derivatives and rates of change

... Law of Cooling (Newton), 326, 985–986, 1002–1004, 1025–1028 Law of Cosines explanation of, 658–659, 709 proof of, 660–661 use of, 661–662 Law of Sines explanation of, 659–660, 709 proof of, 664 ... log of x, 442 Negative numbers, 95 Net change with constant rate of change, 712–714 difference between left- and right-hand sums and, 718–722 explanation of, 712 with nonconstant rate of change, ... explanation of, 903 infinite interval of integration and, 903–907 method of comparison and, 912–914 methods to approach, 908–912 unbounded and discontinuous integrands and, 907–908 use of, 903 Increasing...

Ngày tải lên: 05/07/2014, 18:20

... function and its derivative without being formally introduced to the derivative Part II focuses on rates of change and modeling using linear and quadratic functions Linearity and interpretation of ... structure of this text is intended to facilitate learning deductive and inductive reasoning, learning to use examples and counterexamples, and learning to understand the usefulness of a variety of perspectives ... relationship between meaning and symbolic and graphic representations From the outset the study of functions and the study of calculus are intertwined For example, although the first set of exploratory problems...

Ngày tải lên: 05/07/2014, 18:20

... Basic Vocabulary and Notation 1.3 CHAPTER Functions Are Everywhere N 1.2 Representations of Functions 15 Characterizing Functions and Introducing Rates of Change 49 2.1 Features of a Function: ... Shrinking, Shifting, and Flipping 126 3.4 xiii xiv Contents PART II CHAPTER Rates of Change: An Introduction to the Derivative 139 Linearity and Local Linearity 139 4.1 4.2 4.3 Modeling and Interpreting ... Limits—Methods of Inquiry and a Definition 245 Left- and Right-Handed Limits; Sometimes the Approach Is Critical 7.3 7.4 N CHAPTER 226 A Streetwise Approach to Limits 265 Continuity and the Intermediate and...

Ngày tải lên: 05/07/2014, 18:20

... temperature and the pressure of a gas, an environmental scientist in the relationship between use of pesticides and mortality rate of songbirds, a physician in the relationship between the radius of a ... about it, think of the set of parentheses ( ) as a pair of hands cupped to receive the input The output spurts out of the equal sign 1.2 What Are Functions? Basic Vocabulary and Notation Definition ... expression y = f (x) is shorthand for “y is the output of f corresponding to an input of x” and it is read “y equals f of x.” When we say y = f (x), we mean that y depends on x and is uniquely determined...

Ngày tải lên: 05/07/2014, 18:20

... is the set of all acceptable volumes: [0, B], where B is the volume of the full beaker The volume of a cylinder of height h and radius r is the product of the height and the area of the disk ... 0.04x The graphs of the functions f and g are given below What is the significance of the x-coordinate of the point of intersection of the lines? tax g 500 f cost of car Figure 1.4 REMARKS i The function ... the area of the circle and the function whose input is the radius of a circle and whose output is the area Recall that a “T” sign may sometimes be a landmark indicating a subway station and other...

Ngày tải lên: 05/07/2014, 18:20

... pounds of sugar cane with pounds of panda Question: How many calories does the panda need for one day? C calories · x lbs of panda = Cx calories lbs of panda Question: How many pounds of sugar ... function of the height of the liquid cm 12-h r 12 12 cm h h coffee in a coffee filter Figure 1.11 SOLUTION We can use the volume of a cone to relate the amount of liquid, V , and the height, h, of ... number of pounds of sugar cane it takes to support a pair of pandas, one weighing P pounds and the other weighing Q pounds, for a period of W weeks is a function of 1.3 Representations of Functions...

Ngày tải lên: 05/07/2014, 18:20

... length of the sides of the base and let h denote the height of the box, s and h in inches (a) Express the volume of the box in terms of s and h (b) Express the surface area of the box in terms of ... function of s, the length of the side of the base 30 A rectangle is inscribed in a semicircle of radius R, where R is constant (a) Express the area of the rectangle as a function of the height, h, of ... function of time t, t in seconds 26 Express the area of a circle as a function of: (a) its diameter, d (b) its circumference, c 27 Express the surface area of a cube as a function of the length s of...

Ngày tải lên: 05/07/2014, 18:20

... of the archway by w and the height of the vertical wall of the rectangle y The width and height are given in meters (a) Express y, the height of the side wall, as a function of w, the width of ... typing and a flat rate of $10 per picture The cost of getting a document typed is a function of its length and the number of pictures Write a function that gives a good estimate of the cost of getting ... sign of f may change on either side of a zero or it may remain the same on either side of a zero (There is a lack of consensus on terminology; sometimes one speaks of roots of functions and zeros...

Ngày tải lên: 05/07/2014, 18:20

... Characterizing Functions and Introducing Rates of Change Definition A is inversely proportional to B if A= k B for some constant k As B increases A decreases, and vice versa This type of relationship ... introduced at the beginning of this section: f (x) = x, g(x) = x 2, h(x) = |x|, and j (x) = x (a) Which of functions f , g, h, and j are even? (b) Which of functions f , g, h, and j are odd? (c) Describe ... |2x − (−3)| > (the distance between 2x and −3) > We’ll focus on 2x and solve for x later 68 CHAPTER Characterizing Functions and Introducing Rates of Change –6 –5 –4 –3 –2 –1 Figure 2.16 2x...

Ngày tải lên: 05/07/2014, 18:20

... Characterizing Functions and Introducing Rates of Change (a) The increase in the price of a cup of coffee in Seattle from 1970 to 2000 (b) The percent increase in the price of a cup of coffee in Seattle ... average rate of change of f on [a, b] is the slope of the secant line through the points (a, f (a)) and (b, f (b)) 76 CHAPTER Characterizing Functions and Introducing Rates of Change f(x) slope ... average rate of change of f = f = x y x Geometrically, the average rate of change represents the slope of the line between the two points used This is because the average rate of change of f on [a,...

