1.2 graph, model, expo, ratio1
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1.2 graph, model, expo, ratio1

Course Number: MATH 1014, Spring 2009

College/University: York University

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S E C T I O N 1 . 2 M AT H E M AT I C A L M O D E L S : A C ATA L O G O F E S S E N T I A L F U N C T I O N S |||| 25 A characteristic feature of linear functions is that they grow at a constant rate. For instance, Figure 2 shows a graph of the linear function f x 3x 2 and a table of sample values. Notice that whenever x increases by 0.1, the value of f x increases by 0.3. So f x increases three times as fast...

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E S C T I O N 1 . 2 M AT H E M AT I C A L M O D E L S : A C ATA L O G O F E S S E N T I A L F U N C T I O N S |||| 25 A characteristic feature of linear functions is that they grow at a constant rate. For instance, Figure 2 shows a graph of the linear function f x 3x 2 and a table of sample values. Notice that whenever x increases by 0.1, the value of f x increases by 0.3. So f x increases three times as fast as x. Thus the slope of the graph y 3x 2, namely 3, can be interpreted as the rate of change of y with respect to x. y x y=3x-2 fx 3x 1.0 1.3 1.6 1.9 2.2 2.5 2 0 _2 x 1.0 1.1 1.2 1.3 1.4 1.5 FIGURE 2 V EXAMPLE 1 (a) As dry air moves upward, it expands and cools. If the ground temperature is 20 C and the temperature at a height of 1 km is 10 C, express the temperature T (in C) as a function of the height h (in kilometers), assuming that a linear model is appropriate. (b) Draw the graph of the function in part (a). What does the slope represent? (c) What is the temperature at a height of 2.5 km? SOLUTION (a) Because we are assuming that T is a linear function of h, we can write T We are given that T 20 when h 20 0, so m0 b 1, so 20 10 and the required linear function is 20 10 C km, and this represents b mh b In other words, the y-intercept is b 20. We are also given that T 10 when h T 20 10 0 10 The slope of the line is therefore m T=_10h+20 m1 20 10h 10 T 1 3 h (b) The graph is sketched in Figure 3. The slope is m the rate of change of temperature with respect to height. (c) At a height of h 2.5 km, the temperature is T 10 2.5 20 FIGURE 3 5C M If there is no physical law or principle to help us formulate a model, we construct an empirical model, which is based entirely on collected data. We seek a curve that ts the data in the sense that it captures the basic trend of the data points. 26 |||| CHAPTER 1 FUNCTIONS AND MODELS V EXAMPLE 2 Table 1 lists the average carbon dioxide level in the atmosphere, measured in parts per million at Mauna Loa Observatory from 1980 to 2002. Use the data in Table 1 to nd a model for the carbon dioxide level. S O L U T I O N We use the data in Table 1 to make the scatter plot in Figure 4, where t repre- sents time (in years) and C represents the CO2 level (in parts per million, ppm). C TA B L E 1 370 Year 1980 1982 1984 1986 1988 1990 CO 2 level (in ppm) 338.7 341.1 344.4 347.2 351.5 354.2 Year 1992 1994 1996 1998 2000 2002 CO 2 level (in ppm) 356.4 358.9 362.6 366.6 369.4 372.9 360 350 340 1980 1985 1990 1995 2000 t F I G U R E 4 Scatter plot for the average CO level Notice that the data points appear to lie close to a straight line, so its natural to choose a linear model in this case. But there are many possible lines that approximate these data points, so which one should we use? From the graph, it appears that one possibility is the line that passes through the rst and last data points. The slope of this line is 372.9 2002 and its equation is C or 1 338.7 1980 338.7 34.2 22 1.5545 t 1.5545 1980 C 1.5545 t 2739.21 Equation 1 gives one possible linear model for the carbon dioxide level; it is graphed in Figure 5. C 370 360 350 340 1980 1985 1990 1995 2000 t FIGURE 5 Linear model through first and last data points Although our model ts the data reasonably well, it gives values higher than most of the actual CO2 levels. A better linear model is obtained by a procedure from statistics S E C T I O N 1 . 2 M AT H E M AT I C A L M O D E L S : A C ATA L O G O F E S S E N T I A L F U N C T I O N S |||| 27 N A computer or graphing calculator nds the regression line by the method of least squares, which is to minimize the sum of the squares of the vertical distances between the data points and the line. The details are explained in Section 14.7. called linear regression. If we use a graphing calculator, we enter the data from Table 1 into the data editor and choose the linear regression command. (With Maple we use the t[leastsquare] command in the stats package; with Mathematica we use the Fit command.) The machine gives the slope and y-intercept of the regression line as m 1.55192 b 2734.55 So our least squares model for the CO2 level is 2 C 1.55192t 2734.55 In Figure 6 we graph the regression line as well as the data points. Comparing with Figure 5, we see that it gives a better t than our previous linear model. C 370 360 350 340 FIGURE 6 The regression line 1980 1985 1990 1995 2000 t M V EXAMPLE 3 Use the linear model given by Equation 2 to estimate the average CO2 level for 1987 and to predict the level for the year 2010. According to this model, when will the CO2 level exceed 400 parts per million? S O L U T I O N Using Equation 2 with t 1987, we estimate that the average CO2 level in 1987 2734.55 349.12 was C 1987 1.55192 1987 This is an example of interpolation because we have estimated a value between observed values. (In fact, the Mauna Loa Observatory reported that the average CO2 level in 1987 was 348.93 ppm, so our estimate is quite accurate.) With t 2010, we get C 2010 1.55192 2010 2734.55 384.81 So we predict that the