4.2 Linear Functions 151 plays the role of y. m = velocity time = v T − v 0 T − 0 = 1 T (v T − v 0 ) The v-intercept is v 0 ,sob=v 0 . v(t) = v T − v 0 T t + v 0 . (b) Acceleration = v t = v T −v 0 T . The horse’s acceleration is constant on [0, T ]. PROBLEMS FOR SECTION 4.2 1. The graph of f is given in the figure below. B C D E y=f(x) x y bc ed A Lines L 1 , L 2 , and L 3 are as shown. L 1 is horizontal and L 3 is vertical. A, B, C, D, and E are points on the graph of f . Express your answers to the following questions in terms of b, c, d, e, and the function f . B A C D E y=f(x) x y bc e 1 3 2 d (a) Write the coordinates of points A, B, C, D, and E using functional notation. For instance, point A has coordinates (0, f(0)). 152 CHAPTER 4 Linearity and Local Linearity (b) What is the slope of L 1 ? (c) What is the equation of L 1 ? (d) What is the length of the line segment joining point B and (b,0)? (e) What is the length of the line segment joining points B and D? (f) What is the slope of L 2 ? (g) What is the equation of L 2 ? (h) What is the equation of L 3 ? (i) What is the slope of the line L 3 ? 2. Find the slope of the line through the two points given. (a) (3, −1), (−2, −3) (b) (π,2π), (0, −π) (c) ( √ 2, 3), ( √ 2, 5) (d) ( √ 2, √ 3), (1, √ 3) 3. On the same set of axes, sketch lines through point (0, 1) with the slopes indicated. Label the lines. (a) slope = 0 (b) slope = 1 2 (c) slope = 1 (d) slope = 2 (e) slope = − 1 2 (f) slope = −1 (g) slope = −2 For Problems 4 through 13, find the equation of the line with the given characteristics. 4. Slope − 1 2 , passing through (−2, −3) 5. Slope π, passing through (3, 5) 6. Passing through points (0, a) and (b,0) 7. Passing through points (π,3)and (−π,5) 8. Passing through point ( √ 3, √ 2) and parallel to 3x − 4y = 7 9. Passing through the origin and perpendicular to πx − √ 3y = 12 10. x-intercept of √ π and parallel to the y-axis 11. Perpendicular to y − π = π(x −1) with a y-intercept of 3 12. Horizontal and passing through (− √ π, π 2 ) 13. Vertical and passing through (− √ π, π 2 ) 4.3 Modeling and Interpreting the Slope 153 For Problems 14 through 16, find the slope of the secant line passing through points P and Q, where P and Q are points of the graph of f(x)with the indicated x-coordinates. 14. f(x)= 3 x + 2x; the x-coordinates of P and Q are x = 3 and x = 3 + k (k = 0), respectively. 15. f(x)=x 2 +3x+1; the x-coordinates of P and Q are x = b and x = b + h(h=0), respectively. 16. f(x)=ax 2 + bx + c; the x-coordinates of P and Q are x = k and x = k + h(h=0), respectively. 17. There is a proliferation of telephone-call billing schemes. According to one scheme, a call to anywhere in the United States is billed at 50 cents for the first three minutes and 9.8 cents per minute after that. Express the cost of a call as a function of its duration. 18. From the early 1500s to nearly 1700, the Turkish town of Iznik was famous for its beautiful colored tiles. In the 1990s, tile-making was pursued with renewed vigor in the town. In the late 1990s, a new mosque was built, and the walls, both inside and outside, are currently being covered with the blue and red tiles for which the town is known. If the mosque cost C dollars to construct with an additional T dollars for each tile used, find the total cost as a function of x, where x is the number of tiles used. 4.3 MODELING AND INTERPRETING THE SLOPE Interpreting slope as a rate of change is the key to many applications of calculus to other disciplines. Given the line y = 5x + 5 we can interpret the slope 5 as 5 1 = y x ; for each increase of 1 unit in x, y increases by 5 units. As a second example, consider the line y =− 3 2 x + 7. The slope is − 3 2 . We can interpret this as follows. − 3 2 = −3 2 = y x .Ifxincreases by 2, y will decrease by 3. Alternatively, − 3 2 = 3 −2 = y x .Ifxdecreases by 2, y will increase by 3. Or, − 3 2 = − 3 2 1 = y x .Ifxincreases by 1, y will decrease by 3 2 . To interpret slope in context, we must first clarify which variable is independent (the input) and which variable is dependent (the output). 