an integrative approach to assessment and treatment

Ngày tải lên: 16/03/2014, 03:20

... 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, ... 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 and Bounded Partial Sums Test, 574 Increasing/Decreasing ... 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 use of, 975 Convergence...

Ngày tải lên: 05/07/2014, 18:20

... 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 ... 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 ... between a 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...

Ngày tải lên: 05/07/2014, 18:20

... bones Your understanding should be portable; you ought to be able to take it with you and apply it in a variety of contexts Mathematicians find mathematics exciting and beautiful, and this book may, ... We, the author and your instructors, would like you to leave the course equipped with a toolbox of problem-solving skills and strategies—skills and strategies that you have tried and tested throughout ... 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...

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 ... important and versatile We can attach many different mental pictures and representations to it; our choice of representation will often depend upon context Upon first exposure, the notation 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...

Ngày tải lên: 05/07/2014, 18:20

... numbers and are included in the interval We indicate this on a number line by using filled circles at and 5 (2, 5) denotes the set of all numbers greater than and less than but excluding and themselves ... 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 ... 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

... 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 ... 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 While...

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, ... starting out fresh and tiring at the end First Slice and Tass both run a mile in 1:12 On the same set of axes, graph F (t) and T (t), the distances traveled by First Slice and Tass, respectively, ... 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?...

Ngày tải lên: 05/07/2014, 18:20

... of r (k is a constant.) (d) Suppose the manufacturer decides to switch to plastic lids and bottoms to eliminate the taping problems Assume that custom-made plastic lids and bottoms cost 7k cents ... bikes X miles and walks Y miles to get to work Assume that X, Y , B, H , M, and R are all constants The amount of time it takes the commuter to get to work varies with how long he has to spend at ... The proportionality constant k is the price per pound; at any given time and store it is fixed The proportionality constant may vary with the season and the specific store (a) The cost of a piece...

Ngày tải lên: 05/07/2014, 18:20

... 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 graph of an odd function is said to have symmetry about the origin What does this mean? Graph ... is the distance between x and 3, while |x + 2| (which equals |x − (−2)|) is the distance between x and −2 We can use this geometric interpretation to solve equations and inequalities involving ... Geometric Approach: |2x + 3| > |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...

Ngày tải lên: 05/07/2014, 18:20

... (Show that your answer agrees with the answers you obtained in parts (a), (b), and (c).) (e) Illustrate your answers to parts (a), (b), and (c) by sketching f (x) and drawing secant lines whose ... can be neither even nor odd For example, consider f (x) = x + x Can a function be both even and odd? If your answer is yes, give an example Can you give two examples? 2.3 AVERAGE RATES OF CHANGE ... bottle and discussing how a change in the volume of water changes the height In the case of a cylindrical beaker we saw that the ratio change in height change in volume is constant We refer to this...

Ngày tải lên: 05/07/2014, 18:20

... position and velocity, and, more generally, between amount functions and their corresponding rate functions EXERCISE 2.10 Oil is leaking from a point and spreading evenly in a thin, expanding disk ... 2x + See if you can use your calculator to find the exact values of the zeros and turning points of this function (Some calculators can this.) The answers are: √ √ zeros: x = + and x = − √ turning ... of graphing calculators puts a mountain of information in the palm of your hand and access to it at your fingertips However, there are instances in which it can be difficult to get useful information...

Ngày tải lên: 05/07/2014, 18:20

... than 0.05 (Either use the “zoom” feature of your calculator or change the domain until you can answer this question If your calculator has an equation solver, use that as well and compare the answers ... 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 (b) ... details and a very readable account, consult The History of Mathematics: An Introduction, by David Burton, McGraw-Hill Companies, Inc 1997 17 Due in part to their methods of record keeping and their...

Ngày tải lên: 05/07/2014, 18:20

... to the domains of both f and g N EXAMPLE 3.2 Let f (x) = x and g(x) = x We are familiar with the graphs of f and g We can obtain a rough sketch of f (x) + g(x) = x + x from the graphs of f and ... cases x > and x < separately This is standard protocol for handling absolute values Let f (x) = |x| and g(x) = x Use your knowledge of the graphs of f and g to sketch the graph of h(x) = f (x) + ... outputs of f and g The product is g(x) defined for any x in the domains of both f and g; the quotient is defined for any x in the domains of both f and g provided g(x) is not equal to EXERCISE 3.4...

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 ... Find f (g(x)) and simplify your answer Be sure that your answer is in agreement with the concrete case from part (a) 3x and g(x) = 1−x (a) Find f (g(2)) and g(f (2)) (b) Find f (g(x)) and g(f (x))...

Ngày tải lên: 05/07/2014, 18:20

... (where k is a constant) relate to that of y = f (x)? You might want to break your answer into cases depending upon the sign of k and whether |k| is greater than, less than, or equal to i Does multiplying ... the peaks and valleys change? iii Do the heights of the peaks and valleys change? Alterations to both input and output (a) Let p(x) = − (x + 3)2 − Interpret the graph of p in terms of transformations ... Learning to ask and answer questions that help you retain information is a large part of being a mathematician or scientist More Transformations of Output In addition to shifting, stretching, and...

Ngày tải lên: 05/07/2014, 18:20

... with the graph of a familiar 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 ... 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 y-intercepts ... 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 what you’ve learned in Section...

Ngày tải lên: 05/07/2014, 18:20

... necessary to use the x- and y-intercepts in order to calculate the slope; any two points will N EXERCISE 4.1 Verify that any equation of the form ax + cy = d, where a, c, and d are constants and c ... 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 ... Over 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?...

Ngày tải lên: 05/07/2014, 18:20

... 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 − 10 x-intercept ... time, quantity demanded to be a function of price, and velocity to be a function of time i Suppose s is position, in miles, and t is time, in hours If s = − t + 7, then the slope is s change in ... rate of change of T (w)? Interpret this rate of change in practical terms Economists use demand curves to express the relationship between the price of an item and the number of items demanded by...

Ngày tải lên: 05/07/2014, 18:20

... 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 equilibrium ... than P (2) and a price that is less than P (2) For instance 51 < P (2) < 75, so 51 is a lower bound and 75 is an upper bound We could worse (for instance, using as a lower bound and 75 as an upper ... investor’s money into slow-growing reliable stocks and 30% of the money into high-risk stocks with high growth potential Option R, the riskier option, puts 60% of the money into high-risk stocks and...

Ngày tải lên: 05/07/2014, 18:20