average CO2 level in the year 2010 will be 384.8 ppm. This is an example of extrapolation because we have predicted a value outside the region of observations. Consequently, we are far less certain about the accuracy of our prediction. Using Equation 2, we see that the CO2 level exceeds 400 ppm when 1.55192 t Solving this inequality, we get t 3134.55 1.55192 2019.79 2734.55 400 28 |||| CHAPTER 1 FUNCTIONS AND MODELS We therefore predict that the CO2 level will exceed 400 ppm by the year 2019. This prediction is somewhat risky because it involves a time quite remote from our observations. P O LY N O M I A L S M A function P is called a polynomial if Px an x n an 1 x n 1 a2 x 2 a1 x a0 where n is a nonnegative integer and the numbers a 0 , a 1, a 2 , . . . , a n are constants called the , . If the coefcients of the polynomial. The domain of any polynomial is leading coefcient a n 0, then the degree of the polynomial is n. For example, the function Px 2 x 6 x 4 2 x 3 s2 5 is a polynomial of degree 6. m x b and so it is a linear function. A polynomial of degree 1 is of the form P x a x 2 b x c and is called a quadratic A polynomial of degree 2 is of the form P x function. Its graph is always a parabola obtained by shifting the parabola y a x 2, as we will see in the next section. The parabola opens upward if a 0 and downward if a 0. (See Figure 7.) y 2 y 2 FIGURE 7 0 1 x 1 x The graphs of quadratic functions are parabolas. (a) y=+x+1 (b) y=_2+3x+1 A polynomial of degree 3 is of the form Px ax 3 bx 2 cx d a 0 and is called a cubic function. Figure 8 shows the graph of a cubic function in part (a) and graphs of polynomials of degrees 4 and 5 in parts (b) and (c). We will see later why the graphs have these shapes. y 1 0 1 x y 2 y 20 x 1 x 1 FIGURE 8 (a) y=-x+1 (b) y=x $-3+x (c) y=3x %-25+60x S E C T I O N 1 . 2 M AT H E M AT I C A L M O D E L S : A C ATA L O G O F E S S E N T I A L F U N C T I O N S |||| 29 Polynomials are commonly used to model various quantities that occur in the natural and social sciences. For instance, in Section 3.7 we will explain why economists often use a polynomial P x to represent the cost of producing x units of a commodity. In the following example we use a quadratic function to model the fall of a ball. TA B L E 2 Time (seconds) 0 1 2 3 4 5 6 7 8 9 Height (meters) 450 445 431 408 375 332 279 216 143 61 EXAMPLE 4 A ball is dropped from the upper observation deck of the CN Tower, 450 m above the ground, and its height h above the ground is recorded at 1-second intervals in Table 2. Find a model to t the data and use the model to predict the time at which the ball hits the ground. S O L U T I O N We draw a scatter plot of the data in Figure 9 and observe that a linear model is inappropriate. But it looks as if the data points might lie on a parabola, so we try a quadratic model instead. Using a graphing calculator or computer algebra system (which uses the least squares method), we obtain the following quadratic model: 3 h (meters) 400 h 449.36 0.96 t h 400 4.90 t 2 200 200 0 2 4 6 8 t (seconds) 0 2 4 6 8 t FIGURE 9 FIGURE 10 Scatter plot for a falling ball Quadratic model for a falling ball In Figure 10 we plot the graph of Equation 3 together with the data points and see that the quadratic model gives a very good t. The ball hits the ground when h 0, so we solve the quadratic equation 4.90 t 2 The quadratic formula gives t The positive root is t 9.7 seconds. P OW E R F U N C T I O N S 0.96 t 449.36 0 0.96 s 0.96 2 2 4 4.90 449.36 4.90 9.67, so we predict that the ball will hit the ground after about M A function of the form f x consider several cases. x a, where a is a constant, is called a power function. We 30 |||| CHAPTER 1 FUNCTIONS AND MODELS (i) a n, where n is a positive integer The graphs of f x x n for n 1, 2, 3, 4, and 5 are shown in Figure 11. (These are polynomials with only one term.) We already know the shape of the graphs of y x (a line through the origin with slope 1) and y x 2 [a parabola, see Example 2(b) in Section 1.1]. y= y 1 1 x 0 1 x y 1 0 1 y=x y 1 y=x # y 1 0 y=x $ y 1 y=x % x 0 1 x 0 1 x F I G U R E 1 1 Graphs of =x n for n=1, 2, 3, 4, 5 The general shape of the graph of f x x n depends on whether n is even or odd. If n n is even, then f x x is an even function and its graph is similar to the parabola y x 2. If n is odd, then f x x n is an odd function and its graph is similar to that of y x 3. Notice from Figure 12, however, that as n increases, the graph of y x n becomes atter near 0 and steeper when x 1. (If x is small, then x 2 is smaller, x 3 is even smaller, x 4 is smaller still, and so on.) y y y=x $ y=x ^ y= (1, 1) 0 y=x # (1, 1) (_1, 1) y=x % x 0 x (_1, _1) FIGURE 12 Families of power functions (ii) a 1 n, where n is a positive integer n The function f x x 1 n sx is a root function. For n 2 it is the square root function f x sx , whose domain is 0, and whose graph is the upper half of the n parabola x y 2. [See Figure 13(a).] For other even values of n, the graph of y sx is 3 similar to that of y sx . For n 3 we have the cube root function f x sx whose domain is (recall that every real number has a cube root) and whose graph is shown in n 3 Figure 13(b). The graph of y sx for n odd n 3 is similar to that of y sx . y y (1,1) 0 x 0 (1,1) x FIGURE 13 Graphs of root functions x (a) = x (b) = S E C T I O N 1 . 