154 CHAPTER 4 Linearity and Local Linearity The slope is dependent variable independent variable = change in variable plotted on vertical axis change in variable plotted on horizontal axis . ◆ EXAMPLE 4.6 In this example we consider position to be a function of time, quantity demanded to be a function of price, and velocity to be a function of time. i. Suppose s is position, in miles, and t is time, in hours. If s =− 3 2 t + 7, then the slope is s t = change in position (in miles) change in time (in hours) = velocity (measured in miles per hour). Here the velocity is − 3 2 miles hours , the negative sign indicating direction. S t Figure 4.10 ii. Suppose D is quantity demanded, in bushels, and p is price, in dollars. If D =− 3 2 p + 7, then the slope is D p = change in bushels demanded change in price (in dollars) . For every increase of $2 in price, 3 fewer bushels will be demanded. D p Figure 4.11 4.3 Modeling and Interpreting the Slope 155 Equivalently, for every $2 decrease in price, 3 more bushels will be demanded. Notice that a negative slope is exactly what we would expect in this kind of situation. Students of economics should check the footnote below. 8 iii. Suppose v is an object’s velocity, in meters/second, and t is time measured in seconds. If v =− 3 2 t + 7, then the slope is v t = change in velocity change in time (in seconds) . Every two seconds, the object’s velocity decreases by 3 meters/second. The object is decelerating. V t Figure 4.12 ◆ PROBLEMS FOR SECTION 4.3 1. A photocopying shop has a fixed cost of operation of $6000 per month. In addition, it costs them $0.01 per page they copy. They charge customers $0.07 per page. (a) Write a formula for R(x), the shop’s monthly revenue from making x copies. (b) Write a formula for C(x), the shop’s monthly costs from making x copies. (c) Write a formula for P(x), the shop’s monthly profit (or loss if negative) from making x copies. (Profit is computed by subtracting total costs from the total revenue.) (d) How many copies must they make per month in order to break even? (Breaking even means that the profit is zero; the total costs and total revenue are equal.) (e) Sketch C(x), R(x), and P(x) on the same set of axes and label the break-even point. (f) Find a formula for A(x), the shop’s average cost per copy. (g) Make a table of A(x) for x = 0, 1, 10, 100, 1000, 10000. (h) Sketch a graph of A(x). 8 Economists tend to put price on the vertical axis and quantity on the horizontal axis. We can solve the equation above for p. p =− 2 3 D + 14 3 The slope is − 2 3 , but the interpretation in words is identical to that in part (ii). 156 CHAPTER 4 Linearity and Local Linearity 2. An item costs $1000 this year. This is a 10% increase over the price last year. What was the price last year? (Caution: it was not $900. It would be wise to give last year’s price a name—like “x,” or “P ,” or some other labeling of your choice.) 3. According to a study done by Chester Kyle, Ph.D. (Long Distance Cycling, Rodale Press, Emmaus, PA, 1993), adding 6 pounds to a bicycle slowed the rider down by 22 seconds on a certain 2-mile course. Assume that riding the course without the extra weight took K seconds (actual time not specified). (a) Assuming that this relationship is linear, find an equation for T(w),the time needed to complete the course as a function of the amount of extra weight added. (b) What is the rate of change of T(w)?Interpret this rate of change in practical terms. 4. Economists use demand curves to express the relationship between the price of an item and the number of items demanded by consumers. Below is a demand curve for a certain good. q is the quantity of the good demanded (i.e., the number of items demanded). p is the price per item. (a) Write an equation for the line in terms of p and q. Express p as a function of q. (b) In words, interpret the meaning of the slope of the line. (2400, 700) (3000, 550) p q = = 5. The three most commonly used temperature scales are the Fahrenheit ( ◦ F), the Celsius ( ◦ C), and the Kelvin (K, an absolute temperature) scales. One interval on the Kelvin scale is equal to one degree Celsius. The freezing point of water is 0 ◦ C, which is 32 ◦ F and 273.15 on the Kelvin scale. The boiling point of water is 100 ◦ C and 212 ◦ F. On the Celsius scale, the interval between the freezing and boiling points of water is divided into 100 degrees while on the Fahrenheit scale it is divided into 180 degrees. You have been given more than enough information to answer the following questions! You’ll have to select the