4.1 Making Predictions: An Intuitive Approach to Local Linearity 141
change of 1 minute/day. Certainly it is ridiculous to estimate that on November 6, 1998, the sun will set at 11:04 a.m. (where 5:09 p.m. (1 minute/day)(365 days) = 5:09 p.m. 365 minutes 5:09 p.m. (6 hours 5 minutes) = 11:04 a.m.)!
time
5:11
5:05
5:00
Nov. 4 10 25 301 5 15 25 Dec.
Sunset time Greenwich Mean Time at 30 North Latitude. time given to the nearest minute
Date (1997) measured by days
Figure 4.2 Data from the 1998 World Almanac, pp. 463474.3
EXAMPLE 4.2
SOLUTION
As we can see by looking at the graph in Figure 4.2, over small enough intervals, the datapointseitherlieonalineorlieclosetosomelinethatcanbettedtothedata.However,
the line that ts the data best varies with the interval chosen. When looked at over the entire interval from November 4 to December 25, the graph does not look linear.
In the last example we looked at a discrete phenomenon and made predictions based on the assumption of a constant rate of change over a small interval. In the next example well look at a continuous model.
Brian Younger is a high-caliber distance swimmer; in competition he swam approximately 1 mile, 36 laps of a 25-yard pool.4 If he completes the rst 24 laps in 12 minutes, what might you expect as his time for 36 laps?
Knowing that Brian is a distance swimmer, it is reasonable to assume that he does not tire much in the last third. Assuming a constant speed of 12 laps every 6 minutes (or 120 yards/minute) we might expect him to nish 36 laps in about 18 minutes. We would feel
less condent saying that he could swim 4 miles if given an hour and 12 minutes, or 8 miles if given 2 hours and 24 minutes.
A quantity that changes at a constant rate increases or decreases linearly.
If the rate of change of height with respect to time, height
time
is constant over a certain time interval, then height is a linear function of time on that interval.
3 These times might look suspect to you; the sun begins to set later by December 10, well before the winter solstice. Do not be alarmed: Sunrise gets later and later throughout December and continues this trend through the beginning of January.
4 One lap is 50 yards.
142 CHAPTER 4 Linearity and Local Linearity
If the rate of change of position with respect to time, position
time
is constant, then position is a linear function of time.
Thinkbacktothebottlecalibrationproblem.Foracylindricalbeaker,therateofchange of height with respect to volume is constant; height is a linear function of volume.
height
Cylindrical beaker volume
Figure 4.3
Many of the functions that arise in everyday life (in elds like biology, environmental science, physics, and economics) have the property of being locally linear. What does this mean?
Localmeansnearbyandlinearmeanslikealine.5 Soafunctionislocallylinear if, intheimmediateneighborhoodofanyparticularpointonthegraph, thegraphlookslike a line. This is not to say that the function is linear; we mean that near a particular point the function can be well approximated by a line. In other words, over a small enough interval, the rate of change of the function is approximately constant.
Graphically this means that is locally linear at a point if, when the graph is sufciently magnied around point , the graph looks like a straight line.6 The questions of which line best approximates the function at a particular point and just what we mean by nearby are very important ones, and we will examine them more closely in chapters to come. First, lets look back at Examples 4.1 and 4.2.
In Example 4.1 the runner estimating the time of sunset is assuming local linearity; her assumption leads to a prediction that is only 1 minute off when she predicts just a few days ahead. The function is only locally linear; the idea of locality does not extend from the rst three readings in early November all the way to Christmas day. In Example 4.2 the assumption that the swimmers pace is maintained for another 5 minutes is an assumption of local linearity.
You probably have made many predictions of your own based on the assumption of local linearity without explicitly thinking about it. For instance, if you buy a gallon of milk and you have only a quarter of a gallon left after three days, you might gure that youll be
out of milk in another day. Here youre assuming that you will consume milk at a constant rate of 1 gallon/day. Or, suppose you come down with a sore throat one evening and take
three throat lozenges in four hours. You might take six lozenges to work with you the next day, assuming that youll continue to use them at a rate of 4 lozenge/hour for eight hours.
5 By line we mean straight line.
6 A computer or graphing calculator can help you get a feel for this if you zoom in on point .
4.2 Linear Functions 143
Clearlyyouwouldntexpecttopacksixlozengeswithyoueveryday; youreassumingonly local linearity.
