Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 46 pptx

... panda needs to take in C calories for every pound 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 weighing P pounds and the other weighing Q pounds, for a period o...

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... 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 func...

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... dollars and quantity in thousands of units. (The equilibrium occurs when supply and demand are equal.) (9, 12) 16 12 p q supply demand 4. A moving company charges a minimum of $250 for a move. An ... SLOPE OF A CURVE AND INSTANTANEOUS RATE OF CHANGE In Chapter 4 we looked at linear functions, functions characterized by a constant rate of change. This characteristic is uni...

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... Calculating the Slope of a Curve and Instantaneous Rate of Change 183 the 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-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...

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... a double 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. Denote the radius by r and the height by h. The material used for the top and bottom is...

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... quadratic that we can factor. = x(x − 4)(x + 2) So, the zeros are x = 0, x = 4, and x =−2. ◆ 1 From 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/ or illustrate your answer. 9. A fourth degree polynomial with zeros of multiplicity two at x = 2 and x =−3...

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... noticed this and proceeded to look for the inverse analytically, we would soon realize that there was a problem. Set y = x 2 and then interchange the roles of x and y to obtain an inverse relationship. x ... 1 -to- 1 on its natural domain it is possible to restrict the domain in order to make the function invertible. Note that the domain of f is the range of f −1 and t...

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... Ke x + K = Le x − L, where K and L are constants and 0 <K<L (d) R(1 + n) nx = (P n) x , where P , R, and n are constants (e) 3b x = c x 3 2x , where b and c are constants 5. Solve for x. (a) ... x. SOLUTION Think of what was done to x and undo it. Remember that because you put on your socks and then your shoes, your shoes must come off before your socks. To construct t...

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... coordinate axes u and v because we want x to determine the point P , making sine and cosine functions of x. 19.1 The Sine and Cosine Functions: Definitions and Basic Properties 599 cos x is an even function; ... Example 19.2 and Exercise 19.3 we approximated the values of sin x and cos x for various values of x.Ifxis any integer multiple of π/2, we can evaluate sin x...

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... methods of transforming a difficult integrand into something more manageable. In the final section of this chapter we expand our notion of integration to encompass unbounded integrands and/ or an unbounded ... scientific energy toward understanding it. We build bridges to travel over it, tunnels to travel under it, dams to contain it, plumbing, draining, and irrigation systems...

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... Exercise 29.5 for an outline of the derivation of these reduction formulas. If the powers of sin x and cos x are not positive, it can be useful to express the integrand using tan x and sec x. Integrals ... = 1 2 (1 − cos 2x) and cos 2 x = 1 2 (1 + cos 2x) are useful if the powers of sine and cosine are both positive and even. They are used to reduce powers. On the othe...

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... the same set of axes, graph f(x),P 1 (x), P 2 (x), P 3 (x), and P 4 (x). (c) Use P 1 (x), P 2 (x), P 3 (x), and P 4 (x) to approximate f(0.1) and f(0.3). Compare these approximations to those given ... 34 at which the values of f , f  , and f  can be readily computed. An off-the-cuff approximation of 5 √ 34 is 5 √ 34 ≈ 5 √ 32 = 2; we know that 5 √ 34 is a bit more than...

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... increases. We can determine the “shape” of the trajectories in the xy-plane by using the original differential equations and looking at the relative rates of change of y and x as t varies. The slope of the ... y) where f and g are continuous functions. 9 Suppose we want to analyze the behavior of solution trajectories in the xy-phase-plane. The general direction of a t...

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... know that we must have (2x + )(x + ) and the only question is whether to use − 1and3 or 1and 3and where to place these numbers. 4 If b is to be 5, we’ll want to multiply the 2 by 3, so we have 2x 2 + ... common factor of x − 1 could be extracted. ◆ 3 An expression is said to be factored completely if it is factored and each factor cannot be further factored. 1062 APPENDIX A...

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... (c,0 )and (−c,0 )and the magnitude of the difference of the distances from any point on the hyperbola to the foci is denoted by 2a, then the hyperbola will have x-intercepts of (a,0 )and (−a,0 )and ... has foci at (c,0 )and (−c,0 )and the sum of the distances from any point on the ellipse to the foci is denoted by 2a, then the ellipse will have x-intercepts of (a,0 )and (...

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