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. 410 25 2530 1515 Date (1997) measured by days Dec. Sunset time Greenwich Mean Time at 30° North Latitude. time given to the nearest minute Figure 4.2 Data from the 1998 World Almanac, pp. 463–474. 3 As we can see by looking at the graph in Figure 4.2, over small enough intervals, the data points either lie on a line or lie close to some line that can be fitted to the data. However, the line that fits 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 we’ll look at a continuous model. ◆ EXAMPLE 4.2 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 first 24 laps in 12 minutes, what might you expect as his time for 36 laps? SOLUTION 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 finish 36 laps in about 18 minutes. We would feel less confident 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. Think back to the bottle calibration problem. For a cylindrical beaker, the rate of change of height with respect to volume is constant; height is a linear function of volume. Cylindrical beaker volume height Figure 4.3 Many of the functions that arise in everyday life (in fields like biology, environmental science, physics, and economics) have the property of being locally linear. What does this mean? “Local” means “nearby” and “linear” means “like a line.” 5 So a function is locally linear if, in the immediate neighborhood of any particular point on the graph, the graph “looks like 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 f is locally linear at a point A if, when the graph is sufficiently magnified around point A, 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, let’s 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 first three readings in early November all the way to Christmas day. In Example 4.2 the assumption that the swimmer’s 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 figure that you’ll be out of milk in another day. Here you’re assuming that you will consume milk at a constant rate of 1 4 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 you’ll continue to use them at a rate of 3 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 A. 4.2 Linear Functions 143 Clearly you wouldn’t expect to pack six lozenges with you every day; you’re assuming only local linearity. Many examples of the use of local linearity arise in the fields of finance and economics. Investors lay billions of dollars on the line when they use economic data to project into the future. The question of exactly how far into the future one can, within reason, linearly project any economic function based on its current rate of change, and by how much this projection may be inaccurate, is a matter of intense discussion. Local linearity plays a key role in calculus. The problem of finding 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, let’s discuss them. PROBLEMS FOR SECTION 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 reflection of reality? 2. Cindy quit her job as a manager in Chicago’s corporate world, put on a backpack, and is now traveling around the globe. Upon arrival in Cairo, she spent $34 the first day, including the cost of an Egyptian visa. Over the course of the next four days, she spent a total of $72 on food, lodging, transportation, museum entry fees, and baksheesh (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 she’ll 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 defining 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 S(x) be his weekly salary in dollars, where x 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 D(t) be her odometer reading t hours later. SOLUTION (a) salary = base salary + commission salary = base salary + dollars item (items) S(x) = 250 + 10x The rate of change of S = S x = $10 per item. (b) odometer reading = (initial odometer reading) + (additional distance traveled) odometer reading = (initial reading) + miles hour (hours) D(t) = 4280 + 60t The rate of change of D = D t = 60 miles per hour. ◆ Definition f is a linear function of x if f can be written in the form f(x)=mx + b, where m and b 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 y = mx + b has slope m and y-intercept b. The slope corresponds to the rate of change of y with respect to x. Every point (x 0 , y 0 ) that lies on the graph of the line satisfies the equation. In other words, if (x 0 , y 0 ) lies on the graph of y = mx + b, then y 0 = mx 0 + b. Conversely, every point whose coordinates satisfy the equation of the line lies on the graph of the equation. As discussed in Chapter 1, this is what it means to be the graph of a function. The following are examples of equations of lines and their graphs. 