2 M AT H E M AT I C A L M O D E L S : A C ATA L O G O F E S S E N T I A L F U N C T I O N S |||| 31 y y= 1 0 1 x 1 The graph of the reciprocal function f x x 1 1 x is shown in Figure 14. Its graph has the equation y 1 x, or x y 1, and is a hyperbola with the coordinate axes as its asymptotes. This function arises in physics and chemistry in connection with Boyles Law, which says that, when the temperature is constant, the volume V of a gas is inversely proportional to the pressure P: (iii) a FIGURE 14 V C P The reciprocal function where C is a constant. Thus the graph of V as a function of P (see Figure 15) has the same general shape as the right half of Figure 14. V FIGURE 15 Volume as a function of pressure at constant temperature 0 P Another instance in which a power function is used to model a physical phenomenon is discussed in Exercise 26. R AT I O N A L F U N C T I O N S A rational function f is a ratio of two polynomials: y fx 20 0 2 x Px Qx where P and Q are polynomials. The domain consists of all values of x such that Q x 0. A simple example of a rational function is the function f x 1 x, whose domain is x x 0 ; this is the reciprocal function graphed in Figure 14. The function fx is a rational function with domain x x ALGEBRAIC FUNCTIONS 2x 4 x 2 x2 4 1 FIGURE 16 2x $-+1 = -4 2 . Its graph is shown in Figure 16. A function f is called an algebraic function if it can be constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots) starting with polynomials. Any rational function is automatically an algebraic function. Here are two more examples: fx sx 2 1 tx x4 x 16 x 2 sx x 3 2 sx 1 32 |||| CHAPTER 1 FUNCTIONS AND MODELS When we sketch algebraic functions in Chapter 4, we will see that their graphs can assume a variety of shapes. Figure 17 illustrates some of the possibilities. y 2 x y y _3 1 1 0 5 x 1 0 1 x FIGURE 17 (a) =x x+3 (b) =$ -25 (c) h(x )=x @ ? #(x-2)@ An example of an algebraic function occurs in the theory of relativity. The mass of a particle with velocity v is m0 m fv 1 v2 c2 s where m 0 is the rest mass of the particle and c a vacuum. TRIGONOMETRIC FUNCTIONS N 3.0 10 5 km s is the speed of light in The Reference Pages are located at the front and back of the book. Trigonometry and the trigonometric functions are reviewed on Reference Page 2 and also in Appendix D. In calculus the convention is that radian measure is always used (except when otherwise indicated). For example, when we use the function f x sin x, it is understood that sin x means the sine of the angle whose radian measure is x. Thus the graphs of the sine and cosine functions are as shown in Figure 18. y 2 y _ _ 2 1 _1 0 2 3 2 _ 2 5 2 _ 1 _1 0 2 3 2 3 2 5 2 3 x x (a) =sin x FIGURE 18 (b) that =cosx Notice for both the sine and cosine functions the domain is is the closed interval 1, 1 . Thus, for all values of x, we have 1 or, in terms of absolute values, sin x 1 cos x 1 sin x 1 1 cos x 1 , and the range Also, the zeros of the sine function occur at the integer multiples of ; that is, sin x 0 when x n n an in teger S E C T I O N 1 . 2 M AT H E M AT I C A L M O D E L S : A C ATA L O G O F E S S E N T I A L F U N C T I O N S |||| 33 An important property of the sine and cosine functions is that they are periodic functions and have period 2 . This means that, for all values of x, sin x 2 sin x cos x 2 cos x The periodic nature of these functions makes them suitable for modeling repetitive phenomena such as tides, vibrating springs, and sound waves. For instance, in Example 4 in Section 1.3 we will see that a reasonable model for the number of hours of daylight in Philadelphia t days after January 1 is given by the function Lt 12 2.8 sin 2 t 365 80 y The tangent function is related to the sine and cosine functions by the equation tan x sin x cos x 1 3 _ _ _ 2 2 0 2 3 2 x and its graph is shown in Figure 19. It is undened whenever cos x 0, that is, when x 2, 3 2, . . . . Its range is , . Notice that the tangent function has period : tan x tan x for all x FIGURE 19 y=tan x The remaining three trigonometric functions (cosecant, secant, and cotangent) are the reciprocals of the sine, cosine, and tangent functions. Their graphs are shown in Appendix D. EXPONENTIAL FUNCTIONS The exponential functions are the functions of the form f x a x , where the base a is a x x positive constant. The graphs of y 2 and y 0.5 are shown in Figure 20. In both cases the domain is , and the range is 0, . y y 1 0 1 x 1 0 1 x FIGURE 20 (a) y=2 (b) y=(0.5) Exponential functions will be studied in detail in Section 1.5, and we will see that they are useful for modeling many natural phenomena, such as population growth (if a 1) and radioactive decay (if a 1 . 34 |||| CHAPTER 1 FUNCTIONS AND MODELS y y=logx y=logx LOGARITHMIC FUNCTIONS 1 0 x The logarithmic functions f x log a x, where the base a is a positive constant, are the inverse functions of the exponential functions. They will be studied in Section 1.6. Figure 21 shows the graphs of four logarithmic functions with various bases. In each case the domain is 0, , the range is , , and the function increases slowly when x 1. 1 y=logx y=logx T R A N S C E N D E N TA L F U N C T I O N S FIGURE 21 These are functions that are not algebraic. The set of transcendental functions includes the trigonometric, inverse trigonometric, exponential, and logarithmic functions, but it also includes a vast number of other functions that have never been named. In Chapter 11 we will study transcendental functions that are dened as sums of innite series. EXAMPLE 5 Classify the following functions as one of the types of functions that we have discussed. (a) f x 5x (b) t x x5 (c) h x SOLUTION 1x 1 sx (d) u t 1 t 5t 4 (a) f x 5 x is an exponential function. (The x is the exponent.) (b) t x x 5 is a power function. (The x is the base.) We could also consider it to be a polynomial of degree 5. 1x (c) h x is an algebraic function. 1 sx (d) u t 1 t 5t 4 is a polynomial of degree 4. M 1.2 EXERCISES 3 4 Match each equation with its graph. Explain your choices. 1 2 Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function. 1. (a) f x 5 sx (Dont use a computer or graphing calculator.) 