information you will use. (a) Write a formula for a function that takes as input degrees Celsius and gives as output degrees Fahrenheit. (b) Write a formula for a function that takes as input degrees Fahrenheit and gives as output degrees Celsius. Do this as efficiently as possible! The function you’ve arrived at is the inverse of the function from part (a). (c) Write a formula for a function that takes as input degrees Celsius and gives the temperature on the Kelvin scale as output. (d) Write a formula for a function that takes as input degrees Fahrenheit and gives the temperature on the Kelvin scale as output. Express this function as the composition of two functions from previous parts of this problem. 4.3 Modeling and Interpreting the Slope 157 6. Economists use indifference curves to show all combinations of two goods that give the same (fixed) level of satisfaction to a household. Generally an indifference curve is nonlinear, but for certain combinations of goods it is possible to have a straight-line indifference curve. The following is a linear indifference curve. Let R = the number of units of item 1 and S = the number of units of item 2. S R a c (a) Write an equation for the line in terms of S, R, a, and c. (b) Interpret the meanings of the intercepts. (c) Optional (but suggested for those studying economics): Give an example of two items for which the indifference curve could reasonably be linear. 7. Stories are told about some of the less fair-minded teams of early baseball (e.g., the Baltimore Orioles of the 1890s) freezing baseballs until shortly before game time so that although the cover would feel normal, the core of the ball would be much colder. Then, they would attempt to introduce these balls into play when the opposing team was at bat, working on the assumption that the frozen balls would not travel as far when hit. Experiments have shown that a ball whose temperature is −10 ◦ F would travel 350 feet after a given swing of the bat, while a ball whose temperature is 150 ◦ F would be hit 400 feet by the same swing. Assume this relationship is linear. Let B(T ) be the distance this swing would produce, where T is the temperature in degrees Fahrenheit. (a) Find an equation for B(T ). (b) What is the B-intercept? What is its practical meaning? (c) What is the slope of B(T )? What is its practical meaning? 8. A horseman has some ponies of his own and boards horses for other people. For his own ponies, he orders 9 bales of hay from the supplier. The total number of bales he orders increases linearly with the number of horses he boards. When he boards 6 horses, he orders a total of 36 bales of hay (for these horses and his ponies). Express the number of bales of hay he orders as a function of the number of horses he boards. 158 CHAPTER 4 Linearity and Local Linearity Exploratory Problem for Chapter 4 Thomas Wolfe’s Royalties for The Story of a Novel Thomas Wolfe was an author born in Asheville, North Carolina, in 1900. His first novel, Look Homeward Angel, was a thinly veiled fiction that gave such scathingly accurate portrayals of well-known townspeople that despite the novel’s enthusiastic reception nation- wide, Wolfe felt he received a cool welcome in his hometown. He proceeded to write You Can’t Go Home Again. The problem you are asked to consider involves one of Wolfe’s lesser-known novels. Exploratory Problem: In lieu of a straight royalty percentage on sales of his book, The Story of a Novel, Thomas Wolfe agreed to accept a sliding scale of 10% on the first 3000 copies sold, 12.5% on the next 4500 copies, and 15% on all copies after that. The book was priced at $1.50. (Elizabeth Nowell, Thomas Wolfe: A Biography, Doubleday, Garden City, NY, 1960.) (a) Write a function R(x) that gives Wolfe’s royalties as a function of x, the number of books sold. You will have to write the formula in pieces, because the formula varies depending upon the number sold. Make sure your formula works as you want it to by checking it out on some concrete cases. Does your formula work if Wolfe sells 3001 copies? 7501 copies? (b) Graph R(x). Compare your answers to parts (a) and (b) with those of some of your classmates. (c) Solve R(x) = 1000. What does this equation mean? (d) Graph R  (x), where R  (x) is the slope function—giving the slope at every point on the graph of R. (e) What is the practical meaning of R  (x)? Include units in your answer. 