polynomial functions and their canonical representations

... 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, ... 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 ... 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

... u sAn Integrated Approach to Functions and Their Rates of Change P R E L I M I N A R Y E D I T I O N Trang 3C a l c u l u sAn Integrated Approach to Functions and Their Rates of Change R O B I ... mathematical ideas and representations and making nections between functions and the world around us are important to fostering a con-ceptual framework that will be both sturdy and portable Generating ... derivative of bxis bx Part III also takes up polynomials and optimization Part IV deals more fully with logarithmic and exponential functions and their deriva-tives The number e is revisited

Ngày tải lên: 05/07/2014, 18:20

... velocity over the intervals, say, [2, 2.0001] and [1.9999, 2] to get lower and upper bounds We could get better and better approximations by taking smaller and smaller time intervals Let’s save ourselves ... better and better approximations to the instantaneous velocity at t = 2 by looking at the slopes of secant lines passing through (2, 192) and a second point on the graph of s that gets closer and ... of (2, 192) if h is positive and to the left if h is negative As we take values of h closer and closer to zero, this second point slides along the graph of s closer and closer to (2, 192) The slope

Ngày tải lên: 05/07/2014, 18:20

... x-coordinates 3 and 3 + h, respectively (a) Sketch the graph of f and the secant lines through P and Q for h = 1 and h = 0.1 (b) Find the slope of the secant line through P and Q for h = 1, h = 0.1, and ... x-coordinates 3 and 3 + h, respectively (a) Sketch the graph of f and the secant lines through P and Q for h = 1 and h = 0.1 (b) Find the slope of the secant line through P and Q for h = 1, h = 0.1, and ... shorthand for “the slope of the secant line” and mtanas shorthand for “the slope of the tangent line.”) i) g(4+w)−g(4)w A msecthrough points D and F ii) g(4)−g(4−w)w B msecthrough points B and

Ngày tải lên: 05/07/2014, 18:20

... answer to parts (a) and (b) with pictures (a) [12, 2] (b) [1, 4] (c) [c, d] d > c >0 16 Let f (x) =x12 Let P and Q be points on the graph of f with coordinates (x, f (x)) and (x + x, f (x ... decreasing Where f>0 and fis increasing, the graph of f looks like Figure 5.16(a) Where f>0 and fis decreasing, the graph of f looks like Figure 5.16(b): Where f<0 and fis increasing, ... section and the following, we’ll interpret the derivative function fas the slope function Assertions will be made that should make sense intuitively However, we will not prove these assertions and

Ngày tải lên: 05/07/2014, 18:20

... of the values of a, b, and c We can, for example, set b = 0 and c = 0 and conclude that if f (x) = ax2then f
(x)= 2ax Similarly, we know that the derivative of bx is b, and the derivative of the ... range and domain of [−10, 10], we would get a picture that looks like Figure 6.2(a) and provides little information To get useful information from the graph we need to pick a reasonable domain and ... travel 1000 feet up the river bank i Let h(x) = A(f (x)), where f and A are the functions given above Describe in words the input and output of the function h ii Interpret the statement h(700) =

Ngày tải lên: 05/07/2014, 18:20

... S F O R S E C T I O N 6 11 On the left below are graphs of the quadratic functions f , g, h, and j Match f
, g
, h
, and j
with the lines drawn on the right f x g x y′ x (i) h x j x graph of ... r is a root of multiplicity two, then the polynomial has the factor (x − r) 2 as opposed to simply (x − r). Trang 5other terms and determines the sign and behavior of f x2is always positive, ... constants, a, b, and c.5If one of the points we know happens to be the vertex of the parabola (and we are aware of that), then the parabola is determined by just the vertex and one other point.6

Ngày tải lên: 05/07/2014, 18:20

... Trang 16.3 QUADRATICS AND THEIR GRAPHS Graphing Parabolas Beginning with the basic parabola y = x2and using the results of Section 3.4 on shifting, stretching, shrinking, and flipping, we can ... equation (i) and three arbitrary points we can obtain a system of three linear equations that we can solve for the constants a, b, and c We can use the x-intercepts r1and r2in equation (ii) and use ... our standard parabola y = x2shifted left 2 units and stretched and flipped Therefore, it is of the form y = k(x + 2)2 Find k as in Approach I We expect kto be negative to flip the graph, and indeed

Ngày tải lên: 05/07/2014, 18:20

... that q, the quantity of gas (in gallons) demanded for heating purposes, is given by q = mp + b, where m and b are constants (m negative and b positive) and p is the price of gas per gallon The gas ... not enough buyers and the revenue actually goes down Therefore, if a mo-nopolist is attempting to maximize revenue, the momo-nopolist must look at the demand curve Suppose the demand curve for widgets ... in terms of m and b.) (e) Find R
and graph it 6 A catering company is making elegant fruit tarts for a huge college graduation cele-bration The caterer insists on high quality and will not accept

Ngày tải lên: 05/07/2014, 18:20

... since f (x) =x2 for x = 3 and the arguments given above, both formal and informal, hold here as well
Notice that we’ve established that limx→3f (x) = 32 for each of the functions f (x) drawn below ... other hand, “as x → ∞, x2→ ∞” is a reasonable shorthand for “as x grows without bound, x2grows without bound.” Therefore in this text we use the convention that lim x→∞x2= ∞ is a shorthand for ... Let f (t) =t (3 + t)t (a) Sketch the graph of f (t) (b) What are the domain and range of f (t)? 7.2 LEFT- AND RIGHT-HANDED LIMITS; SOMETIMES THE APPROACH IS CRITICAL
EXAMPLE 7.10 How does f

Ngày tải lên: 05/07/2014, 18:20

... 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

... 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