# SECTION 2-2 Straight Lines

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1 - Straight Lines Engineering. The cross section of a rivet has a top that is an arc of a circle (see the figure). If the ends of the arc are 1 millimeters apart and the top is 4 millimeters above the ends, what is the radius of the circle containing the arc? (B) If the compan decides to position the tower on this circle at a point directl east of town A, how far from town A should the place the tower? Compute answer to one decimal place. Rivet 9. Construction. Town B is located 6 miles east and 1 miles north of town A (see the figure). A local telephone compan wants to position a rela tower so that the distance from the tower to town B is twice the distance from the tower to town A. (A) Show that the tower must lie on a circle, find the center and radius of this circle, and graph. Tower Town A (, ) 96. Construction. Repeat Problem 9 if the distance from the tower to town A is twice the distance from the tower to town B. Town B (6, 1) SECTION - Straight Lines Graphs of First-Degree Equations in Two Variables Slope of a Line Equations of a Line Special Forms Parallel and Perpendicular Lines In this section we will consider one of the most basic geometric figures a straight line. We will learn how to graph straight lines, given various standard equations, and how to find the equation of a straight line, given information about the line. Adding these important tools to our mathematical toolbo will enable us to use straight lines as an effective problem-solving tool, as evidenced b the application eercises at the end of this section. Graphs of First- Degree Equations in Two Variables With our past eperience in graphing equations in two variables, ou probabl remember that first-degree equations in two variables, such as 4 9 have graphs that are straight lines. This fact is stated in Theorem 1. For a partial proof of this theorem, see Problem 80 of the eercises at the end of this section. Theorem 1 The Equation of a Straight Line If A, B, and C are constants, with A and B not both 0, and and are variables, then the graph of the equation A B C Standard Form (1) is a straight line. An straight line in a rectangular coordinate sstem has an equation of this form.

2 116 Graphs and Functions Also, the graph of an equation of the form m b () where m and b are constants, is a straight line. Form (), which we will discuss in detail later, is simpl a special case of form (1) for B 0. This can be seen b solving form (1) for in terms of : A B C B B 0 To graph either equation (1) or (), we plot an two points from the solution set and use a straightedge to draw a line through these two points. The points where the line crosses the aes are convenient to use and eas to find. The intercept* is the ordinate of the point where the graph crosses the ais, and the intercept is the abscissa of the point where the graph crosses the ais. To find the intercept, let 0 and solve for ; to find the intercept, let 0 and solve for. It is often advisable to find a third point as a checkpoint. All three points must lie on the same straight line or a mistake has been made. EXAMPLE 1 Using Intercepts to Graph a Straight Line Graph the equation 4 1. Solution Find intercepts, a third checkpoint (optional), and draw a line through the two (three) points (Fig. 1). FIGURE intercept is (4, 0) (8, ) -intercept is 4 (0, ) Check point 10 Matched Problem 1 Graph the equation 4 1. *If the intercept is a and the intercept is b, then the graph of the line passes through the points (a, 0) and (0, b). It is common practice to refer to both the numbers a and b and the points (a, 0) and (0, b) as the and intercepts of the line.

3 - Straight Lines 117 To check the answer to Eample 1 on a graphing utilit, we first solve the equation for and then graph (Fig. ): FIGURE 10 Slope of a Line If we take two points P 1 ( 1, 1 ) and P (, ) on a line, then the ratio of the change in to the change in as we move from point P 1 to point P is called the slope of the line. Roughl speaking, slope is a measure of the steepness of a line. Sometimes the change in is called the run and the change in is called the rise. DEFINITION 1 Slope of a Line If a line passes through two distinct points P 1 ( 1, 1 ) and P (, ), then its slope m is given b the formula m P (, ) Vertical change (rise) Horizontal change (run) P 1 ( 1, 1 ) 1 Run 1 Rise (, 1 ) For a horizontal line, doesn t change as changes; hence, its slope is 0. On the other hand, for a vertical line, doesn t change as changes; hence, 1 and its slope is not defined: For a vertical line, slope is not defined. In general, the slope of a line ma be positive, negative, 0, or not defined. Each of these cases is interpreted geometricall as shown in Table 1.

