an integrated approach to hf

... chapter to 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 ... Explain briefly. PART II Rates of Change: An Introduction to the Derivative 4 CHAPTER Linearity and Local Linearity 4.1 MAKING PREDICTIONS: AN INTUITIVE APPROACH TO LOCAL LINEARITY Every day we are ... the graph of a famil- iar function and applying appropriate shifts, flips, and stretches. Label all x- and y-intercepts and the coordinates of any vertices and corners. Use exact values, not numerical

Ngày tải lên: 05/07/2014, 18:20

... next four days, she spent a total of $72 on food, lodging, transportation, museum entry fees, and baksheesh (tips). She is going to the bank to change enough money to last for three more days ... the past half hour six customers have walked into the corner delicatessen. How many people might the owner expect to miss if he were to close the deli to run an errand for the next 15 minutes? ... necessary to use the x- and y-intercepts in order to calculate the slope; any two points will do. ◆ EXERCISE 4.1 Verify that any equation of the form ax + cy = d, where a, c, and d are constants and

Ngày tải lên: 05/07/2014, 18:20

... subtracting total costs from the total revenue.) (d) How many copies must they make per month in order to break even? (Breaking even means that the profit is zero; the total costs and total revenue ... Economists use 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 ... Figure 4.10 ii. Suppose D is quantity demanded, in bushels, and p is price, in dollars. If D =− 3 2 p + 7, then the slope is D p = change in bushels demanded change in price (in dollars) . For

Ngày tải lên: 05/07/2014, 18:20

... instant such a rate of change can be determined. For a given function we know how to calculate the average rate of change over an interval. We’ll now tackle the problem of calculating an instantaneous ... worse (for instance, using 0 as a lower bound and 75 as an upper bound), but we could do better! To “do better” means to find a larger lower bound and a smaller upper bound so we can pin down the ... plus an additional $10 for each item sold. How many items must a worker sell over the course of a week in order to have the job at Company B to be to her advantage? Please give all possible answers.

Ngày tải lên: 05/07/2014, 18:20

... close to zero. As h gets closer and closer to zero, the average rate of change over the tiny interval approaches the instantaneous rate of change at x = a. The instantaneous rate of change of ... this corresponds to the slope of the secant line through (a, f(a))and (b, f(b)). The instantaneous rate of change of f at x = a can be approximated by the average rate of change of f on [a, a ... instantaneous velocity at t = 2 is less negative than −64.16. We’ve put a higher floor on the instantaneous velocity at t = 2. First we said that the instantaneous velocity was greater than −80

Ngày tải lên: 05/07/2014, 18:20

... attention to another point. (b) Goal: We want to estimate the value of f  (1); that is, we want to approximate the slope of the tangent line to the graph at the point (1, −3). i. Sketch the tangent ... Goal: We want to estimate the value of f  (2), i.e., to approximate the slope of the tangent line to the graph at the point (2, 0). To this end, do the following: i. Sketch the tangent line to the ... (1)? (c) Goal: We want to estimate the value of f  (0); i.e., we want to approximate the slope of the tangent line to the graph at the point (0, −4). i. Sketch the tangent line to the graph at

Ngày tải lên: 05/07/2014, 18:20

... computing derivatives is to simplify the difference quotient to the point that we can factor an h from the numerator to cancel with the h in the denominator (assuming that h = 0) and, once done, the ... here; to make progress we must cancel the h in the denominator. This means getting rid of the square roots. We’re working with an expression, so our options are to multiply by 1 (multiply numerator ... value(s) of x is the slope of the tangent line to g equal to the average rate of change of g on the interval indicated? Illustrate your answer to parts (a) and (b) with pictures. (a) [ 1 2 ,

Ngày tải lên: 05/07/2014, 18:20

... 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 with h going to zero. She wants ... is increasing? Use words to say precisely what you mean. The real purpose of this essay: These are issues that are important for you to understand. We want you to put together what you have learned ... point to get arbitrarily close to the instant in question and found the instantaneous velocity. Adopting a graphical perspective provided us with a new insight; the problem of finding the instantaneous

Ngày tải lên: 05/07/2014, 18:20

... will use to fence off a rectangular tomato plot along the south wall of her house. What should the dimensions of the garden be in order for her to have the most area to grow her tomatoes? 217 ... Derivative Function Answers to Selected Exercises ANSWERS TO EXERCISE 5.5 a, b, c, and d are equal to f  (1). PROBLEMS FOR SECTION 5.4 1. Which of the following are equal to g  (3)? A sketch ... of London. 15 Cloistered in his small hometown, Newton developed calculus in order to understand physical phenomena. The language of fluxions he used to explain his ideas reflected his scientific

Ngày tải lên: 05/07/2014, 18:20

... 229 5. Refer to Problem 4 for your answers to this question. (a) How many solutions are there to 2(x − 3) 2 − 5 =−6? (b) How many solutions are there to −4(x + 1) 2 + 3 =−6? (c) How many solutions ... c, we can solve a system of three simultaneous linear equations 4 for the three unknown constants, a, b, and c. 5 If one of the points we know happens to be the vertex of the parabola (and we ... ways to approach this problem. 2. Quadratics arise naturally in analyzing projectile motion due to the physics of falling objects. In this problem you will work on understanding why. Suppose an

