several stocks and their rates of return

... 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 ... Test for convergence of series, 1129–1131 explanation of, 965–968, 977 proof of, 968 use of, 968–969, 972 Integrands antiderivative of, 805–806 definition of, 783 explanation of, 729, 770 expressed ... 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

... 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 N J G O T ... set of problem-solving skills, and strengthening the students as learners of mathematics and science An integrated course offers freedom, new possibilities, and an invigorating freshness of outlook ... 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

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... 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 ... graph of f with 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 ... to the process of taking point Q closer and closer to P and computing the slope of PQ Trang 414 If we have a formula for f we can get quite good numerical estimates of the slope of the tangent

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... EXERCISE 5.2 Use the limit definition of derivative to find the derivatives of x 2 and 3x. Conclude that the derivative of x 2 + 3x is the sum of the derivatives of x 2 and 3x. Notice that so far the ... Derivative Function (a) How is the graph of √ x − 1 related to that of √ x? (b) How is the graph of the derivative of √ x − 1 related to that of the derivative of √ x? Illustrate with a rough sketch. ... sign of f ' graph of f f ' x rough sketch of f ' f '=0 at x=a, b, c, d a b c d ◆ ◆ EXAMPLE 5.7 Given the graph of f , produce a rough sketch of f  . a x f __ sign of f

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... graph of f doesn’t change its slope. EXERCISE 5.3 Given the graph of f , sketch the graph of f  . Include a number line indicating both the sign of f  and the direction of the graph of f . ... relationship between average rate of change and instantaneous rate of change, and between secant lines and tangent lines. Your classmate is unclear why the definition of derivative involves some limit ... graphs of functions and their derivatives. Can you give labels to the graphs making the con- nections between function and derivative clear? . Identify relationships between the graph of s and the

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... The derivative of a quadratic function is a linear function. ii. This formula for f  is valid regardless of the values of a, b, and c. We can, for example, set b = 0 and c = 0 and conclude that ... derivative of bx is b, and the derivative of the constant c is 0. iii. The value of c does not affect the value of the derivative. This makes geometric sense; c only shifts the graph vertically and ... gis increasing for x<0 his decreasing for x<9/4 graph of f graph of f ′ + – –1 graph of g graph of g′ + – 0 graph of h graph of h′ – + 2.25 parabolas with derivative f  parabolas with

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... 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 f graph of g graph of h graph of j y′ x (ii) y′ x (iii) y′ x (iv) For each of the ... constants, a, b, and c. 5 If 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. ... in Example 6.1 efficiently, and exactly. Question: How can we find the x-coordinate of the vertex of the parabola? Answer: The x-coordinate of the vertex is the zero of the derivative function,

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... x-intercepts of 3 and −2; maximum value of 1 18 x-intercepts of π and 3π ; y-intercept of 6 19 x-intercepts of π and 3π ; y-intercept of −2 20 Vertex at (1, 5); y-intercept of 1 Trang 721 Vertex at (−2, ... 2 12 The set of parabolas with x-intercepts at x = 0 and x = 3 13 The set of parabolas with x-intercepts at x = −5 and x = 1 14 The set of parabolas with vertex (2, 3) 15 The set of parabolas ... negative None of this action will affect the zeros, so we can nail them down with peace of mind.) The x-intercepts are −2 and 3, so −2 and 3 must be zeros of the function The quadratic must be of the

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... the males and females He has 510 meters of fencing to make his corral (a) Let x = the width of the corral and y = the length of the corral Use the fact that Troy has only 510 meters of fencing ... 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 ... Backbone: Limits and Continuity AND A DEFINITION The idea of taking a limit is at the heart of calculus The limiting process allows us to move from calculating average rates of change to determining

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... the derivative of some function K(x) evaluated at x = a (a) Determine K(x) and the value of a (b) Evaluate the limit Trang 4SOLUTION (a) A derivative is the limit of the slope of the secant lines, ... date of hire is D = 1.5 Nelson is thinking of offering his brother a starting salary of $40,000 Adjust S(D) to define S(1.5) appropriately 22 Let f (t) =t (3 + t)t (a) Sketch the graph of f ... Limits—Methods of Inquiry and a Definition 251beginning of this section, we will not be applying this definition much in practice; it is the meaningbehind the definition that is of primary importance

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... 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 and Their Graphs ... 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 ... 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:...

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... 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 ... We can use the volume of a cone to relate the amount of liquid, V , and the height, h, of the liquid V = π r 2h Here r is the radius of the top of the cone of liquid, not of the conical filter ... 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...

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... 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 ... center row of yellow tulips Each row of flowers extends from one side of the plot to the other Express the length of a row of flowers as a function of its distance from the center line of yellow...

Ngày tải lên: 05/07/2014, 18:20

... we return to the 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 ... 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 ... 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) Where...

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 ... h(t) denote the height of a rocketship t seconds after takeoff (a) Express the average rate of change of height of the rocket betweeen and 2.01 seconds after takeoff in terms of the function h (b) ... is measured in seconds, amount of water in liters, and height of the water in centimeters Time Amount of Water Height of Water 0 1.5 4 The average rate of change of volume with respect to time...

Ngày tải lên: 05/07/2014, 18:20

... 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 ... 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 to know ... infinitely many of them Let’s return to our function f If we try to graph f , the best we can is along the lines of the figure below The domain of this function is the set of all real numbers, and the...

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... 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 ... 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 ... 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 negative)...

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