Ngày tải lên: 05/07/2014, 18:20

... area changed? What is the average rate of change of area with respect to radius? ii When the radius increases from feet to feet, by how much has the area changed? What is the average rate of change ... ratio of integers The decimal expansion of an irrational number is infinite and nonrepeating Some √ √ examples of irrational numbers are π, 2, and 5.14 The set of real numbers is the set of all ... what values of t is |f (t)| < 1? (f ) What are the values of k between −3 and for which f (t) = k has exactly one solution? 90 CHAPTER Characterizing Functions and Introducing Rates of Change f...

Ngày tải lên: 05/07/2014, 18:20

... square feet.) Rate of change : change in area = change in radius A A(3) − A(1) 8π = = = 4π r 3−1 ii Change in area: A(5) − A(3) = 25π − 9π = 16π Rate of change : change in area = change in radius ... Functions and Introducing Rates of Change (a) Describe the trip in words Include where the trip started and ended and how fast (and in what direction) we traveled (b) Graph velocity as a function of ... 16π = = = 8π r 5−3 100 CHAPTER Characterizing Functions and Introducing Rates of Change Notice that as r increases the rate of change of area with respect to the radius also increases; increasing...

Ngày tải lên: 05/07/2014, 18:20

... quotient of the functions I and P If h(x) = f (x) · g(x), then the output of h is the product of the outputs of f and g If j (x) = f (x) , then the output of j is the quotient of the outputs of f and ... sum of functions f and g, h(x) = f (x) + g(x), then the output of h corresponding to an input of x1 is the sum of f (x1) and g(x1) In terms of the graphs, the height of h at x1 is the sum of the ... heights of the graphs of f and g at x1 An analogous statement can be made for subtraction The domain of h is the set of all x common to the domains of both f and g N EXAMPLE 3.2 Let f (x) = x and...

Ngày tải lên: 05/07/2014, 18:20

... Composition of Functions 113 iii Once we know A as a function of r and know r as a function of t we can find A as a function of t using composition of functions A(t) = π t, so each hour the area of the ... x- and y-intercepts of f (x) and g(x) affect the intercepts of f (x)g(x)? State this as a general rule 17 For what values of x (if any) is f (x) g(x) undefined? 18 How the x- and y-intercepts of ... (fg)(x), and domains f g (x), and find their 39 f (x) = ax + b and g(x) = cx + d 40 f (x) = 3x + and g(x) = 5x − 41 f (x) = 2x + and g(x) = x − 42 f (x) = 43 f (x) = x+1 √ and g(x) = x and g(x)...

Ngày tải lên: 05/07/2014, 18:20

... −1, and −1 < k < i What is the effect on the x-intercepts? ii Do the locations of the peaks and valleys change? iii Do the heights of the peaks and valleys change? Alterations to both input and ... How does the graph of y = cf (x) + k (where c and k are constants) relate to that of y = f (x)? In particular, let f (x) = x On the same set of axes sketch the graphs of f (x) and h(x), where h(x) ... innermost part and remember the order of operations rules (b) Let q(x) = − x−4 + Interpret the graph of q in terms of transformations of a familiar function Where are the vertical and horizontal...

Ngày tải lên: 05/07/2014, 18:20

... average rate of change of the function k over the interval [6, + h] (c) Suppose that the average rate of change of the function k over the interval [6, + h] is −5 The functions f , g, and h are ... 2), h(x) = 2k(x) i Which of the following must also be equal to −5? A The average rate of change of the function f over the interval [6, + h] B The average rate of change of the function g over ... rate of change of the function h over the interval [6, + h] ii One of the functions f , g, and h has an average value of −10 on the interval [6, + h] Which is it? Explain briefly P A R T II C H Rates...

Ngày tải lên: 05/07/2014, 18:20

... rate of change of D = = 60 miles per hour N 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 ... slope of the line Verification: Suppose (x1, y1) and (x2, y2) 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 ... 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...

Ngày tải lên: 05/07/2014, 18:20

... 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 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 Economists use demand curves to express the ... curve Let R = the number of units of item and S = the number of units of item S c a R (a) Write an equation for the line in terms of S, R, a, and c (b) Interpret the meanings of the intercepts (c)...

Ngày tải lên: 05/07/2014, 18:20

... the value of x.) Inflation in Turkey has caused prices of small everyday items to be measured in tens of thousands of lira One day I went to a market and purchased one container of yogurt and two ... the equation of the supply and demand curves shown in the figure below (b) What are the equilibrium price and quantity? Assume price is measured in dollars and quantity in thousands of units (The ... average, between hours and 10, what is the rate of increase of temperature with respect to time? In other words, what is the average rate of change of temperature between hour and hour 10? (b) What...

Ngày tải lên: 05/07/2014, 18:20

... a height of 256 ft and given no initial velocity |h| is a handy notation for “the size of h” or “the magnitude of h.” 5.1 Calculating the Slope of a Curve and Instantaneous Rate of Change 175 ... instantaneous rate of change of f at x = a can be approximated by the average rate of change of f on [a, a + h] for h very close to zero As h gets closer and closer to zero, the average rate of change over ... instantaneous rate of change at x = a The instantaneous rate of f (a + h) − f (a) for h small ≈ (a + h) − a change of f at x = a f (a + h) − f (a) The instantaneous rate of = lim , h→0 change of f at x...

Ngày tải lên: 05/07/2014, 18:20