretation olds for te smbol - q, wic represents negative infinit. NOTE An inequalit in te form 6 or 7 indicates te set of real numbers tat are eiter less tan or greater tan. Te union smbol can be used to write tis inequalit in interval notation as (- q, ) (, q). Table.5 Interval Notation Inequalit Interval Notation Grap -<< (-, ) -< open interval (-, ] - alf-open interval [-, ] >- closed interval (-, q) infinite interval (- q, ] infinite interval - or > (- q, -] (, q) infinite intervals - q 6 6 q (- q, q) (entire number line) infinite interval MAKING CONNECTIONS Points and Intervals Te epression (, 5) as two possible meanings. It ma represent te ordered pair (, 5), wic can be plotted as a point on te -plane, or it ma represent te open interval To alleviate confusion, prases like te point (, 5) or te interval (, 5) ma be used.

4 66_cpp-75.qd /6/8 4:8 PM Page 59.5 Functions and Teir Rates of Cange 59 Increasing, Decreasing, and Endpoints Tere can be confusion as to weter to include te endpoints of an interval wen stating were a function is increasing or decreasing. For eample, is te grap of ƒ() =, sown in Figure.9, increasing on [, q) or just on (, q)? Te definition of increasing and decreasing allows us to include as part of te interval I were ƒ is increasing, because if we let =, ten ƒ() 6 ƒ( ) wenever 6 Tus ƒ() = is increasing on Similarl, we can sow tat ƒ() =. [, q). is decreasing on (- q, ] b letting =. Do not confuse tese concepts b saing tat function ƒ bot increases and decreases at te point (, ). Te concepts of increasing and decreasing appl onl to intervals of te real number line and not to individual points. Decreasing: (, ] f () = Increasing: [, ) Note tat it is not incorrect to sa tat ƒ() = is increasing on ( q, ) because ƒ() = also increases on [, q), (, q), and [, ]. However, we generall give te largest interval possible, [, q), wen stating were a function is increasing or decreasing. (Reference: J. Stewart, Essential Calculus, 7, p. 8.) NOTE Some definitions of increasing and decreasing functions require tat te -interval be an open interval. An open interval does not include te endpoints. For eample, wit tese definitions, te grap of ƒ() = is decreasing for (- q, ), not (- q, ], and increasing for (, q), not [, q). EXAMPLE Figure.9 Determining were a function is increasing or decreasing Use te grap of ƒ() = 4 - (sown in Figure.9) and interval notation to identif were ƒ is increasing or decreasing. 4 f() = Figure.9 Moving from left to rigt on te grap of ƒ, te -values decrease until = -, increase until =, and decrease tereafter. Tus ƒ() = 4 - is decreasing on (- q, -], increasing on [-, ], and decreasing again on [, q). In interval notation ƒ is decreasing on (- q, -] [, q). Now Tr Eercise

5 66_cpp-75.qd /6/8 4:8 PM Page 6 6 CHAPTER Introduction to Functions and Graps Average Rate of Cange A nonlinear function can increase on one interval of its domain and decrease on anoter interval of its domain. Te graps of nonlinear functions are not lines, so tere is no notion of a single slope. See Eample. Te slope of te grap of a linear function gives its rate of cange. Wit a nonlinear function we speak of an average rate of cange. Suppose tat te points (, ) and (, ) lie on te grap of a nonlinear function ƒ. See Figure.9. Te slope of te line L passing troug tese two points represents te average rate of cange of ƒ from to. Te line L is referred to as a secant line. If different values for and are selected, ten a different secant line and a different average rate of cange usuall result. = f () L (, ) Distance (miles) m = 8 (.5, ) (, ) = f () (, ) Time (ours) Figure.9 Figure.94 In applications te average rate of cange measures ow fast a quantit is canging over an interval of its domain, on average. For eample, suppose te grap of te function ƒ in Figure.94 represents te distance in miles tat a car as traveled on a straigt igwa (under construction) after ours. Te points (.5, ) and (, ) lie on tis grap. Tus after.5 our te car as traveled miles and after our te car as traveled miles. Te slope of te line passing troug tese two points is m = - = = 8 Tis means tat during te alf our from.5 to our te average rate of cange, or average velocit, was 8 miles per our. Tese ideas lead to te following definition. Average Rate of Cange Let (, ) and (, ) be distinct points on te grap of a function ƒ. Te average rate of cange of ƒ from to is -. - Tat is, te average rate of cange from to equals te slope of te line passing troug (, ) and (, ). NOTE If, ten average rate of cange equals ƒ( ) ƒ( ) = ƒ().

6 66_cpp-75.qd /6/8 4:8 PM Page 6.5 Functions and Teir Rates of Cange 6 If ƒ is a constant function, its average rate of cange is. For a linear function defined b ƒ() = a + b, te average rate of cange is a, te slope of its grap. Te average rate of cange for a nonlinear function varies. Table.6 Year Population EXAMPLE Calculating and interpreting average rates of cange Table.6 lists te U.S. population in millions for selected ears. (a) Calculate te average rates of cange in te U.S. population from 8 to 84 and from 9 to 94. Interpret te results. (b) Illustrate our results from part (a) grapicall. (a) In 8 te population was 5 million, and in 84 it was 7 million. Terefore te average rate of cange in te population from 8 to 84 was =.. Population (millions) 5 m =.4 5 L 5 m =. L Year Figure.95 In 9 te population was 76 million, and in 94 it was million. Terefore te average rate of cange in te population from 9 to 94 was =.4. Tis means tat from 8 to 84 te U.S. population increased, on average, b. million per ear and from 9 to 94 te U.S. population increased, on average, b.4 million per ear. (b) Tese average rates of cange can be illustrated grapicall b sketcing a line L troug te points (8, 5) and (84, 7) and anoter line L troug te points (9, 76) and (94, ), as depicted in Figure.95. Te slope of L is. and te slope of L is.4. Now Tr Eercises 5 and 59 EXAMPLE 4 Modeling braking distance for a car Higwa engineers sometimes estimate te braking distance in feet for a car traveling at miles per our on wet, level pavement b using te formula ƒ() = 9. (Source: L. Haefner, Introduction to Transportation Sstems.) (a) Evaluate ƒ() and ƒ(6). Interpret tese results. (b) Calculate te average rate of cange of ƒ from to 6. Interpret tis result. (a) ƒ() = 9 () = and ƒ(6) = 9 (6) = 4. At miles per our te braking distance is feet, and at 6 miles per our te braking distance is 4 feet. (b) Te average rate of cange of ƒ from to 6 is =. Braking distance increases, on average, b feet for eac -mile-per-our increase in speed between and 6 miles per our. Now Tr Eercise 6

