By James Stewart
Stewart's CALCULUS: strategies AND CONTEXTS, FOURTH version deals a streamlined method of instructing calculus, targeting significant ideas and aiding people with targeted definitions, sufferer motives, and thoroughly graded difficulties. CALCULUS: strategies AND CONTEXTS is very looked simply because this article bargains a stability of thought and conceptual paintings to meet extra innovative courses in addition to people who are more well-off instructing in a extra conventional model. each one identify is only one part in a entire calculus direction application that rigorously integrates and coordinates print, media, and expertise items for profitable educating and studying.
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Additional info for Single Variable Calculus: Concepts and Contexts (Stewart's Calculus Series)
Sample text
We will use a similar idea in Chapter 5 to find areas of regions of the type shown in Figure 3. We will approximate the desired area A by areas of rectangles (as in Figure 4), let the width of the rectangles decrease, and then calculate A as the limit of these sums of areas of rectangles. y y y (1, 1) y (1, 1) (1, 1) (1, 1) y=≈ A 0 FIGURE 3 1 x 0 1 4 1 2 3 4 1 x 0 1 x 0 1 n 1 x FIGURE 4 The area problem is the central problem in the branch of calculus called integral calculus. The techniques that we will develop in Chapter 5 for finding areas will also enable us to compute the volume of a solid, the length of a curve, the force of water against a dam, the mass and center of gravity of a rod, and the work done in pumping water out of a tank.
1 0 FIGURE 5 1 FIGURE 6 The notation for intervals is given in Appendix A. x SOLUTION (a) We see from Figure 6 that the point ͑1, 3͒ lies on the graph of f , so the value of f at 1 is f ͑1͒ 3. 7. (b) We see that f ͑x͒ is defined when 0 ഛ x ഛ 7, so the domain of f is the closed interval ͓0, 7͔. Notice that f takes on all values from Ϫ2 to 4, so the range of f is Խ ͕y Ϫ2 ഛ y ഛ 4͖ ͓Ϫ2, 4͔ 14 CHAPTER 1 FUNCTIONS AND MODELS y EXAMPLE 2 Sketch the graph and find the domain and range of each function.
Its graph is always a parabola obtained by shifting the parabola y ax 2, as we will see in the next section. The parabola opens upward if a Ͼ 0 and downward if a Ͻ 0. ) y y 2 2 x 1 0 FIGURE 7 The graphs of quadratic functions are parabolas. 1 x (b) y=_2≈+3x+1 (a) y=≈+x+1 A polynomial of degree 3 is of the form P͑x͒ ax 3 ϩ bx 2 ϩ cx ϩ d a 0 and is called a cubic function. Figure 8 shows the graph of a cubic function in part (a) and graphs of polynomials of degrees 4 and 5 in parts (b) and (c).