Ngày tải lên: 05/07/2014, 18:20

... that we can solve for the constants a, b, and c. We can use the x-intercepts r 1 and r 2 in equation (ii) and use the other known point to solve for k. Similarly, if we know the vertex and use ... form f(x)= 1 2 (x − 4) 2 − 2 and wanted to find the x-intercepts, we would have to solve a quadratic equation. Rather than multiplying out, it is more efficient to take advantage of the perfect square ... standard parabola y = x 2 shifted 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 k to be negative to flip the graph, and

Ngày tải lên: 05/07/2014, 18:20

... by: Nand Wadhwani, Executive Director, Rehydration Project Tel: +506 656-0504 Prepared by: Nand Wadhwani, Max Pitman and Ami Mody Name of Project: Health Education to Villages: an integrated approach ... families and communities, and individuals who work hard to contribute to those communities Too many of these individuals die before having a fair chance at life, and many more live, but are left to ... diarrhoea and water-related diseases, there is no time to wait It is the moral responsibility of governments, private organisations, and civilians – of anyone who can help – to work toward ending...

Ngày tải lên: 22/03/2014, 15:21

... application mode, topography and meteorology, crop factors and land and water use) Results from risk models will provide Vietnamese planning authorities with tools for guiding future landuse development, ... been successfully transferred to research teams in two universities (Hanoi University of Science and University of Agriculture and Forestry, Ho Chi Minh City), and to a national analytical institute ... strategies in order to minimise their impact on the environment Analysis of pesticide residue Develop an data and input information into integrated system to GIS technology to produce risk assess...

Ngày tải lên: 22/06/2014, 13:20

... 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, 715–718 overview of, 711–712 Newton, ... Instantaneous rate of change calculation of, 170–178, 208 derivative and, 209–211, 711, 983 explanation of, 170, 176, 523 velocity and, 170, 173 Instantaneous velocity average velocity to approximate, ... Conjecture, 335 Constant factor property, 738 Constant function, Constant Multiple Rule, 291–292 Constant of proportionality, 306n Constant rate of change calculating net change in case of, 712–714...

Ngày tải lên: 05/07/2014, 18:20

... An 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 ROBIN J GOTTLIEB H A R V A R D U N I V E R S I T Y Sponsoring Editor: Laurie Rosatone Managing ... offer “Exploratory Problems.” These are integral to the text, and some are referred to in later sections Exploratory problems can be incorporated into the course in many ways, but the bottom line ... 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 in devising an...

Ngày tải lên: 05/07/2014, 18:20

... cases and to use numerical, graphical, and analytic methods to investigate problems We ask that you think about the answers you get, compare them with what you expect, and decide whether your answers ... mathematical ideas and tools To learn mathematics successfully you need to actively involve yourself in your studies and work thoughtfully on problems To otherwise would be like trying to learn to be a ... questions Like a branching road, a single inquiry can lead down multiple paths A path may meander, may lead to a dead-end, detour into fascinating terrain, or steer a straight course toward your destination...

Ngày tải lên: 05/07/2014, 18:20

... vessel and blood pressure, an economist in the relationship between the quantity of an item purchased and its price, a grant manager in the relationship between funds allocated to a program and ... mathematical language in perspective Consider the word “subway,” meaning an underground railway The word conveys a meaning; we associate it with the physical object A Bostonian, a New Yorker, and a Tokyo ... chances are that you’re at a bus stop The symbol takes on a life of its own when a Bostonian says, “I’ll take the ‘T’ downtown.” On the other hand, New Yorkers look for an “M” (for Metropolitan...

Ngày tải lên: 05/07/2014, 18:20

... stocks, and the rest on bonds Let s be the amount he puts into slow-growth stocks, 2s be the amount he puts into riskier stocks, and b be the amount he puts into bonds Then s + 2s + b = 2000 We can ... g is 1 -to- 1 because every output is used only once The function f is not 1 -to- 1 because one output corresponds to more than one input (infinitely many in this case) N Chances are that many of ... times may indicate a bus stop; we can discern the meaning by the context Similarly, A(r) may sometimes be used to indicate the area function and other times be used to indicate the output associated...

Ngày tải lên: 05/07/2014, 18:20

... discussions, so we’ll pause for a moment to clarify the meaning of 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 ... want to confuse 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 ... unit analysis to help us We’re looking for the number of pounds of sugar cane it takes to support a panda weighing x pounds for one day In order to use unit analysis constructively we’ll need to...

Ngày tải lên: 05/07/2014, 18:20

... For instance, suppose you were to calculate the value, V , of the change in your pocket V is a function of q, d, n, and p, where q, d, n, and p are the number of quarters, dimes, nickels, and pennies, ... 1 -to- 1 and a function g that is not 1 -to- 1 1.3 Representations of Functions 39 The graph of a function is given in Problems 16 through 22 Determine the range and domain Is the function 1 -to- 1? ... function is the range of f ; if f maps a to b then its inverse function maps b to a.16 EXERCISE 1.4 Given the graph of a function, how can we determine whether the function is 1 -to- 1? 15 The conventions...

Ngày tải lên: 05/07/2014, 18:20