Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 12 ppt

... details and a very readable account, consult The History of Mathematics: An Introduction, by DavidBurton, McGraw-Hill Companies, Inc. 1997.17Due in part to their methods of record keeping and their ... climate, the ancient Egyptians and Babylonians haveleft modern historians more evidence of their mathematical development than have the people of other ancient civilizations. Thewritings of the Greek ... problem.94 CHAPTER 2 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)...

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 5 pptx

... panda needs to take in C calories for everypound of the panda’s weight.i. The number of pounds of sugar cane it takes to support a panda for a day is a function of the weight, x, of the panda. ... constants S, N, and C.ii. The number of pounds of sugar cane it takes to support a pair of pandas, one weighingP pounds and the other weighing Q pounds, for a period of W weeks is a function of 28 ... sowe’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 symbols...

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 16 pptx

... 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 ... average rate of change of the function g over the interval [6, 6 + h]C. The average rate of change of the function h over the interval [6, 6 + h]ii. One of the functions f , g, and h has an average ... section of this chapter to the pocketful of functions you’ve been introduced to (the identity, squaring, reciprocal, and absolutevalue functions) , graph the following functions. Label any asymptotes...

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 19 pptx

... dollars and quantity in thousands of units. (The equilibrium occurs when supply and demand are equal.)(9, 12) 16 12 pqsupplydemand4. A moving company charges a minimum of $250 for a move. An ... rate of change over an interval.We’ll now tackle the problem of calculating an instantaneous rate of change. To ﬁnd an average rate of change we need two data points; an instant provides us with ... reliable stocks and 30% of the money into high-risk stocks with high growth potential. Option R, the riskieroption, puts 60% of the money into high-risk stocks and 40% into low-growth stocks.If...

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 21 pptx

... Calculating the Slope of a Curve and Instantaneous Rate of Change 183the slope of the secant line through P and Q. By choosing Q successively closer to P ,guess the slope of the tangent line at P ... Let P and Q be points on the graph of f with x-coordinates3 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 ... Let P and Q be points on the 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...

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 27 ppt

... quantity of gas (in gallons) demanded for heating purposes, is givenby q = mp + b, where m and b are constants (m negative and b positive) and p is theprice of gas per gallon. The gas company ... distance.The distance between f(x )and L is |f(x)−L|.Similarly, the distance between x and3 is|x−3|.Toshow that x is within some distance δ (delta for distance) of 3 but not equal to 3, wewrite 0 < |x ... enough to 3. We can write this morecompactly: If f(x)is within  of L, then the distance between f(x )and L is less than .Let’s use absolute values to express this distance.The distance between...

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 28 ppt

... 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(x)=1xbehave as x approaches zero?SOLUTION ... hit the valueL any number of times.EXERCISE 7.2 Show, using the methods of Example 7.3, that limx→3k = k for any constant k. Concludethat limx→ck = k for c and k any constants.EXERCISE ... follows.DefinitionsIf the values of f(x) stay arbitrarily close to L provided x is sufﬁciently close to abut greater than a, we writelimx→a+f(x)=L and read this as “the limit of f(x)as x approaches a from...

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 29 ppt

... limx→af(x), try values of x close to a but not too close. And then makeyour guess and take your chances. You’ll win some and you’ll lose some. But frequently,thoughtful graphical and numerical investigation ... =−3, x = 1, and x = 5 but doesn’t take care of the removable discon-tinuities at x =− 1and x = 0. The latter reminds us that f(a)must be deﬁned and equal to the left- and right-hand limits. In ... derivative of 2xat x = 0.It is given bylimh→02h− 1h.5Chapter 11 will give you more insight into polynomials and their factorization.7.2 Left- and Right-Handed Limits; Sometimes the Approach...

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 39 pptx

... adouble thickness of aluminum.(a) Give an expression for the volume of the can.(b) Give an expression for the amount of material used. (Remember that the top and bottom of the can are two layers ... ?14. A can for mandarin oranges is a cylinder with volume of 250 cubic centimeters. Denotethe radius by r and the height by h. The material used for the top and bottom is strongerthan that ... value of the function? The absolute minimumvalue?20. An artist wants to frame an 8-inch by 10-inch watercolor landscape with a mauve mat.She wants to use a total of 200 square inches of matting...

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 41 pptx

... quadratic that we can factor.= x(x − 4)(x + 2)So, the zeros are x = 0, x = 4, and x =−2.◆1From Howard Eves, An Introduction to the History of Mathematics, and David Burton, A History of Mathematics.11.2 ... or not the answer to the problem is unique. Explain and/ orillustrate your answer.9. A fourth degree polynomial with zeros of multiplicity two at x = 2 and x =−3, and ay-intercept of −2.10. ... graphs of xkin Figure 11.6, we can see why the names “even” and “odd” were chosen for these types of symmetry. Functions of the form xk, where k is even,have even symmetry, and functions of...

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