4 118 Graphs and Functions TABLE 1 Geometric Interpretation of Slope Line Slope Eample Rising as moves from left to right Positive Falling as moves from left to right Negative Horizontal 0 Vertical Not defined In using the formula to find the slope of the line through two points, it doesn t matter which point is labeled P 1 or P, since changing the labeling will change the sign in both the numerator and denominator of the slope formula: For eample: 7 7 In addition, it is important to note that the definition of slope doesn t depend on the two points chosen on the line as long as the are distinct. This follows from the fact that the ratios of corresponding sides of similar triangles are equal. EXAMPLE Finding Slopes Sketch a line through each pair of points and find the slope of each line. (A) (, 4), (, ) (B) (, ), (1, ) (C) ( 4, ), (, ) (D) (, 4), (, ) Solutions (A) (B) (, ) (, ) (, 4) (1, ) m ( 4) ( ) m 1 ( ) 6

5 - Straight Lines 119 (C) (D) ( 4, ) (, ) (, 4) (, ) m ( 4) m 4 ; 7 0 slope is not defined Matched Problem Find the slope of the line through each pair of points. Do not graph. (A) (, ), (, ) (B) (, 1), (1, ) (C) (0, 4), (, 4) (D) (, ), (, 1) Equations of a Line Special Forms Let us start b investigating wh m b is called the slope intercept form for a line. EXPLORE-DISCUSS 1 (A) Graph b for b,, 0,, and simultaneousl in the same coordinate sstem. Verball describe the geometric significance of b. (B) Graph m 1 for m, 1, 0, 1, and simultaneousl in the same coordinate sstem. Verball describe the geometric significance of m. (C) Using a graphing utilit, eplore the graph of m b for different values of m and b. As ou can see from the above eploration, the constants m and b in m b () have special geometric significance, which we now eplicitl state. If we let 0, then b, and we observe that the graph of equation () crosses the ais at (0, b). The constant b is the intercept. For eample, the intercept of the graph of is. To determine the geometric significance of m, we proceed as follows: If m b, then b setting 0 and 1, we conclude that both (0, b) and (1, m b) lie on the graph, which is a line. Hence, the slope of this line is given b Slope 1 (m b) b m 1 1 0

6 10 Graphs and Functions Thus, m is the slope of the line given b m b. Now we know wh equation () is called the slope intercept form of an equation of a line. Theorem Slope Intercept Form m b m Rise Run Slope b intercept -intercept b m b Rise Run EXAMPLE Using the Slope Intercept Form (A) Write the slope intercept equation of a line with slope and intercept. (B) Find the slope and intercept, and graph. 4 Solutions (A) Substitute m and b in m b to obtain b -intercept FIGURE 4 (B) The intercept of 4 is, so the point (0, ) is on the graph. The slope of the line is 4, so when the coordinate of (0, ) increases (runs) b 4 units, the coordinate changes (rises) b. The resulting point (4, ) is easil plotted, and the two points ield the graph of the line. In short, we start at the intercept, and move 4 units to the right and units down to obtain a second point. We then draw a line through the two points, as shown in Figure. Matched Problem Write the slope intercept equation of the line with slope and intercept 1. Graph the equation. In Eample we found the equation of a line with a given slope and intercept. It is also possible to find the equation of a line passing through a given point with a given slope or to find the equation of a line containing two given points. Suppose a line has slope m and passes through a fied point ( 1, 1 ). If the point (, ) is an other point on the line, then 1 1 m 1

7 - Straight Lines 11 that is, 1 m( 1 ) (4) We now observe that ( 1, 1 ) also satisfies equation (4) and conclude that (4) is an equation of a line with slope m that passes through ( 1, 1 ). We have just obtained the point slope form of the equation of a line. Theorem Point Slope Form An equation of a line through a point P 1 ( 1, 1 ) with slope m is 1 m( 1 ) Remember that P(, ) is a variable point and P 1 ( 1, 1 ) is fied. P(, ) P 1 ( 1, 1 ) The point slope form is etremel useful, since it enables us to find an equation for a line if we know its slope and the coordinates of a point on the line or if we know the coordinates of two points on the line. In the latter case, we find the slope first using the coordinates of the two points; then we use the point slope form with either of the two given points. EXAMPLE 4 Using the Point Slope Form (A) Find an equation for the line that has slope and passes through the point (, 1). Write the final answer in the standard form A B C. (B) Find an equation for the line that passes through the two points (4, 1) and ( 8, ). Write the final answer in the slope intercept form m b. Solutions (A) Let m and ( 1, 1 ) (, 1). Then 1 m( 1 ) 1 [ ( )] 1 ( ) 4 7 or 7