1 –1 1 2 f x Figure 4.4 f(x)=2x−1 m=2, b =−1 4.2 Linear Functions 145 f x Figure 4.5 f(x)= √ 10 m = 0, b = √ 10 –2 –112 1 1 f x Figure 4.6 f(x)=−0.5x m =−0.5, b =0 The slope of a line is the ratio rise run ,or change in dependent variable change in independent variable . If y is a linear function of x, then the slope is change in y change in x ,or y x . We will now verify that if y = mx + b, then the constant m is the slope of the line. Verification: Suppose (x 1 , y 1 ) and (x 2 , y 2 ) are two distinct points on the graph of y = mx + b. We want to show that the rate of change of y with respect to x is m, regardless of our choice of points. Since y = mx + b, the points (x 1 , y 1 ) and (x 2 , y 2 ) can be written as (x 1 , mx 1 + b) and (x 2 , mx 2 + b). 146 CHAPTER 4 Linearity and Local Linearity ∆y ∆ x (x 2 , y 2 ) = (x 2 , mx 2 + b) (x 1 , y 1 ) = (x 1 , mx 1 + b) Figure 4.7 Therefore, change in y change in x = y x = y 2 − y 1 x 2 − x 1 = (mx 2 + b) − (mx 1 + b) x 2 − x 1 = mx 2 − mx 1 x 2 − x 1 = m(x 2 − x 1 ) x 2 − x 1 = m We have shown that the slope of the line y = mx + b is indeed m. The slope of a line is a fixed constant, regardless of how it is computed. To verify that b is the y-intercept of the line, we set x = 0. Then y = m · 0 + b = b. (Remember that the y-intercept is the value of the function on the y-axis, i.e., when x = 0.) ◆ EXAMPLE 4.4 3x + 2y = 7 is the equation of a line. Find the slope and the x- and y-intercepts. SOLUTION Put the equation into the form y =− 3 2 x + 7 2 . We can then read off the y-intercept as 7 2 and the slope as − 3 2 . Find the x-intercept by setting y = 0: 3x + 2(0) = 7 3x = 7 x = 7 3 . Alternatively, begin with 3x + 2y = 7 and find the x-intercept by setting y = 0 and solving for x: ( 7 3 ,0).Find the y-intercept by setting x = 0 and solving for y: (0, 7 2 ). (The x- and y-intercepts can be useful in graphing the line.) Given any two points, you can find the slope by computing y x . For instance, given the points ( 7 3 ,0)and (0, 7 2 ), the slope is computed as follows. y x = 7 2 − 0 0 − 7 3 = 7 2 − 7 3 = 7 2 − 3 7 =− 3 2 4.2 Linear Functions 147 y x 1 12 3 4 –1 –2 2 3 5 7 2 7 3 Figure 4.8 It is not necessary to use the x- and y-intercepts in order to calculate the slope; any two points will do. ◆ EXERCISE 4.1 Verify that any equation of the form ax + cy = d, where a, c, and d are constants and c = 0 is a linear equation. In other words, verify that it can be written in the form y = mx + b.If ax + cy = d, what is y x ? Lines and Linear Equations EXERCISE 4.2 Graph the following lines. Use three sets of axes, one for each of the Parts I, II, and III. I. a) y = x b) y = 2x c) y = 3x d) y = 0.5x e) y =−2x f) y =−0.5x II. a) y = 2x b) y = 2x + 1c)y=2x−2 III. a) y = 3b)x=4c)x=0d)y=0 Notice that the results of this exercise are consistent with the principles of stretching, shrinking, flipping, and shifting discussed in Section 3.4. From this exercise you can observe how the constant m corresponds to the steepness of the line. m = 0 ⇒ The line is horizontal. m>0 ⇒ The line rises from left to right, so the function is increasing. The more positive m is, the steeper the rise of the line. m<0 ⇒ The line falls from left to right, so the function is decreasing. The more negative m is, the steeper the fall of the line. The closer m is to zero, the less steep the line. Vertical Lines A vertical line is the graph of an equation of the form x = constant. Notice that a vertical line is not the graph of a function of x. One x-value is mapped to infinitely many y-values; a vertical line certainly fails the vertical line test! 148 CHAPTER 4 Linearity and Local Linearity x=3 x yy slope undefined x y=1 slope = 0 Figure 4.9 Slope = y x , so the slope of a vertical line is undefined, while the slope of a horizontal line is zero. Parallel and Perpendicular Lines If L 1 and L 2 are nonvertical lines with slopes m 1 and m 2 , respectively, then L 1 is parallel to L 2 if and only if m 1 = m 2 , i.e., their slopes are equal. L 1 is perpendicular to L 2 if and only if m 1 = −1 m 2 , i.e., their slopes are negative reciprocals. Fundamental Observation A line is completely determined by either i. the slope of the line and any one point on the line, or ii. any two points on the line. This geometric observation can be translated as follows. If there is a linear relationship between two variables, then that relationship can be completely determined provided i. one data point is known and the rate of change of one variable with respect to the other is known, or ii. two data points are known. Finding the Equation of a Line i. Suppose we know the slope of the line, m, and a point on the line, (x 1 , y 1 ).Wecanfind the equation of the line by using either of the following two methods. . Method 1: The equation of any nonvertical line can be put into the