11.1 Average Rate of Change

Transcription

1 11.1 Average Rate of Cange Question 1: How do you calculate te average rate of cange from a table? Question : How do you calculate te average rate of cange from a function? In tis section, we ll examine ow quantities cange wit respect to eac oter. Tis is a topic tat is not entirely unfamiliar. On a long trip in a car, you may be interested in knowing ow te distance traveled canges wit respect to ow muc time as elapsed. By knowing ow many miles per our your veicle as averaged, you get an idea of ow long it will take you to arrive at your destination. Auto manufacturers are also interested in ow far a veicle will travel, but as te amount of gasoline tat is in te tank canges. By comparing te distance traveled to te amount of gasoline consumed, tey get an idea of ow efficient te veicle is. Veicles tat acieve a iger miles per gallon are more efficient tan tose tat acieve a lower miles per gallon. An insurance analyst migt be interested in knowing ow te percentage of people wo are driving uninsured varies as te percentage of people wo are unemployed canges. By understanding ow tese percentages vary wit respect to eac oter, tey can better understand te risk of being in an accident wit a person wo is uninsured. 1

2 Question 1: How do you calculate te average rate of cange from a table? We quantify ow one quantity canges wit respect to anoter using te average rate of cange. Average Rate of Cange Te average rate of cange of f wit respect to x from x a to x b is defined as Average rate of cange of f Cange in f wit respect to x over a, b Cange in x Since te numerator and denominator eac contain a difference of two values, te quotient or te rigt side of te definition is often called a difference quotient. Te variable x describes one of te quantities we are interested in comparing and te variable f describes te oter quantity. In general, one of te quantities is tougt to depend on te oter. Te quantity tat depends on te oter corresponds to te dependent variable f and te variable x, te independent variable, corresponds to te oter. Often te coice of wic quantity is wic is not very clear cut. In many of tose cases, we can use te units on te comparison to determine ow te average rate of cange sould be computed. For instance, suppose te auto manufacturer is interested in te miles per gallon tat its veicles acieve. Tis means tey wis to calculate ow te miles cange wit respect to te cange in te gallons in te tank. In tis case, tey would tink of tis average rate of cange as Average rate of cange of distance wit respect to a cange in gallons over ab, Cange in distance Cange in gallons

3 Te numerator of tis difference quotient is in miles and te denominator is in gallons yielding units of miles per gallon on te average rate of cange. In tis case, we would tink of te distance traveled as being dependent on te number of gallons in te tank of te automobile. A general guideline to use is wen we discuss te average rate of cange of quantity f wit respect to a quantity x, te cange in te quantity f is in te numerator of te difference quotient. Te cange in te quantity x is in te denominator of te difference quotient. Te average rate of cange can be a positive number or a negative number. If te average rate of cange is a positive number, te quantity corresponding to te numerator of te difference quotient increases as te quantity in te denominator increases. On te oter and, if te quantity in te numerator decreases as te quantity in te denominator increases, ten te average rate of cange is negative. Te average rate of cange can be viewed in many different ways. To make it as simple as possible, it is useful to work wit te definition Average rate of cange of f Cange in f wit respect to x over a, b Cange in x in all situations, and ten adapt tis definition to specific applications. Keep in mind tat te numerator and denominator eac describe canges in some quantities. Te units on te average rate of cange correspond to te units on te quantities in te numerator and denominator. 3

4 Example 1 Find te Average Rate of Cange from a Table Te table below defines te relationsip y f( x). x f ( x ) Use tis table to compute te average rates of cange below. a. Find te average rate of cange of f wit respect to x over 0, 4. Solution Apply te definition of te average rate of cange to give Average rate of cange of f Cange in f wit respect to x over 0,4 Cange in x f(4) f(0) Find te cange in te numerator and denominator over [0, 4] Put in te function values from te table b. Find te average rate of cange of f( x ) from x to x 5. Solution In tis part, te interval is defined wit sligtly different prasing. By saying, from x to x 5, te interval over wic te average rate of cange is being found is being defined to be,5. Using te definition for average rate of cange yields 4

5 Average rate of cange of wit respect to x over,5 f Cange in f Cange in x f(5) f() Find te cange in te numerator and denominator over [, 5] Put in te function values from te table Note tat wat you are doing is calculating te slope between te ordered pairs,5 and 5, 6. In bot parts, te average rate of cange is negative indicating tat f ( x ) decreases as x increases. Example Find te Average Rate of Cange from a Table Te average price for a ticket to a movie teater in Nort America for selected years is sown in te table below. Year Price ($) (Source: National Association of Teater Owners, In eac part, calculate te indicated average rate of cange. a. Find te average rate of cange of ticket price wit respect to time over te period 1987 to Solution Use te definition of average rate of cange to write as Average rate of cange of P Cange in P wit respect to t over 1987,1999 Cange in t 5