8 1 Graphs and Functions (B) First, find the slope of the line b using the slope formula: m 1 1 ( 1) Now let ( 1, 1 ) be either of the two given points and proceed as in part A we choose ( 1, 1 ) (4, 1): 1 m( 1 ) ( 1) 1 ( 4) 1 1 ( 4) You should verif that using ( 8, ), the other given point, produces the same equation. Matched Problem 4 (A) Find an equation for the line that has slope and passes through the point (, ). Write the final answer in the standard form A B C. (B) Find an equation for the line that passes through the two points (, 1) and (7, ). Write the final answer in the slope intercept form m b. EXAMPLE Business Markup Polic A sporting goods store sells a fishing rod that cost \$60 for \$8 and a pair of crosscountr ski boots that cost \$80 for \$106. (A) If the markup polic of the store for items that cost more than \$0 is assumed to be linear and is reflected in the pricing of these two items, write an equation that relates retail price R to cost C. (B) Use the equation to find the retail price for a pair of running shoes that cost \$40. (C) Check with a graphing utilit. Solutions (A) If the retail price R is assumed to be linearl related to cost C, then we are looking for an equation whose graph passes through (C 1, R 1 ) (60, 8) and (C, R ) (80, 106). We find the slope, and then use the point slope form to find the equation. m R R C C R R 1 m(c C 1 ) R 8 1.(C 60) R 8 1.C 7 R 1.C 10

9 - Straight Lines 1 10 (B) R 1.(40) 10 \$8 (C) The check is shown in Figure FIGURE 4 0 Matched Problem The management of a compan that manufactures ballpoint pens estimates costs for running the compan to be \$00 per da at zero output and \$700 per da at an output of 1,000 pens. (A) Assuming total cost per da C is linearl related to total output per da, write an equation relating these two quantities. (B) What is the total cost per da for an output of,000 pens? The simplest equations of lines are those for horizontal and vertical lines. Consider the following two equations: 0 a or a () 0 b or b (6) In equation (), can be an number as long as a. Thus, the graph of a is a vertical line crossing the ais at (a, 0). In equation (6), can be an number as long as b. Thus, the graph of b is a horizontal line crossing the ais at (0, b). We summarize these results as follows: Theorem 4 Vertical and Horizontal Lines Equation Graph a (short for 0 a) Vertical line through (a, 0) (Slope is undefined.) b (short for 0 b) Horizontal line through (0, b) (Slope is 0.) a b b a

10 14 Graphs and Functions EXAMPLE 6 Graphing Horizontal and Vertical Lines Graph the line and the line. Solution Matched Problem 6 Graph the line 4 and the line. The various forms of the equation of a line that we have discussed are summarized in Table for convenient reference. TABLE Equations of a Line Standard form A B C A and B not both 0 Slope intercept form m b Slope: m; intercept: b Point slope form 1 m( 1 ) Slope: m; Point: ( 1, 1 ) Horizontal line b Slope: 0 Vertical line a Slope: Undefined EXPLORE-DISCUSS Determine conditions on A, B, and C so that the linear equation A B C can be written in each of the following forms, and discuss the possible number of and intercepts in each case. 1. m b, m 0. b. a Parallel and Perpendicular Lines From geometr, we know that two vertical lines are parallel to each other and that a horizontal line and a vertical line are perpendicular to each other. How can we tell

11 - Straight Lines 1 when two nonvertical lines are parallel or perpendicular to each other? Theorem, which we state without proof, provides a convenient test. Theorem Parallel and Perpendicular Lines Given two nonvertical lines L 1 and L with slopes m 1 and m, respectivel, then L 1 L if and onl if m 1 m L 1 L if and onl if m 1 m 1 The smbols and mean, respectivel, is parallel to and is perpendicular to. In the case of perpendicularit, the condition m 1 m 1 also can be written as m 1 m 1 or m 1 1 m Thus: Two nonvertical lines are perpendicular if and onl if their slopes are the negative reciprocals of each other. EXAMPLE 7 Parallel and Perpendicular Lines Given the line: L: and the point P(, ), find an equation of a line through P that is: (A) Parallel to L (B) Perpendicular to L Write the final answers in the slope intercept form m b. Solutions First, find the slope of L b writing in the equivalent slope intercept form m b: Thus, the slope of L is. The slope of a line parallel to L is the same,, and the slope of a line perpendicular to L is. We now can find the equations of the two lines in parts A and B using the point slope form.