3. (a) y x2 (b) y x5 y (c) y g h x8 (b) t x x4 (d) r x (f) t x (b) y (d) y t4 (f) y x s1 x2 x3 x2 1 x (c) h x (e) s x 2. (a) y x9 tan 2 x x x 10 x 2t 6 6 6 log10 x x2 sx 1 f 0 x (c) y (e) y x 10 cos sin S E C T I O N 1 . 2 M AT H E M AT I C A L M O D E L S : A C ATA L O G O F E S S E N T I A L F U N C T I O N S |||| 35 4. (a) y (c) y 3x x3 y (b) y (d) y F 3x 3 sx 12. The manager of a weekend ea market knows from past g f x experience that if he charges x dollars for a rental space at the market, then the number y of spaces he can rent is given by the equation y 200 4 x. (a) Sketch a graph of this linear function. (Remember that the rental charge per space and the number of spaces rented cant be negative quantities.) (b) What do the slope, the y-intercept, and the x-intercept of the graph represent? 13. The relationship between the Fahrenheit F and Celsius C G 5. (a) Find an equation for the family of linear functions with slope 2 and sketch several members of the family. (b) Find an equation for the family of linear functions such that f 2 1 and sketch several members of the family. (c) Which function belongs to both families? 6. What do all members of the family of linear functions temperature scales is given by the linear function F 9 C 32. 5 (a) Sketch a graph of this function. (b) What is the slope of the graph and what does it represent? What is the F-intercept and what does it represent? 14. Jason leaves Detroit at 2:00 P M and drives at a constant speed fx 1 m x 3 have in common? Sketch several members of the family. 7. What do all members of the family of linear functions west along I-96. He passes Ann Arbor, 40 mi from Detroit, at 2:50 P M. (a) Express the distance traveled in terms of the time elapsed. (b) Draw the graph of the equation in part (a). (c) What is the slope of this line? What does it represent? 15. Biologists have noticed that the chirping rate of crickets of a fx c x have in common? Sketch several members of the family. 8. Find expressions for the quadratic functions whose graphs are shown. y f (4,2) 0 3 x (_2,2) y (0,1) 0 x (1,_2.5) g certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 113 chirps per minute at 70 F and 173 chirps per minute at 80 F. (a) Find a linear equation that models the temperature T as a function of the number of chirps per minute N. (b) What is the slope of the graph? What does it represent? (c) If the crickets are chirping at 150 chirps per minute, estimate the temperature. 16. The manager of a furniture factory nds that it costs $2200 9. Find an expression for a cubic function f if f 1 6 and f 1 f0 f2 0. 10. Recent studies indicate that the average surface tempera- ture of the earth has been rising steadily. Some scientists have modeled the temperature by the linear function T 0.02 t 8.50, where T is temperature in C and t represents years since 1900. (a) What do the slope and T -intercept represent? (b) Use the equation to predict the average global surface temperature in 2100. 11. If the recommended adult dosage for a drug is D (in mg), to manufacture 100 chairs in one day and $4800 to produce 300 chairs in one day. (a) Express the cost as a function of the number of chairs produced, assuming that it is linear. Then sketch the graph. (b) What is the slope of the graph and what does it represent? (c) What is the y-intercept of the graph and what does it represent? 17. At the surface of the ocean, the water pressure is the same as then to determine the appropriate dosage c for a child of age a, pharmacists use the equation c 0.0417D a 1 . Suppose the dosage for an adult is 200 mg. (a) Find the slope of the graph of c. What does it represent? (b) What is the dosage for a newborn? the air pressure above the water, 15 lb in2. Below the surface, the water pressure increases by 4.34 lb in2 for every 10 ft of descent. (a) Express the water pressure as a function of the depth below the ocean surface. (b) At what depth is the pressure 100 lb in2 ? 36 |||| CHAPTER 1 FUNCTIONS AND MODELS 18. The monthly cost of driving a car depends on the number of miles driven. Lynn found that in May it cost her $380 to drive 480 mi and in June it cost her $460 to drive 800 mi. (a) Express the monthly cost C as a function of the distance driven d, assuming that a linear relationship gives a suitable model. (b) Use part (a) to predict the cost of driving 1500 miles per month. (c) Draw the graph of the linear function. What does the slope represent? (d) What does the y-intercept represent? (e) Why does a linear function give a suitable model in this situation? 19 20 For each scatter plot, decide what type of function you (b) Find and graph a linear model using the rst and last data points. (c) Find and graph the least squares regression line. (d) Use the linear model in part (c) to estimate the ulcer rate for an income of $25,000. (e) According to the model, how likely is someone with an income of $80,000 to suffer from peptic ulcers? ( f ) Do you think it would be reasonable to apply the model to someone with an income of $200,000? ; 22. Biologists have observed that the chirping rate of crickets of a certain species appears to be related to temperature. The table shows the chirping rates for various temperatures. Temperature (F) 50 55 60 65 70 Chirping rate (chirps min) 20 46 79 91 113 Temperature (F) 75 80 85 90 might choose as a model for the data. Explain your choices. 