11.2 Instantaneous Rate of Change
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1 11. Instantaneous Rate of Cange Question 1: How do you estimate te instantaneous rate of cange? Question : How do you compute te instantaneous rate of cange using a limit? Te average rate of cange is useful for calculating ow quantities cange wit respect to eac oter. It allows us to quantify tese canges and to understand weter one quantity is increasing or decreasing wit respect to anoter quantity. Unfortunately, te average rate of cange as its limitations. Tese limitations can be illuminated by calculating te average rate of cange of te Dow Jones Industrial Average on May 6, 010 wit respect to time. Te Dow Jones Industrial Average (DJIA) is an index tat tracks 30 large companies on te New York Stock Excange (NYSE). On tis date, te stock market was preoccupied wit news of a debt crisis in Greece. Te NYSE opened at 9:30AM and te Dow Jones Industrial Average was at a level of points. It closed at 4:00PM at a level of points. Let s look at ow te Dow Jones Industrial Average canged tat day by calculating its average rate of cange wit respect to time. To find tis rate, we must calculate te cange in te index tat day, for trading, t. DJIA, and te cange in time tat te NYSE was open Te cange in te index is calculated by subtracting te opening level of te index from te closing level of te index, DJIA In te language of te stock excange, te Dow Jones Industrial Average lost points on May 6, 010. Since te NYSE is open from 9:30AM to 4:00PM, te cange in time is 6 ½ ours or 390 minutes. We could use ours or minutes to calculate te average rate of cange, but in tis case we ll use minutes and set t 390 1
2 Using tese canges, we can calculate te average rate of cange, DJIA points t 390 minute 0.95 points per minute Tis means tat, on average, te Dow Jones Industrial Index dropped sligtly less tan one point for eac minute te NYSE is open on May 6, 010. Tis may not seem like muc, but in general te Dow Jones Industrial Average canges very little from day to day. On May 6, 010, te Dow Jones Industrial Average suffered a flas cras. A flas cras occurs wen te index drops a large amount over a very sort period of time. During te flas cras on May 6, 010, te DJIA dropped over 900 points in a matter of minutes. Sortly after tis drop, te index recovered tese losses. Figure 1 - Tis grap sows te steep drop in te DJIA before recovering on May 6, 010. Te average rate of cange we found earlier was calculated over te span of te entire trading day or 390 minutes. It takes into account te opening and closing levels of te Dow Jones Industrial Average, but noting else over te course of te trading day. It
3 sows wat appen, on average, during te day, but not wat appen in a ten minute period around :45PM. To calculate ow fast te DJIA was dropping during te flas cras, we need to calculate te instantaneous rate of cange of te Dow Jones Industrial Average wit respect to time. Like te average rate of cange, te instantaneous rate of cange measures fast two quantities are canging wit respect to eac oter. As te term instantaneous indicates, te instantaneous rate of cange measures ow fast one quantity canges wen anoter quantity canges by a very small amount. In te case of te Dow Jones Industrial Average, we would like to calculate te instantaneous rate of cange of te wit respect to time at te eigt of te flas cras. We ll do tis by calculating ow muc te index canges over a very sort period of time. Tis will tells us ow fast te Dow Jones Industrial Average was dropping at te instant te flas cras occurred. 3
4 Question 1: How do you estimate te instantaneous rate of cange? Te average rate of cange of f wit respect to x is computed using a difference quotient, Average rate of cange of f Cange in f wit respect to x over a, b Cange in x Te same difference quotient can be used to compute te instantaneous rate of cange of f wit respect to x as long as we make te cange in te denominator very small. Ideally, we would like tere to be no cange in x. But tis is not possible since it would result in division by zero. However, we can estimate te instantaneous rate of cange by making te cange in te denominator as small as possible: Instantaneous rate of cange of Cange in f f wit respect to x over a, b Small cange in x Te smaller te cange in te denominator, te better te estimate is of te instantaneous rate of cange. Example 1 Estimate te Instantaneous Rate of Cange On May 6, 010, te Dow Jones Industrial Average (DJIA) dropped points or 9.% from te close of trading on May 5, 010. During te flas cras, te DJIA dropped according to te table below. At te time, tis drop was te largest point drop during any day in istory on te NYSE. Twenty minutes after dropping to a level of points, te index recovered around 600 points of te loss. Tis loss drove te NYSE to develop new trading curbs called circuit breakers. Tese circuit breakers dictate tat trading will be alted on any stock on te S&P Index tat canges by 10% in a five minute period. 4
5 Estimate te instantaneous rate of cange of te DJIA minutes after 1PM. Solution Te data in te table corresponds to te Dow Jones Industrial Average at various times after 1PM on May 6, 010. Te average rate of cange over several different intervals is calculated using te definition of average rate of cange, Minutes after 1PM DJIA (points) Average rate of cange of DJIA wit respect to time DJIA time For instance, te average rate of cange of te Dow Jones Industrial Average over te interval 1.7,107.0 is DJIA points time minutes points minutes 8.8 points per minute An interval of lengt minutes is certainly not an instant or even a reasonable approximation of an instant in time. Te average rate of cange of te Dow Jones Industrial Average over te interval 90.0,107.0 is 5
6 DJIA points time minutes points 17.0 minutes 43.0 points per minute Even toug te drop in points is not as steep as te previous interval, te average rate of cange is greater since te interval is muc sorter. Te average rate of cange of te Dow Jones Industrial Average over te interval 103.3,107.0 is DJIA points time minutes points 3.7 minutes points per minute Te endpoint on te rigt of te interval is fixed, but te left endpoint canges in eac of tese rates. To approximate an instant, we must make te endpoint on te left side of te interval as close as possible to t Te best approximation for te instantaneous rate of cange is te average rate of cange over te interval 106.7,107.0, DJIA points time minutes points 0.3 minutes points per minute For tis table, an instant is approximated by an interval tat is 0.3 minutes long and an instantaneous rate of cange of points per 6