form y = mx + b. If we know the slope, we know m. Our job is to find b. Because (x 1 , y 1 ) is a point on the line, (x 1 , y 1 ) satisfies the equation y = mx + b. Therefore, y 1 = mx 1 + b, where x 1 , y 1 , and m are constants. We can solve for b, the only unknown, and then write y = mx + b. . Method 2: The slope = y x . Point (x 1 , y 1 ) is a fixed point on the line. Therefore, if (x, y) lies on the line, m = y − y 1 x − x 1 . This is written y − y 1 = m(x − x 1 ) 4.2 Linear Functions 149 to make it clear that the domain is all real numbers. Solving for y will put this into the form y = mx + b. The equation y − y 1 = m(x − x 1 ) is called the point-slope form of the equation of a line. ii. Suppose we know two points on a line: (x 1 , y 1 ) and (x 2 , y 2 ).Wecanfind the slope of the line: y x = y 2 − y 1 x 2 − x 1 = m. Knowing the slope and a point (two points, actually), we can continue as described in part i. ◆ EXAMPLE 4.5 Find the equation of the line passing through points (2, −3) and (−4, 5). SOLUTION Calculate the slope. m = −3 − 5 2 − (−4) = −8 6 =− 4 3 , so y =− 4 3 x + b. Use the fact that point (2, −3) lies on the line to find b. −3 =− 4 3 (2) + b ⇒ b =−3+ 8 3 =− 9 3 + 8 3 =− 1 3 y =− 4 3 x − 1 3 . Alternatively, after finding m =− 4 3 we can use Method 2 above. − 4 3 = y + 3 x − 2 y + 3 =− 4 3 (x − 2) We leave it up to you to verify that the equations are equivalent. ◆ The Equation of a Line Slope-intercept form of a line Point-slope form of a line y = mx + by−y 1 =m(x − x 1 ) where m is the slope and b is the y-intercept where m is the slope and (x 1 , y 1 ) is a point on the line 150 CHAPTER 4 Linearity and Local Linearity Scientists and social scientists often make general models in which constants or pa- rameters are represented by letters. 7 Cases in which the slope and/or points of a line are not given as numbers but as letters representing constants typically are stumbling blocks for students trying to determine the equation of the line. Unless you think carefully and clearly, it is easy to lose track of the main features of the equation: the two variables, the constants that are given, and the unknown constants that you are trying to find. If you plan to consider slope, you must decide which variable will play the role of the inde- pendent variable and which will be treated as the dependent variable. (If you’re accustomed to x’s and y’s, then determine which variable will play the role of x and which will play the role of y.) By taking inventory, keeping your goal firmly in mind, and determining the steps you’ll take to reach that goal, you ought to be able to proceed fearlessly. Try the following exercises. EXERCISE 4.3 i. Find the equation of the line through the points (2π, π 2 ) and π, 1 π . ii. Find the equation of the line through the point (a, c 2 ) having slope p. iii. Find the equation of the line through (m, n) with slope b. CAUTION Here m and b are playing nonstandard roles. To avoid confusion, try a strategic maneuver; use something such as y = Ax + C as your “reference” equation to avoid getting the m’s and b’s in the problem confused with the slope and the y-intercept.) EXERCISE 4.4 Over the interval [0, T ] a horse’s velocity, v, is a linear function of time, t . (a) At time t = 0, the horse’s velocity is v 0 . At time t = T, his velocity is v T . Find v(t) on the interval [0, T ]. Note: v 0 , T, and v T are all constants. (b) What can you say about the horse’s acceleration? Answers to Selected Exercises Answers to Exercise 4.3 i. y = π 3 −1 π 2 x − π 2 + 2 π ii. y = px + c 2 − pa iii. y = bx + n − mb Answers to Exercise 4.4: (a) v(t) = mt + b because v is a linear function of t. Our job is to find m and b.Weare given two points, (0, v 0 ) and (T, v T ). (Points are written in the form (time, velocity) because time is the independent variable.) In other words, t plays the role of x and v 7 Physicists use g to denote acceleration due to the force of gravity. Biologists might denote the carrying capacity of an ecosytem for a certain species by a letter, such as C, without specifying the value of C. Chemists use r for Avogadro’s number, the number of molecules in a mole. . graph of y = mx + b. We want to show that the rate of change of y with respect to x is m, regardless of our choice of points. Since y = mx + b, the points (x 1 , y 1 ) and (x 2 , y 2 ) can be. necessary to use the x- and y-intercepts in order to calculate the slope; any two points will do. ◆ EXERCISE 4.1 Verify that any equation of the form ax + cy = d, where a, c, and d are constants and. linearity. Many examples of the use of local linearity arise in the fields of finance and economics. Investors lay billions of dollars on the line when they use economic data to project into the future.