19. (a) y (b) y Chirping rate (chirps min) 140 173 198 211 0 x 0 x 20. (a) y (b) y (a) Make a scatter plot of the data. (b) Find and graph the regression line. (c) Use the linear model in part (b) to estimate the chirping rate at 100 F. ; 23. The table gives the winning heights for the Olympic pole vault competitions in the 20th century. Year 1900 1904 1908 1912 1920 1924 1928 1932 1936 1948 1952 Height ( ft) 10.83 11.48 12.17 12.96 13.42 12.96 13.77 14.15 14.27 14.10 14.92 Year 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 0 x 0 x Height ( ft) 14.96 15.42 16.73 17.71 18.04 18.04 18.96 18.85 19.77 19.02 19.42 ; 21. The table shows (lifetime) peptic ulcer rates (per 100 population) for various family incomes as reported by the National Health Interview Survey. Income $4,000 $6,000 $8,000 $12,000 $16,000 $20,000 $30,000 $45,000 $60,000 Ulcer rate (per 100 population) 14.1 13.0 13.4 12.5 12.0 12.4 10.5 9.4 8.2 (a) Make a scatter plot of these data and decide whether a linear model is appropriate. (a) Make a scatter plot and decide whether a linear model is appropriate. (b) Find and graph the regression line. (c) Use the linear model to predict the height of the winning pole vault at the 2000 Olympics and compare with the actual winning height of 19.36 feet. (d) Is it reasonable to use the model to predict the winning height at the 2100 Olympics? S E C T I O N 1 . 3 N E W F U N C T I O N S F RO M O L D F U N C T I O N S |||| 37 ; 24. A study by the U S Ofce of Science and Technology in 1972 estimated the cost (in 1972 dollars) to reduce automobile emissions by certain percentages: Reduction in emissions (%) 50 55 60 65 70 Cost per car (in $) 45 55 62 70 80 Reduction in emissions (%) 75 80 85 90 95 Cost per car (in $) 90 100 200 375 600 ; 26. The table shows the mean (average) distances d of the planets from the sun (taking the unit of measurement to be the distance from the earth to the sun) and their periods T (time of revolution in years). Planet Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune d 0.387 0.723 1.000 1.523 5.203 9.541 19.190 30.086 T 0.241 0.615 1.000 1.881 11.861 29.457 84.008 164.784 Find a model that captures the diminishing returns trend of these data. ; 25. Use the data in the table to model the population of the world in the 20th century by a cubic function. Then use your model to estimate the population in the year 1925. Year 1900 1910 1920 1930 1940 1950 Population (millions) 1650 1750 1860 2070 2300 2560 Year 1960 1970 1980 1990 2000 Population (millions) 3040 3710 4450 5280 6080 (a) Fit a power model to the data. (b) Keplers Third Law of Planetary Motion states that The square of the period of revolution of a planet is proportional to the cube of its mean distance from the sun. Does your model corroborate Keplers Third Law? 1.3 N E W F U N C T I O N S F RO M O L D F U N C T I O N S In this section we start with the basic functions we discussed in Section 1.2 and obtain new functions by shifting, stretching, and reecting their graphs. We also show how to combine pairs of functions by the standard arithmetic operations and by composition. T R A N S F O R M AT I O N S O F F U N C T I O N S By applying certain transformations to the graph of a given function we can obtain the graphs of certain related functions. This will give us the ability to sketch the graphs of many functions quickly by hand. It will also enable us to write equations for given graphs. Lets rst consider translations. If c is a positive number, then the graph of y f x c is just the graph of y f x shifted upward a distance of c units (because each y-coordinate is increased by the same number c). Likewise, if t x f x c , where c 0, then the value of t at x is the same as the value of f at x c (c units to the left of x). Therefore, the graph of y f x c is just the graph of y f x shifted c units to the right (see Figure 1). VERTIC AL AND HORIZONTAL SHIFTS Suppose c y y y y fx fx fx fx c, shift the graph of y c, shift the graph of y c , shift the graph of y c , shift the graph of y 0. To obtain the graph of f x a distance c uni ts upward f x a distance c uni ts downward f x a distance c uni ts to the right f x a distance c uni ts to the left

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York University - MATH - 1014
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2.1T H E TA N G E N T A N D V E L O C I T Y P RO B L E M S In this section we see how limits arise when we attempt to nd the tangent to a curve or the velocity of an object.T H E TA N G E N T P RO B L E Mt(a) PtCThe word tangent is derived from the
York University - MATH - 1014
88|C H A P T E R 2 L I M I T S A N D D E R I VAT I V E S2.2THE LIMIT OF A FUNCTION Having seen in the preceding section how limits arise when we want to nd the tangent to a curve or the velocity of an object, we now turn our attention to limits in gen
York University - MATH - 1014
S E C T I O N 2 . 3 C A L C U L AT I N G L I M I T S U S I N G T H E L I M I T L AW S|9937. (a) Evaluate the function f xx2 2 x 1000 for x 1, 0.8, 0.6, 0.4, 0.2, 0.1, and 0.05, and guess the value of lim x 2 2x 1000the origin several times. Comment o
York University - MATH - 1014
SECTION 2.5 CONTINUITY|1192.5CONTINUITY We noticed in Section 2.3 that the limit of a function as x approaches a can often be found simply by calculating the value of the function at a. Functions with this property are called continuous at a. We will
York University - MATH - 1014
130|C H A P T E R 2 L I M I T S A N D D E R I VAT I V E S55. Prove that f is continuous at a if and only ifhl062. If a and b are positive numbers, prove that the equationlim f ahfa x3a 2x 21x3b x2056. To prove that sine is continuous, we ne
York University - MATH - 1014
S E C T I O N 2 . 7 D E R I VAT I V E S A N D R AT E S O F C H A N G E|143;(b) Graph v t if v* 1 m s and t 9.8 m s2. How long does it take for the velocity of the raindrop to reach 99% of its terminal velocity? and y 0.1 on a common screen, e discover
Rochester - PHARM - 121
