polynomial functions and their canonical representations

... consider functions for which the sets A and B are sets of real numbers The set A is called the domain of the function 1.1 P.7 VALUE AND RANGE The number f(x) is the value of f at x and is read ... curve twice, at (a, b) and (a, c), then the curve can’t represent a function because a function can’t assign two different values to a 1.1 P.40 PIECEWISE-DEFINED FUNCTIONS The functions in the following ... different formulas in different parts of their domains 1.1 P.41 Example A function f is defined by: 1 − x if x ≤1 f ( x)  if x >1 x Evaluate f(0), f(1), and f(2) and sketch the graph 1.1 P.42 Example...

Ngày tải lên: 26/08/2016, 15:16

... change and, 209–211, 711, 983 of inverse trigonometric functions, 703–706 limit definition of, 190, 191 local linearity and, 279–284 of logarithmic functions, 467–471 meaning and notation and, 208–209 ... basic principles of, 964–965 Comparison Test and, 969–971 Integral Test and, 965–969, 972 Limit Comparison Test and, 971–972 Ratio and Root tests and, 972–976 summary of, 977 transition to, 962 ... 712–714 linear functions and, 143–144, 169 Constant solutions See Equilibrium solutions Continuity definition of, 270, 273 Extreme Value Theorem and, 271 Intermediate Value Theorem and, 271 Sandwich...

Ngày tải lên: 05/07/2014, 18:20

... mathematical ideas and representations and making connections between functions and the world around us are important to fostering a conceptual framework that will be both sturdy and portable Generating ... derivative of bx is bx Part III also takes up polynomials and optimization Part IV deals more fully with logarithmic and exponential functions and their derivatives The number e is revisited here ... through VIII and sections of Part IX, although the composition of the syllabus is, of course, at the instructor’s discretion Part I provides an introduction to functions and their representations...

Ngày tải lên: 05/07/2014, 18:20

... 11.3 Polynomial Functions and Their Graphs N 373 391 EXPLORATORY PROBLEMS FOR CHAPTER 11: Functions and Their Graphs: Tinkering with Polynomials and Rational Functions 404 11.4 Rational Functions ... Functions and Their Graphs CHAPTER 12 406 Inverse Functions: A Case Study of Exponential and Logarithmic Functions 421 Inverse Functions: Can What Is Done Be Undone? 421 12.1 What Does It Mean for f and ... of Sums, Products, Quotients, and Power Functions 290 Contents PART III CHAPTER Exponential, Polynomial, and Rational Functions with Applications 303 Exponential Functions 303 9.1 9.2 Exponential...

Ngày tải lên: 05/07/2014, 18:20

... pictures and representations to it; our choice of representation will often depend upon context Upon first exposure, the notation and representations may be confusing, but as you use the language of functions ... mathematical notation and usage may provide lucid relief 4 CHAPTER Functions Are Lurking Everywhere Exploratory Problems for Chapter Calibrating Bottles From The Language of Functions and Graphs: An ... Shell Centre for Mathematical Education 1.2 What Are Functions? Basic Vocabulary and Notation 1.2 WHAT ARE FUNCTIONS? BASIC VOCABULARY AND NOTATION N EXAMPLE 1.1 The following table describes...

Ngày tải lên: 05/07/2014, 18:20

... undefined; Equality of Functions The functions f and g are equal if: f and g have the same domain, and f (x) = g(x) for every x in the domain −x For example, the functions f (x) = x x and g(x) = x − ... x be the list price of a car and let f and g be the tax functions in states and 2, respectively We can describe the input-output relationship of the functions f and g using formulas f (x) = 500 ... 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 times...

Ngày tải lên: 05/07/2014, 18:20

... an “if and only if” statement Language and Logic: An Interlude “A if and only if B” means “A and B are equivalent statements.” Using symbols we write A ⇔ B Specifically, “P is a square if and only ... searching for a way to relate h and r h success! relate r and 2-h using a triangle with hypotenuse Figure 1.12 We can relate r and h by looking at a cross-sectional slice and using a right triangle ... 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 27 W , the number of weeks...