12 16 Graphs and Functions (A) Parallel (m ): (B) Perpendicular (m ): 1 m( 1 ) 1 m( 1 ) ( ) ( ) 9 19 Matched Problem 7 Given the Line L: 4 and the point P(, ), find an equation of a line through P that is: (A) Parallel to L (B) Perpendicular to L Write the final answers in the slope intercept form m b. Answers to Matched Problems 1.. (A) m 0 (B) m 1 (C) m 4 (D) m is not defined (A) 4 (B) 1. (A) C (B) \$, (A) 1 (B) 4

13 - Straight Lines 17 EXERCISE - A. In Problems 1 6, use the graph of each line to find the intercept, intercept, and slope. Write the slope-intercept form of the equation of the line Graph each equation in Problems 7 0, and indicate the slope, if it eists. Check our graphs in Problems 7 0 b graphing each on a graphing utilit In Problems 1 4, find the equation of the line with the indicated slope and intercept. Write the final answer in the standard form A B C, A Slope 1; intercept 0. Slope 1; intercept 7. Slope ; intercept 4 4. Slope ; intercept 6

14 18 Graphs and Functions B In Problems 8, find the equation of the line passing through the given point with the given slope. Write the final answer in the slope-intercept form m b.. (0, ); m 6. (4, 0); m 7. (, 4); m 8. (, ); m In Problems 9 4, find the equation of the line passing through the two given points. Write the final answer in the slope-intercept form m b or in the form c. 9. (, ); (4, ) 0. ( 1, 4); (, ) 1. (, ); (, ). (0, ); (, ). ( 4, ); (0, ) 4. (, 4); (, 6) In Problems 46, write an equation of the line that contains the indicated point and meets the indicated condition(s). Write the final answer in the standard form A B C, A 0.. (, 1); parallel to 7 6. (, ); parallel to 4 7. (0, 4); parallel to 9 8. (, 0); parallel to (, ); parallel to ais 40. (, 1); parallel to ais 41. (4, ); perpendicular to 4 4. ( 1, ); perpendicular to 4. (, 0); perpendicular to (0, ); perpendicular to 1 4. (, ); perpendicular to ais 46. (1, 7); perpendicular to ais In Problems 47 0, classif the quadrilateral ABCD with the indicated vertices as a trapezoid, a parallelogram, a rectangle, or none of these. 47. A(, ); B(8, 7); C(10, 1); D( 4, 6) 48. A(, ); B(, 4); C(6, 10); D(4, 4) 49. A(0, ); B(4, 1); C(1, ); D(, ) 0. A( 6, ); B(, 7); C(, 4); D( 4, 1) 1. Find the equation of the perpendicular bisector of the line segment joining ( 4, ) and (, 4) b using the pointslope form of the equation of a line.. Solve Problem 1 b using the distance between two points formula, and compare the results. 4 Problems 8 are calculus-related. Recall that a line tangent to a circle at a point is perpendicular to the radius drawn to that point (see the figure). Find the equation of the line tangent to the circle at the indicated point. Write the final answer in the standard form A B C, A 0. Graph the circle and the tangent line on the same coordinate sstem.., (, 4) , ( 8, 6). 0, (, ) 6. 80, ( 4, 8) 7. ( ) ( 4) 169, (8, 16) 8. ( ) ( 9) 89, ( 1, 6) C 9. (A) Graph the following equations in the same coordinate sstem: 6 6 (B) From our observations in part A, describe the famil of lines obtained b varing C in A B C while holding A and B fied. (C) Verif our conclusions in part B with a proof. 60. (A) Graph the following two equations in the same coordinate sstem: (B) Graph the following two equations in the same coordinate sstem: 1 1 (C) From our observations in parts A and B, describe the apparent relationship of the graphs of A B C and B A C. (D) Verif our conclusions in part C with a proof.