7 minute at te time immediately prior to wen te Dow Jones Industrial Average began to rise again. As te average rate of cange is computed over smaller and smaller intervals near te lowest point on te grap, it gets more and more negative since te Dow Jones Industrial Average dropped faster and faster before recovering. 7
8 Question : How do you compute te instantaneous rate of cange using a limit? If te quantities being compared in an average rate of cange are given by data, we can estimate te instantaneous rate of cange using a difference quotient. By using te data nearest te point at wic te instantaneous rate of cange is desired, we calculate te difference quotient. In te case of te instantaneous rate of cange of f wit respect to x, Instantaneous rate of cange of Cange in f f wit respect to x over a, b Small cange in x Wen te quantity in te numerator of te difference quotient is given by a formula, we do not ave to settle for an estimate of te instantaneous rate of cange. If te quantity in te numerator is given by a function f ( x ), we can write te average rate of cange as Average rate of cange of f f ( a) f( a) wit respect to x over a, a were describes te magnitude of te cange in te denominator. We can use tis expression to write a corresponding definition for te instantaneous rate of cange. For te instantaneous rate of cange, we want te cange to be as small as possible. Altoug we cannot let tis cange be zero, we can do te next best ting using a limit. Te instantaneous rate of cange of f ( x ) wit respect to x at x a is Instantaneous rate of cange of f f ( a) f( a) lim wit respect to at x x a 0 8
9 By using a limit as te magnitude of te cange approaces zero, we are able to find te instantaneous rate of cange by taking te limit of te average rate of cange f ( a) f( a). Example Find te Instantaneous Rate of Cange A small toolmaker estimates te annual total revenue TR( x ) from selling a quantity of x bearing presses to be TR x x x ( ) 00 dollars a. Find te instantaneous rate of cange of annual total revenue wit respect to te quantity of bearing presses sold wen 50 bearing presses are sold annually. Solution Start by rewriting te definition of instantaneous rate of cange for te function TR( x ) wit a 50, Instantaneous rate of (50 ) (50) TR x x TR TR lim cange of ( ) at 50 0 To calculate te limit, we need to find te revenue function values in te numerator of te difference quotient: TR(50) TR(50 ) Multiply Remove parenteses Simplify 9
10 Tese function values are substituted into te difference quotient to yield TR(50 ) TR(50) lim lim lim lim 0 lim Remove parenteses and simplify Factor numerator Reduce 100 As approaces 0, approaces 100 b. Explain wat te instantaneous rate of cange in part a tells you about ow revenue is canging as more bearing presses are sold. Solution To elp us understand wat an instantaneous rate of cange means in tis context, let s examine te units associated wit tis rate. For any rate, te units are te units on te dependent variable divided on te units on te independent variable. In tis rate, te units on te dependent variable are dollars and te units on te independent variable are bearing presses. So te units on te rate are units on te dependent variable dollars units on te independent variable bearing presses An instantaneous rate of 100 dollars 1 bearing presses means tat selling one additional bearing press will increase revenue by 100 dollars. 10
11 Example 3 Find te Instantaneous Rate of Cange Apple is very successful in translating expenditures on researc and development into sales of electronic products like Ipods, Ipones and Ipads. Based on data from 001 troug 010, te annual sales at Apples (in millions of dollars) can be modeled by SR R R ( ) were R is te amount, in millions of dollars, spent annually on researc and development. (Modeled from Apple Annual Reports) a. Find te instantaneous rate of cange of sales wit respect to researc and development spending wen annual researc and development spending is 1000 million dollars. Solution For tis function, te instantaneous rate of cange of sales wit respect to researc and development spending wen 1000 million dollars is spent on researc and development is Instantaneous rate S(1000 ) S(1000) lim of cange 0 We can find te function value S (1000) by substituting 1000 into te function, S(1000) Te function value S(1000 ) is found by replacing R wit 1000, 11
12 S(1000 ) In simplifying tis function value, take care to square te binomial 1000 properly. It is very common for students to write incorrectly Instead, write te square as te product of two binomials and multiply te terms Using te function values in te definition of instantaneous rate of cange leads to S(1000 ) S(1000) lim lim 0 0 lim lim lim Note tat only one term in te limit contains. As gets smaller, te term gets smaller and te constant term does not cange. We compute te limit to be R lim
13 b. Explain wat te instantaneous rate of cange in part a tells you about sales and spending on researc and development. Solution Te units on te variables elp us to determine te units on te instantaneous rate. By dividing te units on te variables, units on te dependent variable millions of dollars of sales units on te independent variable millions of dollars of researc and development we get te units on te instantaneous rate of cange. An instantaneous rate of million dollars of sales per million dollars of researc and development means tat a one million dollar increase in researc and development leads to an increase in sales of million dollars. Alternately, we could also say tat a one dollar increase in researc and development leads to an increase in sales of dollars since te factor of millions in te numerator and denominator of te units can be reduced. 13
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