Introduction: AnthonyGiardina(1950)wasborninthewesternsuburbsofBostoninthecityofWaltham, MA.HisparentswereItalianimmigrants;hismother,Angela,wasahomemaker,andhisfather, Felix,workedasasalesmantomovethefamilyfromaworkingclassneighborhoodtoanupper middlecla
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Introduction: Anthony Giardina (1950- ), contemporary American novelist and short story writer, playwright, and college professor of creative writing, was born in the western suburbs of Boston in the city of Waltham, MA. His parents were Italian immigrant
Rochester - PHARM - 121
Introduction: Anthony Giardina (1950- ), contemporary American novelist and short story writer, playwright, and college professor of creative writing, was born in the western suburbs of Boston in the city of Waltham, MA. His parents were Italian immigrant
Rochester - PHARM - 121
Introduction: AnthonyGiardina(1950)wasborninthewesternsuburbsofBostoninthecityofWaltham, MA.HisparentswereItalianimmigrants;hismother,Angela,wasahomemaker,andhisfather, Felix,workedasasalesmantomovethefamilyfromaworkingclassneighborhoodtoanupper middlecla
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Thucydides (c. 460-404 B.C.) Thucydides Olori filii de bello Peloponnesiaco libri octo. Geneva: Henricus Stephanus (Henri Estienne II), 1564.The publisher and editor of this edition of Thucydides History of the Peloponnesian War was Henri II Estienne (15
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Taped Interviews by Lowell Fewster Second Gift of Prof. Dean Harper, 7/23/08[Each real: .65 mil tensilized polyester; 3 in. reel: 30 min. at 3/34 ips recording both directions] Ashford, Laplois Negro Leaders Interview No. 017 Cooper, Dr. Walter Negro L
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Taped Interviews by Lowell Fewster Second Gift of Prof. Dean Harper, 7/23/08[Each real: .65 mil tensilized polyester; 3 in. reel: 30 min. at 3/34 ips recording both directions] Ashford, Laplois Negro Leaders Interview No. 017 Cooper, Dr. Walter Negro L
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T. Richard Long- Correspondence with students in service Adams, Ross Jr. Austin, Alan York Berger, Curtis J. Bruckel, (William) Bill J. Brunner, Bob Caccamise, Charles Davidson, William Frederick Derby, Robert (Bob) E. Filsinger, Robert S. Foultz, Bill Fo
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Stewart, Susan. Columbarium. Chicago: University of Chicago Press, 2004. Cloth. $25. Hoeing $. Abe Books from The Legacy Company, Williamsburg, IA. Notified JL Stewart, Susan. Euripedes Andromache. Oxford University Press, 2001. Paperback. $4.99. Hoeing $
Rochester - LIB - 125
Stewart,Susan.Columbarium.Chicago:UniversityofChicagoPress,2004.Cloth.$25.Hoeing $.AbeBooksfromTheLegacyCompany,Williamsburg,IA.NotifiedJL Stewart,Susan.EuripedesAndromache.OxfordUniversityPress,2001.Paperback.$4.99. Hoeing$.AbeBooksfromMagersandQuinnBook
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Saturday July 2 This day has been the counterpart of yesterday and worse as we are weaker. Had dreadful headache all day. Took some ale and ice and _ a small piece of ham for breakfast. Rain is falling again. Stayed in the upper saloon. Mrs. Morris finall
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Sabbath July 3rd Bright and beautiful. Everything is changed. The sun has brought healing in his wings. We have all been out walling or sitting in the sun and the change has given _ to everything. At 11 o clock, the bell called us down to church. Rev _, a
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D.73 Perkins (Dexter) Papers Box 3 Date 1937 1937 1939 1941 1944 1946 1947 1947-53 1949 1950 1951 1951 1953 1953 1953 1956 1957 1957 1960 1960 1963 1964 1964 1969 1973-74 Title The Constitution and the American Spirit The Love of Knowledge, the Guide of L
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Coetzee,J.M.DiaryofaBadYear.NY:Viking,2007.Cloth.$20.96.YBPyellowslipord. 2/19/08.Hoeing$.NotifiedJL. Lee,LiYoung.BehindMyEyes.NY:W.W.Norton,2008.Cloth.$20.96.YBPyellowslipord. 2/19/08.Hoeing$.NotifiedJL Longenbach,James.TheArtofthePoeticLine.St.Paul,MN:G
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Helen with Bob Hopkins (center) and Jim ForsythHelen with the late David DuttonHelen with Bob Hopkins (left), Jim Forsyth (rear), and David Dexter , right ( the late)Helen with Dan Dutton (left) and Rudolf Kingslake (rear)Helen with Davis Dexter and u
Rochester - LIB - 125
Old Don Lyon Files 1980s (in two boxes) Box I Admission TV Tips Admissions Radio Spots 1967-1983 CAMEROS 83 CAMEROS Lect 84 CASE Entry Meliora Community-University Event Calendar Current Res. Beat 1979 GSM Annual Economic Round-up for Corporate Executives
Rochester - LIB - 125
PROJECT BUDGETCATALOGUE EXPENSES: Printing Specifications: Format of book is 9.25 x 12.25 soft cover book; cover is 4-color process with gloss laminate on one side; interior pages are 4 color process with spot gloss varnish throughout; binding is Smythe
Rochester - LIB - 125
Morrow, Bradford. A lmanac Branch . New York: Simon & Schuster, 1991. Hardcover. 1st edition. $8. Hoeing $. Ordered 1/22/08 via ABE books from Craig Hokenson Bookseller, Dallas, TX. Morrow, Bradford. A riels Crossing . New York: Viking Penguin, 2002. Hard
Rochester - LIB - 125
MaterialsDeposited7/15/08byNanJohnsontoDepartmentofRareBooksandSpecial CollectionsLessing,Doris.DocumentsRelatingtoTheSentimentalAgentsintheVolyenEmpire.London:Jonathan CapeLtd,1983. Lessing,Doris.TheMarriagesBetweenZonesThree,Four,andFive(AsNarratedbyth
Rochester - LIB - 125
Koethe, John. Constructor: Poems. Harper Collins, NY, 1999. 1st ed. Hardcover. $10. Hoeing $ $. Ord. 1/15/08 via ABE Books. Koethe, John. Falling Water: Poems. Harper Perennial, NY, 1997. Softcover. $7. Hoeing $. Ord. 1/15/08 via ABE Books. Koethe, John.