Ngày tải lên: 05/07/2014, 18:20

... one place, then the function is 1-to-1 Functions: The Grand Scheme In this text we will be looking at functions of one variable—but not all functions are functions of one variable For instance, ... can tell that the relationships represented in Figures 1.16(a)–(c), 1.17(a) and (b), and 1.18(b) and (d) are, in fact, functions The test for a function is that every input must have only one output ... any of the functions from part (a) 1-to-1? If so, which ones? In Problems through find the domain of each of the functions (a) f (x) = x+2 (b) g(x) = x +4 x 1.3 Representations of Functions (a)...

Ngày tải lên: 05/07/2014, 18:20

... the pile as a function of its (a) height (b) radius 1.3 Representations of Functions 45 50 In Durham, England, the circular plots of land at the center of the roundabouts17 are often meticulously ... gas and tolls and I estimate that each day costs C cents in wear and tear on the car I have no other expenses Express my daily profit as a function of h, the number of hours I work (A, w, G, and ... create and I minutes to insert (a) What is the estimated amount of time it will take for this typist to create a document N words long and containing Z illustrations? 1.3 Representations of Functions...

Ngày tải lên: 05/07/2014, 18:20

... four functions we 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 ... decreasing on (−∞, 0) and decreasing on (0, ∞) (d) j is undefined and discontinuous at x = (e) j is 1-to-1 Answers to Exercise 2.8 (a) g and h are even functions (b) f and j are odd functions (c) The ... EXERCISE 2.5 Look at each of the functions f (x) = x, g(x) = x 2, h(x) = |x|, and j (x) = and answer the following questions x one by one (a) What are the domain and range of the function? (b)...

Ngày tải lên: 05/07/2014, 18:20

... Characterizing Functions and Introducing Rates of Change T Temperature (°C) (11, 12) 11 t (time) (6, –3) Suppose we want to determine how fast the temperature is increasing between a.m and 11 a.m ... [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 = ∆y ∆x (b, f(b)) ∆y = f(b) ... answer 80 CHAPTER Characterizing Functions and Introducing Rates of Change (d) [1, + h] (Show that your answer agrees with the answers you obtained in parts (a), (b), and (c).) (e) Illustrate your...

Ngày tải lên: 05/07/2014, 18:20

... amount functions and their corresponding rate functions EXERCISE 2.10 Oil is leaking from a point and spreading evenly in a thin, expanding disk We can measure the radius of the disk and want ... position graph and some information about relative position from the velocity graph As we proceed, we will develop a deeper understanding of the connections between position and velocity, and, more ... farthest south? ix Does she begin and end the trip in the same place? SOLUTION First we will answer these question colloquially, and then we will use the notation of functions introduced in Chapter...

Ngày tải lên: 05/07/2014, 18:20

... –20) 94 CHAPTER Characterizing 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 ... Inc 1997 17 Due in part to their methods of record keeping and their hot dry climate, the ancient Egyptians and Babylonians have left modern historians more evidence of their mathematical development ... Characterizing Functions and Introducing Rates of Change death one of their members who revealed to the outside world this dreadful contradiction in beliefs.23 While rational numbers (positive, zero, and...

Ngày tải lên: 05/07/2014, 18:20

... COMPOSITION OF FUNCTIONS Whereas the addition, subtraction, multiplication, and division of functions is simply the addition, subtraction, multiplication, and division of the outputs of these functions, ... g undo one another If h(g(x)) = x and g(h(x)) = x, then h and g are called inverse functions (We first introduced the topic of inverse functions in Section 1.3 and will discuss it in detail in ... Multiplication, and Division of Functions SOLUTION 105 (a) Prices of $0 and $p1 will yield no revenue (In fact, any price above $p1 will yield no revenue.) If widgets are free there is no revenue, and likewise...