15 - Straight Lines 19 Sketch the graphs of the equations in Problems Describe the relationship between the graphs of m b and m b. (See Problems 61 and 6.) 68. Describe the relationship between the graphs of m b and m b. (See Problems 6 and 64.) 69. Prove that if a line L has intercept (a, 0) and intercept (0, b), then the equation of L can be written in the intercept form In Problems 70 and 71, write the equation of the line with the indicated intercepts in the standard form A B C, A (, 0) and (0, ) 71. (, 0) and (0, 7) 7. Let P 1 ( 1, 1 ) P 1 ( 1, m 1 b) P (, ) P (, m b) P (, ) P (, m b) be three arbitrar points that satisf m b with 1. Show that P 1, P, and P are collinear; that is, the lie on the same line. [Hint: Use the distance formula and show that d(p 1, P ) d(p, P ) d(p 1, P ).] This proves that the graph of m b is a straight line. APPLICATIONS 7. Boiling Point of Water. At sea level, water boils when it reaches a temperature of 1 F. At higher altitudes, the atmospheric pressure is lower and so is the temperature at which water boils. The boiling point B in degrees Fahrenheit at an altitude of feet is given approimatel b (A) Complete Table 1. TABLE 1 B ,000 10,000 1,000 0,000,000 0,000 B a b 1 a, b 0 (B) Based on the information in the table, write a brief verbal description of the relationship between altitude and the boiling point of water. 74. Air Temperature. As dr air moves upward, it epands and cools. The air temperature A in degrees Celsius at an altitude of kilometers is given approimatel b (A) Complete Table. TABLE 0 A A 9 (B) Based on the information in the table, write a brief verbal description of the relationship between altitude and air temperature. 7. Car Rental. A car rental agenc computes dail rental charges for compact cars with the equation c 0. where c is the dail charge in dollars and is the dail mileage. Translate this algebraic statement into a verbal statement that can be used to eplain the dail charges to a customer. 76. Installation Charges. A telephone store computes charges for phone installation with the equation c where c is the installation charge in dollars and is the time in minutes spent performing the installation. Translate this algebraic statement into a verbal statement that can be used to eplain the installation charges to a customer. Merck & Co., Inc., is the world s largest pharmaceutical compan. Problems 77 and 78 refer to the data in Table, taken from the compan s 199 annual report. TABLE Sales Net income 1 4 Selected Financial Data (billion \$) for Merck & Co., Inc \$.9 \$ \$6. \$ Sales Analsis. A mathematical model for Merck s sales is given b where 0 corresponds to \$7.7 \$ \$8.6 \$ \$9.7 \$.4

16 10 Graphs and Functions (A) Complete Table 4. Round values of to one decimal place. (B) Sketch the graph of and the sales data on the same aes. (C) Use the modeling equation to estimate the sales in 199. In 000. (D) Write a brief verbal description of the compan s sales from 1988 to Income Analsis. A mathematical model for Merck s income is given b where 0 corresponds to (A) Complete Table. Round values of to one decimal place. TABLE Net income TABLE 4 Sales (B) Sketch the graph of the modeling equation and the income data on the same aes. (C) Use the modeling equation to estimate the income in 199. In 000. (D) Write a brief verbal description of the compan s income from 1988 to Phsics. The two temperature scales Fahrenheit (F) and Celsius (C) are linearl related. It is known that water freezes at F or 0 C and boils at 1 F or 100 C. (A) Find a linear equation that epresses F in terms of C. (B) If a European famil sets its house thermostat at 0 C, what is the setting in degrees Fahrenheit? If the outside temperature in Milwaukee is 86 F, what is the temperature in degrees Celsius? (C) What is the slope of the graph of the linear equation found in part A? (The slope indicates the change in Fahrenheit degrees per unit change in Celsius degrees.) Phsics. Hooke s law states that the relationship between the stretch s of a spring and the weight w causing the stretch is linear (a principle upon which all spring scales are constructed). For a particular spring, a -pound weight causes a stretch of inches, while with no weight the stretch of the spring is 0. (A) Find a linear equation that epresses s in terms of w. (B) What weight will cause a stretch of.6 inches? (C) What is the slope of the graph of the equation? (The slope indicates the amount of stretch per pound increase in weight.) 81. Business Depreciation. A cop machine was purchased b a law firm for \$8,000 and is assumed to have a depreciated value of \$0 after ears. The firm takes straight-line depreciation over the -ear period. (A) Find a linear equation that epresses value V in dollars in terms of time t in ears. (B) What is the depreciated value after ears? (C) What is the slope of the graph of the equation found in part A? Interpret verball. 8. Business Markup Polic. A clothing store sells a shirt costing \$0 for \$ and a jacket costing \$60 for \$9. (A) If the markup polic of the store for items costing over \$10 is assumed to be linear, write an equation that epresses retail price R in terms of cost C (wholesale price). (B) What does a store pa for a suit that retails for \$40? (C) What is the slope of the graph of the equation found in part A? Interpret verball. 8. Flight Conditions. In stable air, the air temperature drops about F for each 1,000-foot rise in altitude. (A) If the temperature at sea level is 70 F and a commercial pilot reports a temperature of 0 F at 18,000 feet, write a linear equation that epresses temperature T in terms of altitude A (in thousands of feet). (B) How high is the aircraft if the temperature is 0 F? (C) What is the slope of the graph of the equation found in part A? Interpret verball. 84. Flight Navigation. An airspeed indicator on some aircraft is affected b the changes in atmospheric pressure at different altitudes. A pilot can estimate the true airspeed b observing the indicated airspeed and adding to it about % for ever 1,000 feet of altitude. (A) If a pilot maintains a constant reading of 00 miles per hour on the airspeed indicator as the aircraft climbs from sea level to an altitude of 10,000 feet, write a linear equation that epresses true airspeed T (miles per hour) in terms of altitude A (thousands of feet). (B) What would be the true airspeed of the aircraft at 6,00 feet? (C) What is the slope of the graph of the equation found in part A? Interpret verball.