Rochester - LIB - 125
PROJECT STATEMENT The Department of Rare Books, Special Collections & Preservation (RBSCP) is partnering with three scholars (Eugenia Ellis, Jean France, Jonathan Massey) to present a substantial reconsideration of Claude Bragdon (1866-1946), a leading in
Rochester - LIB - 125
T he air cooled as Ragnaros fell into the longest sleep. Grompf always loved the feeling of gnome between his toes. Billions of ludicrous lions prowled outside of the elves camp They watched in horror as their charge devoured the orc.
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And from thence to ascend the _ to a [larger] which is some 70 miles from Moville as t he mouth and the bay at the head and which London day is outdated. The Irish wash, which we [started] first, is rocky and the hills treetop bleak and sterile. We saw no
Rochester - LIB - 125
And from thence to ascend the _ to a [larger] which is some 70 miles from Moville as the mouth and the bay at the head and which London day is outdated. The Irish wash, which we [started] first, is rocky and the hills treetop bleak and sterile. We saw no
Rochester - LIB - 125
Thursday June 30 Awoke with headache and feel listless. Lewis came in our room early feeling somewhat sick and laid in my berth until lunch time. Lemmie and I sat in the dining room reading and writing. All on board were more or less sick and the very atm
Rochester - LIB - 125
Thursday June 30 Awoke with headache and feel listless. Lewis came in our room early feeling somewhat sick and laid in my berth until lunch time. Lemmie and I sat in the dining room reading and writing. All on board were more or less sick and the very atm
Rochester - LIB - 125
Delta Psi Notebooks T i tle Untitled- hall matters . in College Politics Some Facts about Minneapolis Wisconsin Brother Untitled- In deciding this debate I find that I m ust mingle a great deal of criticism with a very lit tle praise To the Iota Cap. Of D
Rochester - LIB - 125
Cornelis de Kiewiet PapersBox 1: Personal Correspondence; 1956-1961-Bills paid; 1959 Stocks; 1959 Income Tax; 1958 Bills paid; 1958 Travel Expenses; 1959 Insurance; C.W. de Kiewiet Work Folder Tax; 1959 Resignation Biographical Information Colgate Roch
Rochester - LIB - 125
Ay754.B86 DC801.N68.L32 1888 DG70.P78 A21b DK215.8.S57 E672.A1p E727.P34c E727.U59n F593.M32t F1863.5 A12w HD2731.B71r HE1051.B71r HG9711.S53f HQ754.B43m LD586.A1y LD2872.5.c67r PR2944.097s PS1358.C68s PS2242.H25t TG145.H37g U17.T69s U103H18e 1862 U130.C8
Rochester - LIB - 125
BerloveBooksBaringGould,WilliamS.SherlockHolmesofBakerStreet:ALifeoftheWorldssFirstConsulting Detective.NewYork:BramhallBooks,1962. Birkett,Jeremy&Richardson,John.LillieLangtry:HerLifeinWordsandPictures.Dorset,England: RupertShuff,Ltd.,1979. Bronston,Sam
Rochester - LIB - 125
Beecher, John. A ll Brave Sailors. The Story of the SS Booker T. Washington. New York: L. B. Fisher, 1945. 1st ed. Hardcover. $45.00. Hoeing $. Ord. 4/25/2008 via ABE Books. From: Charles Seluzicki Fine & Rare Books, Portland, OR. Beecher, John. A nd I Wi
Rochester - LIB - 125
Beecher,John.AllBraveSailors.TheStoryoftheSSBookerT.Washington.NewYork:L.B. Fisher,1945.1sted.Hardcover.$45.00.Hoeing$.Ord.4/25/2008viaABEBooks.From: CharlesSeluzickiFine&RareBooks,Portland,OR. Beecher,John.AndIWillBeHeard:TwoTalkstotheAmericanPeople.NewY
Rochester - LIB - 125
BACKUS (TRUMAN JAY) PAPERS Class of 1864 Diary, 1861 Property of Truman J. Backus, University of Rochester. A.P. Tuesday, January 1, 1861 Commenced the year with eyes open! J.E.G. and I kept private Watch meeting at his house - arose at 8:30 A.M after the
Rochester - LIB - 154
wHi Melissa, Possible Lincoln/Seward Collection items that could be utilized to compare Douglass with Lincoln on slavery, freedom and future of America by Professor Hudsons students might include the following from the Lincoln files. 10.1.1856 Sectionalis
Rochester - LIB - 154
Lessons of LifeAutobiography Written for the 1860 Presidential Campaign, Circa June 1860 (16-20) His background and limited initial formal education. Rail-splitter. Elected to office by votes of the people (except once). The Life and Times of Frederick D