Ngày tải lên: 05/07/2014, 18:20

... x−3 and g(x) = x In Problems 39 through 43, find (f + g)(x), (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 ... h(x) = f (g(x)) and j (x) = g(f (x)) What are the domains of h and j ? 35 f (x) = x + and g(x) = 36 f (x) = x+2 √ x and g(x) = x − 118 CHAPTER Functions Working Together 37 f (x) = x and g(x) = −2x ... 24 Two brothers, Max and Eli, are experimenting with their walkie-talkies (A walkietalkie is a combined radio transmitter and receiver light enough to allow the user to walk and talk at the same...

Ngày tải lên: 05/07/2014, 18:20

... 122 CHAPTER Functions Working Together Decompose the functions in Problems through by finding functions f (x) and g(x), f (x) = x and g(x) = x, such that h(x) = f (g(x)) ... Chapter Flipping, Shifting, Shrinking, and Stretching: Exercising Functions In these exercises you will experiment with altering the input and output of functions and draw conclusions about the effects ... + 17 h(x) = 4π 2x + 3π x + In Problems 18 through 20, find functions f , g, and h such that k(x) = f (g(h(x))) and f (x) = x, g(x) = x, and h(x) = x 18 k(x) = √ 32 x +4 19 k(x) = 20 k(x) = √ (...

Ngày tải lên: 05/07/2014, 18:20

... the pocketful of functions you’ve been introduced to (the identity, squaring, reciprocal, and absolute value functions) , graph the following functions Label any asymptotes and x- and y-intercepts ... –2) Graph the functions in Problems 10 through 18 by starting with the graph of a familiar function and applying appropriate shifts, flips, and stretches Label all x- and y-intercepts and the coordinates ... (x) = |x| Graph the functions on the same set of axes 19 g(x) = (x − 3)2 − and f (g(x)) 20 g(x) = −(x + 2)2 + and f (g(x)) 21 g(x) = |x − 2| − and f (g(x)) 22 Let f (x) = x and g(x) = x Using...

Ngày tải lên: 05/07/2014, 18:20

... Lines If L1 and L2 are nonvertical lines with slopes m1 and m2, respectively, then L1 is parallel to L2 if and only if m1 = m2, i.e., their slopes are equal L1 is perpendicular to L2 if and only ... Alternatively, begin with 3x + 2y = and find the x-intercept by setting y = and solving for x: ( , 0) Find the y-intercept by setting x = and solving for y: (0, ) (The x- and y-intercepts can be useful ... m(x − x1) where m is the slope and b is the y-intercept where m is the slope and (x1, y1) is a point on the line 150 CHAPTER Linearity and Local Linearity Scientists and social scientists often...

Ngày tải lên: 05/07/2014, 18:20

... 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 ... 14 f (x) = x + 2x; the x-coordinates of P and Q are x = and x = + k (k = 0), respectively 15 f (x) = x + 3x + 1; the x-coordinates of P and Q are x = b and x = b + h (h = 0), respectively 16 f ... Passing through points (0, a) and (b, 0) Passing through points (π , 3) and (−π, 5) √ √ Passing through point ( 3, 2) and parallel to 3x − 4y = Passing through the origin and perpendicular to π x...

Ngày tải lên: 05/07/2014, 18:20

... quantity? Assume price is measured in dollars and quantity in thousands of units (The equilibrium occurs when supply and demand are equal.) p supply 16 (9, 12) demand q 12 A moving company charges a minimum ... Convert the prices of yogurt and honey into dollars (a) Determine the equation of the supply and demand curves shown in the figure below (b) What are the equilibrium price and quantity? Assume price ... OF A CURVE AND INSTANTANEOUS RATE OF CHANGE In Chapter we looked at linear functions, functions characterized by a constant rate of change This characteristic is unique to linear functions; the...

Ngày tải lên: 05/07/2014, 18:20