6 From te table we know te price of a ticket in 1987 was $3.91 and te price of a ticket in 1999 was $5.06. Te average rate of cange of P wit respect to t from t 1987 to t 1999 is Average rate of cange of P wit Cange in P respect to t over 1987,1999 Cange in t Te numerator of tis quotient is a difference in prices and corresponds to a cange of 1.15 dollars. Te denominator is a difference in years and corresponds to a cange of 1 years. Te difference quotient is Average rate of cange of P wit 1.15 dollars respect to t over 1987, years If we round tis average rate of cange to te nearest cent, we get approximately 0.10 dollars per year. Tis tells us tat eac year from 1987 troug 1999, te ticket prices rose by an average of about 0.10 dollars or 10 cents. b. Find te average rate of cange of ticket price wit respect to time over te period 1999 to 009. Solution Te price of a ticket in 1999 was $5.06 and $7.50 in 009. Te averate rate of cange of P wit respect to t from t 1999 to t 009 is Average rate of cange of P wit Cange in P respect to t over 1999, 009 Cange in t dollars 10 years 0.4 dollars per year 6

7 Tis means tat ticket prices increased by an average of approximately 0.4 dollars or 4 cents eac year from 1999 to 009. c. Were ticket prices increasing faster during te period from 1987 to 1999 or during te period 1999 to 009? Solution Te time period wit te greater average rate of cange corresponds to te period in wic te prices are increasing faster. Since te average rate of cange of price from 1999 to 009 is approximately 0.4 dollars per year and te average rate of cange of price from 1987 to 1999 is approximately 0.10 dollars, prices are rising faster from 1999 to 009. Te numerator and denominator of te difference quotient is often symbolized using te greek symbol capital delta,. For te average rate of cange of price wit respect to time, we could symbolize te difference quotient as Cange in price Cange in time P t were we tink of te symbol as indicating cange in. Te symbol P corresponds to a cange in price since P represents price. Tis symbol elps us to economize on te amount of writing we need to do in order to indicate an average rate of cange. In 1.1Question 1Example 1and 1.1Question 1Example, tere were only two rows of data in te table. It was fairly easy to decide wat numbers go in te numerator of te difference quotient and wic numbers go in te denominator of te difference quotient. In te next example, tere are several columns of data and we ll need to examine te average rate of cange to determine ow te difference quotient is formed. 7

8 Example 3 Find te Average Rate of Cange from a Table During te years 003 troug 007, te percentage of Americans unemployed and te percentage of Americans driving witout auto insurance bot dropped according to te table below: Year Unemployment (%) Uninsured Motorists (%) (Source: Insurance Researc Council) a. Find te average rate of cange of uninsured wit respect to unemployment over te period 003 troug 007. Solution We are interested in ow te percent uninsured canges as te percent uninsured canges. Te time period is not a part of te average rate of cange except to reference te values for te percent unemployed and te percent of motorists tat are unemployed. We could tink of te values in te table as function values, owever in tis case we ll simply tink of te average rate as a ratio of canges. Te average rate of cange of te percent uninsured wit respect to te percent unemployed is computed as 8

9 Average rate of cange of uninsured wit Cange in te percent uninsured respect to unemployment from 003 to 007 Cange in te percent unemployed Te years serve to reference te particular data we ll use to compute te cange, but are not oterwise involved in te calculation. Using te percent uninsured and unemployed in 003 (6% unemployed and 14.9% uninsured) and 007 (4.6% unemployed and 13.8 uninsured), we get Average rate of cange of uninsured wit Cange in te percent uninsured respect to unemployment from 003 to 007 Cange in te percent unemployed Te unit on te numerator of te difference quotient is percent uninsured and te unit on te denominator of te difference quotient is percent unemployed. Tis means te units on te average rate of cange is percent uninsured per percent unemployed. b. In a new release, an official wit te Insurance Researc Coucil was quoted as saying, "If te unemployment rate goes up by 1 percent, we would anticipate tat te percentage of people wo are uninsured would go up by tree-fourts of 1 percent." Use te data from in 003 and 007 to support tis statement. Solution In part a, we calculated te average rate of cange of te percent uninsured wit respect to te percent unemployed as 0.79 over 9

10 tis time period. Te units on tis number are percent uninsured per percent unemployed. Tink of te average rate as 0.79 percent uninsured 1 percent unemployed Ten te rate can be interpreted as a 1 percent cange in unemployment leads to a 0.79 percent cange in te percent of motorists tat are uninsured. Altoug tis is not exactly a cange of tree-fourts of 1 percent, it is close enoug to be consistent. In popular media, a prase like treefourts of 1 percent is more palatable tan te number 0.79 percent. It is interesting to note tat even toug te percents are bot decreasing, te average rate of cange is interpreted in terms of increases. We could ave also interpreted te average rate as 0.79 percent uninsured 1 percent unemployed In tis case we would say tat a 1 percent drop in te percent unemployed leads to a 0.79 percent drop in te percent of motorists uninsured. 10