7 66_cpp-75.qd /6/8 4:8 PM Page 6 6 CHAPTER Introduction to Functions and Graps f( + ) f() Figure.96 = f () (, f()) ( +, f( + )) + L Te Difference Quotient Te difference quotient often occurs in calculus. It uses function notation to calculate te average rate of cange of a function ƒ in general. Consider te grap of = ƒ() sown in Figure.96, and let be a real number. Te points (, ƒ() ) and ( +, ƒ( + ) ) denote te coordinates of two points on tis grap. (Figure.96 sows 7.) Te line L passes troug tese two points and is a secant line. Te slope m of L is m = ƒ( + ) - ƒ() ( + ) - Tis epression, written in function notation, is called te difference quotient and is equal to te average rate of cange of ƒ from to +. = ƒ( + ) - ƒ(). Difference Quotient Te difference quotient of a function ƒ is an epression of te form ƒ( + ) - ƒ(), were Z. EXAMPLE 5 Calculating a difference quotient Let ƒ() = -. (a) Find ƒ( + ). (b) Find te difference quotient of ƒ and simplif te result. Algebra Review To square a binomial, see Capter R (page R-8). (a) To calculate ƒ( + ), substitute ( + ) for in te epression -. ƒ( + ) = ( + ) - ( + ) (b) Te difference quotient can be calculated as follows. ƒ( + ) - ƒ() = = ( - ) = + - = ( + - ) = + - ƒ() = - Square te binomial; appl te distributive propert. Substitute. Combine like terms. Factor out. Simplif. Now Tr Eercise 75 EXAMPLE 6 Calculating a difference quotient Let te distance d in feet tat a raceorse runs in t seconds be d(t) = t for t. (a) Find d(t + ). (b) Find te difference quotient of d and simplif te result.

8 66_cpp-75.qd /6/8 4:8 PM Page 6.5 Functions and Teir Rates of Cange 6 (c) Evaluate te difference quotient for t = 7 and =.. Interpret our results. (d) Evaluate te difference quotient for t = 4 and =. Ten sketc a grap tat illustrates te result. (a) To calculate d(t + ), substitute (t + ) for t in te epression t. d(t + ) = (t + ) Substitute (t + ) for t. = (t + t + ) Square te binomial. = t + 4t + Distributive propert Distance (feet) d (, ) d(t) = t 5 m = 8 5 (4, ) (5, 5) t Time (seconds) Figure.97 (b) d(t + ) - d(t) = t + 4t + - t Substitute for d(t + ) and d(t). 4t + = Combine like terms. (4t + ) = Factor out. = 4t + Simplif. (c) If t = 7 and =., ten te difference quotient becomes 4t + = 4(7) + (.) = 8.. Te average rate of cange, or average velocit, of te orse from 7 seconds to 7 +. = 7. seconds is 8. feet per second. (d) If t = 4 and =, ten 4t + = 4(4) + () = 8. Tus te slope of te line passing troug (4, ) and (5, 5) is m = 8. See Figure.97. NOTE Because t = 4 and d = (4) =, te first point is (4, ). Because t = 4 and =, it follows tat t + = 4 + = 5 so d = (5) = 5. Te second point is (5, 5). Now Tr Eercise 8.5 Putting It All Togeter Te following table summarizes te basic concepts in tis section. Concept Eplanation Eamples Interval notation An efficient notation for writing inequalities 6 is equivalent to (- q, 6]. 7 is equivalent to (, q). 6 5 is equivalent to (, 5]. continued on net page

9 66_cpp-75.qd /5/8 : PM Page CHAPTER Introduction to Functions and Graps continued from previous page Concept Eplanation Eamples Increasing and decreasing ƒ increases on an interval if, wenever 6, ten ƒ( ) 6 ƒ( ). ƒ decreases on an interval if, wenever 6, ten ƒ( ) 7 ƒ( ). 4 4 ƒ is increasing on (- q, -] [, q). ƒ is decreasing on [-, ]. Average rate of cange of ƒ from to Difference quotient If (, ) and (, ) are distinct points If ƒ() =, ten te average rate of cange from on te grap of ƒ, ten te average rate of = to = is cange from to equals = - -. because ƒ() = 7 and ƒ() =. Tis means tat, on average, ƒ() increases b units for eac unit increase in wen. Calculates average rate of cange of ƒ If ƒ() =, ten te difference quotient equals from to +. ( + ) - ƒ( + ) - ƒ() =, Z =. Interval Notation Eercises 6: Epress eac of te following in interval notation.. Ú { -} 7. { - } Ú - 9. { 6 or Ú }. { - or Ú }..5 Eercises