17 - Functions Oceanograph. After about 9 hours of a stead wind, the height of waves in the ocean is approimatel linearl related to the duration of time the wind has been blowing. During a storm with 0-knot winds, the wave height after 9 hours was found to be feet, and after 4 hours it was 40 feet. (A) If t is time after the 0-knot wind started to blow and h is the wave height in feet, write a linear equation that epresses height h in terms of time t. (B) How long will the wind have been blowing for the waves to be 0 feet high? Epress all calculated quantities to three significant digits. 86. Oceanograph. As a diver descends into the ocean, pressure increases linearl with depth. The pressure is 1 pounds per square inch on the surface and 0 pounds per square inch feet below the surface. (A) If p is the pressure in pounds per square inch and d is the depth below the surface in feet, write an equation that epresses p in terms of d. (B) How deep can a scuba diver go if the safe pressure for his equipment and eperience is 40 pounds per square inch? 87. Medicine. Cardiovascular research has shown that above the 10 cholesterol level, each 1% increase in cholesterol level increases coronar risk %. For a particular age group, the coronar risk at a 10 cholesterol level is found to be and at a level of 1 the risk is found to be (A) Find a linear equation that epresses risk R in terms of cholesterol level C. (B) What is the risk for a cholesterol level of 60? (C) What is the slope of the graph of the equation found in part A? Interpret verball. Epress all calculated quantities to three significant digits. 88. Demographics. The average number of persons per household in the United States has been shrinking steadil for as long as statistics have been kept and is approimatel linear with respect to time. In 1900, there were about 4.76 persons per household and in 1990, about.. (A) If N represents the average number of persons per household and t represents the number of ears since 1900, write a linear equation that epresses N in terms of t. (B) What is the predicted household size in the ear 000? Epress all calculated quantities to three significant digits. SECTION - Functions Definition of a Function Functions Defined b Equations Function Notation Application A Brief Histor of the Function Concept The idea of correspondence plas a central role in the formulation of the function concept. You have alread had eperiences with correspondences in everda life. For eample: To each person there corresponds an age. To each item in a store there corresponds a price. To each automobile there corresponds a license number. To each circle there corresponds an area. To each number there corresponds its cube. One of the most important aspects of an science (managerial, life, social, phsical, computer, etc.) is the establishment of correspondences among various tpes of phenomena. Once a correspondence is known, predictions can be made. A chemist can use a gas law to predict the pressure of an enclosed gas, given its temperature. An engineer can use a formula to predict the deflections of a beam subject to different loads. A computer scientist can use formulas to compare the efficienc of algorithms for sorting data stored in a computer. An economist would like to be able to predict interest rates, given the rate of change of the mone suppl. And so on.

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