4.4 Applications of Linear Models: Variations on a Theme 159 4.4 APPLICATIONS OF LINEAR MODELS: VARIATIONS ON A THEME Linear models are used in many contexts. Sometimes they are used to express linear or piecewise linear relationships and other times to deduce information about relationships that, while not actually linear, are locally linear. The examples in this section direct our attention to each of these topics: simultaneous linear equations, piecewise linear functions, and approximating nonlinear behavior locally using linear approximations. ◆ EXAMPLE 4.7 i. Salesman A gets a base salary of $250 per week plus an additional $10 for every item he sells. Salesman B gets a base salary of $200 per week plus a commission of $20 for each item sold. At what number of sales does salesman B’s salary scheme yield a higher weekly income than salesman A’s salary arrangement? 9 SOLUTION Let A(x) and B(x) be the weekly wages (in dollars) of salesmen A and B, respectively, where x denotes the number of items sold in the week. A(x) = 250 + 10x and B(x) = 200 + 20x Graphs of A(x) and B(x) are drawn below. x A B $ 325 300 275 250 225 200 123456 Figure 4.13 We are most likely interested in an integer answer; if we draw very accurate graphs using appropriate scales this information is readily available. If one assumes that x can take on only positive integer values, then it may be reasonable to check salaries for various x- values numerically. However, the quickest way to answer the question is to find out at what value of x the salaries are equal. 9 Because these items are discrete (let’s say they are selling radios, for instance), we could specify that x must be a nonnegative integer and draw our graph as a discrete set of points. In practice it is often simpler to use a continuous model to study phenomena that are actually discrete. (Economists, biologists, and others do it all the time.) Calculus deals with continuous functions, so in this course most of our models will be continuous; however, when solving a problem you will want to interpret your continuous model in a way that makes sense in context. 160 CHAPTER 4 Linearity and Local Linearity 250 + 10x = 200 + 20x 50 = 10x 5 = x So, if a salesperson can sell more than five items per week, then salesman B’s salary scheme pays more. ii. Saleswoman C has been offered the following arrangement. Her base salary is $220 per week. For each of the first six items she sells during the week she will get $5 commission; for each additional item after that she will earn $25 commission. Let x = the number of items she sells each week. Write a function C(x) giving her weekly salary. Make a continuous model with domain [0, ∞). Before looking at the solution, spend some time trying this problem on your own. As a spot-check for errors, check your answer for x = 0, x = 1, x = 6, and x = 7 to make sure that your function “works.” If it does not work, see where it goes astray. 300 280 260 240 220 2468 x (items) $ (Weekly salary) Figure 4.14 SOLUTION C(x) =  220 + 5x for 0 ≤ x ≤ 6, 250 + 25(x − 6) for x>6, i.e., C(x) =  220 + 5x for 0 ≤ x ≤ 6, 100 + 25x for x>6. For 0 ≤ x ≤ 6 we have the standard scheme. At x = 6, the saleswoman is making $250. In a sense, that is the woman’s new base rate if she sells more than six items. She gets $25 commission on the number of items over six. The number of items over six is (x − 6), not x. Therefore, for x>6wehave250+25(x − 6). Common Errors (a) Suppose you tried C(x) =  220 + 5x for 0 ≤ x ≤ 6, 220 + 25x for x>6. This model erroneously indicates that once the seventh item is sold, each item, including the first six, earns a $25 commission. . use demand curves to express the relationship between the price of an item and the number of items demanded by consumers. Below is a demand curve for a certain good. q is the quantity of the. slope of the secant line passing through points P and Q, where P and Q are points of the graph of f(x)with the indicated x-coordinates. 14. f(x)= 3 x + 2x; the x-coordinates of P and Q are x = 3 and. points (π,3 )and (−π,5) 8. Passing through point ( √ 3, √ 2) and parallel to 3x − 4y = 7 9. Passing through the origin and perpendicular to πx − √ 3y = 12 10. x-intercept of √ π and parallel to the