Manyexamplesoftheuseoflocallinearityariseintheeldsofnanceandeconomics. Investors lay billions of dollars on the line when they use economic data to project into the future.Thequestionofexactlyhowfarintothefutureonecan,withinreason,linearlyproject anyeconomicfunctionbasedonitscurrentrateofchange, andbyhowmuchthisprojection may be inaccurate, is a matter of intense discussion.
Local linearity plays a key role in calculus. The problem of nding the best linear approximation to a function at a given point is a problem at the heart of calculus. In order to work on this keystone problem, one must be very comfortable with linear functions. So, before going on, lets discuss them.
P R O B L E M S F O R S E C T I O N 4 . 1
1. Lucia has decided to take up swimming. She begins her self-designed swimming program by swimming 20 lengths of a 25-yard pool. Every 4 days she adds 2 lengths to her workout. Model this situation using a continuous function. In what way is this model not a completely accurate reection of reality?
2. Cindy quit her job as a manager in Chicagos corporate world, put on a backpack, and is now traveling around the globe. Upon arrival in Cairo, she spent $34 the rst day, including the cost of an Egyptian visa. Over the course of the next four days, she spentatotalof$72onfood,lodging,transportation,museumentryfees,andbaksheesh (tips). She is going to the bank to change enough money to last for three more days in Cairo. How much money might she estimate shell need? Upon what assumptions is this estimate based?
3. It is 10:30 a.m. Over the past half hour six customers have walked into the corner delicatessen. How many people might the owner expect to miss if he were to close the deli to run an errand for the next 15 minutes? Upon what assumption is this based?
Suppose that between 9:30 a.m. and 11:30 a.m. he had 24 customers. Is it reason-able to assume that between 11:30 a.m. and 1:30 p.m. he will have 24 more customers? Why or why not?
4.2 LINEAR FUNCTIONS
The dening characteristic of a linear function is its constant rate of change.
EXAMPLE 4.3 For each situation described below, write a function modeling the situation. What is the rate of change of the function?
(a) A salesman gets a base salary of $250 per week plus an additional $10 commission for every item he sells. Let be his weekly salary in dollars, where is the number of items he sells during the week.
144 CHAPTER 4 Linearity and Local Linearity
(b) A woman is traveling west on the Massachusetts Turnpike, maintaining a speed of 60 miles per hour for several hours. Her odometer reads 4280 miles when she passes the Allston/Brighton exit. Let be her odometer reading hours later.
SOLUTION (a)
(b)
salary base salary commission salary base salary dollarsitems
250 10
The rate of change of $10 per item.
odometer reading initial odometer readingadditional distance traveled odometer reading initial readingmileshours
4280 60
The rate of change of 60 miles per hour.
D e f i n i t i o n
is a linear function of if can be written in the form ( +, where ( and + are constants.
The graph of a linear function of one variable is a straight nonvertical line; conversely, any straight nonvertical line is the graph of a linear function. As we will show, the line
( + has slope ( and -intercept +. The slope corresponds to the rate of change of with respect to . Every point 0, 0 that lies on the graph of the line satises the equation. In other words, if 0, 0 lies on the graph of ( +, then 0 (0 +. Conversely, every point whose coordinates satisfy the equation of the line lies on the graph oftheequation.AsdiscussedinChapter1,thisiswhatitmeanstobethegraphofafunction.
The following are examples of equations of lines and their graphs.
f
2
1
x
1 1
Figure 4.4 2 1 (2, + 1
4.2 Linear Functions 145
f
x
Figure 4.5 10 (0, + 10
f
1
x 2 1 1 2
1
Figure 4.6 0.5 (0.5, + 0
The slope of a line is the ratio
rise
run
or
change in dependent variable
change in independent variable
If is a linear function of , then the slope is
change in
change in
or
We will now verify that if ( +, then the constant ( is the slope of the line. VeriÞcation: Suppose 1, 1 and 2, 2 are two distinct points on the graph of
( +. We want to show that the rate of change of with respect to is (, regardless of our choice of points.
Since ( +, the points 1, 1 and 2, 2 can be written as 1, (1 + and 2, (2 +.
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