11 Question : How do you calculate te average rate of cange from a function? Wen a function is given by a formula, we modify te definition of average rate of cange: Average Rate of Cange Te average rate of cange of f ( x ) wit respect to x from x a to x b is defined as Average rate of cange of f f ( b) f( a) wit respect to x over a, b b a Close examination of tis definition reveals tat it is simply te same definition as before, but wit Cange in f f( b) f( a) Cange in xba so tat we can write Average rate of cange of f Cange in f f( b) f( a) wit respect to x over a, b Cange in x b a Instead of using te data values to calculate te canges in te difference quotient, we use te function s formula to get te values in te numerator of te difference quotient. In te context of te grap of a function, te average rate of cange of a function can be visualized as te slope of a line tat passes troug two points on te function. Tis line, called a secant line, can be drawn on a grap of a function so tat we can quantify te value of te slope of te line. 11

12 Figure 1 - Te average rate of cange of f(x) wit respect to x over [a, b] is equal to te slope of a line of a secant line. A secant line passing troug te points a, f( a ) and b, f( b ) as a vertical rise of f( b) f( a) and a orizontal run of b a. Te slope of between te points on te secant line is f f() b f() a x ba Example 4 Find te Average Rate of Cange from a Function s Formula Te annual sales (in millions of dollars) at Apple from 001 troug 010 can be modeled by St ( ) e 0.84t were t is te number of years since 000. Find te average rate of cange of sales wit respect to time over te period from 001 to 010. (Modeled from Apple Annual Reports) 1

13 Solution Using te definition of te average rate of cange, Average rate of cange of S S(10) S(1) wit respect to t over 1, Te function values in te numerator are computed from te function s formula, S (1) e S (10) e Tis leads to te average rate of cange, Average rate of cange of S wit respect to t over 1, million dollars 9 years million dollars per year Eac decimal is written to tree decimal places since te decimals in te original function were written to tree decimal places. Tis average rate tells us tat te sales increased by an average of about million dollars or $5,973,085,000 in eac year from 001 troug 010. On a grap of te sales function St (), te average rate of cange of sales from t 1 to t 10 may be visualized as te slope of te line connecting te points at 1, S (1) and 10, S (10). 13

14 Figure - Te slope of te line connecting te sales at t = 1 and t = 10 is te same as te average rate of cange between tese points. In calculus, one difference quotient in particular comes up very often. Te average rate of cange of f( x ) wit respect to x from x a to x a is Average rate of cange of f f( a) f( a) wit respect to x over a, a Tis is te same definition we ave been using wit functions given by formulas, but wit b a. Te value of is te orizontal separation of te two points on te secant line. Tis difference quotient will be very important in defining te instantaneous rate of cange in Section

15 Figure 3 - An alternate description of average rate of cange. Example 5 Find te Difference Quotient Let f( x ) be defined by f( x) x 3x a. Find and simplify te difference quotient Solution Tis difference quotient is f( ) f() f( a) f( a) wit a. Find te function values in te numerator of te difference quotient. Te value f () is found by replacing x wit in te function, f () 3 15

16 Similarly, f ( ) is found by replacing x wit, f( ) Square Remove te parenteses Simplify Substitute tese function values into te difference quotient to yield f( ) f() 1 1 Remove te parenteses Simplify te numerator Factor te numerator Reduce b. Evaluate te difference quotient in part a wen 0.5. Solution Te expression for te difference quotient in part a, 1, makes it easy to evaluate te difference quotient for any value of. It enables us to find te average rate of cange of f ( x ) wit respect to x from x and any oter x value. Set 0.5 to give f( 0.5) f() Tis value is te average rate of cange of f ( x ) wit respect to x from x to x

17 Care must be taken to calculate f ( a ) in te difference quotient. Te most common mistake in simplifying te function value f ( a ) is to assume it is te sum of te values of te function. For instance, in te previous example f ( ) f() f( ). Te powers on te factors correspond to multiplication. Tey must be worked out wen calculating f ( a ). Example 6 Find te Difference Quotient Find and simplify te difference quotient g(3 ) g(3) for te function gt () t 3 Solution To find te difference quotient, we must evaluate gt () at 3 and 3. Te value 3 g(3) 3 7 is easy to calculate. However g(3 ) is more callenging: g To simplify te expression on te rigt, we need to cube te quantity g Multiply 33 Multiply 3 times eac term Remove te parenteses Simplify 17

18 Now we can put te function values into te difference quotient and simplify: g(3 ) g(3) Simplify te numerator Factor Reduce 18