Rochester - LIB - 154
CornelisWillemDeKiewietPersonalandFinancialPapersBox1:PersonalCorrespondence,19561961 BillsPaid1959 Stocks1959 IncomeTax1958 BillsPaid1958 TravelExpense1959 Insurance WorkFolderTax,1959 Resignation BiographicalInformation ColgateRochesterDivinitySchool
Rochester - LIB - 154
#Standard Jet DB#n#b` Ugr@?#h~ 1y0c F Nbm7#"( #cfw_6#Sb#C+93y[v#|*|# # f_ u$ g#'De#Fx#`bT#4.0# # # # # # # # # #
Rochester - LIB - 154
Rochester - LIB - 154
CFB account book 1920-1941 [A.B81 47:1] 1920 Wilkinsons (Jo & Katherine) for boys 1921 " (Jo, Marion & Katherine) for boys $ 500 $10001922 " (Jo, Marion & Katherine) for boys $ 800 June, 1922 from A.S. for H.B.? $ 100 [Spent 6497.81; income from work, ro
Rochester - LIB - 154
Clemente, Vince April 28, 1932 Clifton, Lucille June 27, 1936 Conners, Peter Freudenberger, Nell 1975 Anthony Giardina 1950 Ignatow, David February 17, 1914 Krapf, Norbert 1943 Leavitt, David June 23, 1961 Mason, David December 11, 1954 Sleigh, Tom 1953 Z
Rochester - LIB - 154
BRAGDON RESEARCHER, Ann Garadello (?) THURSDAY 4/5 AROUND NOON The owners of a Bragdon home in Oswego are supposed to be coming in to look at the drawings of the house this Thursday around noon. It is the Sloan house, tube 17 in D. 87, the Bragdon archite
Rochester - LIB - 154
BRAGDON RESEARCHER, THURSDAY 4/5 AROUND NOON The owners of a Bragdon home in Oswego are supposed to be coming in to look at the drawings of the house this Thursday around noon. It is the Sloan house, tube 17 in D. 87, the Bragdon architectural files. In a
Rochester - LIB - 154
Alexander Hooker Papers, 1804-1865 Box I Folder 1 Horace A. Hooker Correspondence Folder 2 Alexander Hooker Personal Correspondence (indexed), 1804-1809 Folder 3 Alexander Hooker- Personal Correspondence (indexed), 1810-1815 Folder 4 Alexander Hooker- Per
Rochester - LIB - 154
Box 37 Folder 1 History Fifty years of NYS Division NYS Division The Seventies NYS Division- The Eighties Folder 2- Calendars of AAUW, 1996-2000 Folder 3- College/Union Representation Materials, 1994-2004 Folder 4- Educational Equity, 1998-2002 Folder 5-
Rochester - LIB - 154
Box 26 Folder 1- Financial Guidelines and Memos to Branch Treasurers- 1974-1996 Folder 2 Board of Directors Meetings- 1984-1989 Folder 3 Treasurers Report 1961-1991, not complete Folder 4 Financial Reports- 1985-1987 Folder 5- Money Market Folder 6- Budge
E. Kentucky - IDK - 23432
_/ 120 + _/ 30= _ (-_ for lateness)= _Literary Analysis Drafting PacketThesis/ Argument Suggested revisions:Name_On TimeRevision Credit 0 Reason One Suggested revisions: 1 2 3 4 5On TimeRevision Credit 0 Reason Two Suggested revisions: 1 2 3 4 5On
E. Kentucky - IDK - 23432
Warmup#3 Whatisthepurposeoflearning aboutanalyticalwritinginhigh school? Howwillithelpyouwithyour futuresuccess?AnalysisDefinedAdaptedbyAlesiaWilliamsfromthe originalbyNatalieBedell,WritingCluster ClicktoeditMastersubtitlestyle LeaderforLouisvilleMaleHi
E. Kentucky - IDK - 23432
AP US HISTORY Colonial History (1600-1763) 1. Separatist vs. non-Separatist Puritans Radical Calvinists against the Church of England; Separatists (Pilgrims) argued for a break from the Church of England, led the Mayflower, and established the settlement
E. Kentucky - IDK - 23432
Chapter 9 The Confederation and the Constitution 1776-1790 The Pursuit of Equality The Continental Army officers formed an exclusive hereditary order called the Society of the Cincinnati. Virginia Statue for Religious Freedom- created in 1786 by Thomas Je
Berkeley - UGBA - 08564
Units sold Sales revenue Less cost of goods sold Gross margin Less operating expenses Net operating incomeThis Year 200,000 $1,000,000 $700,000 $300,000 $210,000 $90,000Last Year 160,000 $800,000 $560,000 $240,000 $198,000 $42,0005.112 Butler Sales Com
Berkeley - UGBA - 08564
Indirect labor Insurance, factory Lubricants for machines Direct labor Depreciation, factory Purchases of raw materials Utilities, factory$220,000 $30,000 $12,500 $210,000 $25,000 $240,000 $10,000 Beginning $12,000 $25,000 $30,000 Ending $14,000 $30,000