Stewart calculus concepts and contexts
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Stewart calculus concepts and contexts
CODCtPIS MilCUBĨÍXĨSA Preview of Calculuswww.pdfgrip.coniCalculus is fundamentally different from the mathematics that you have studied previously. Ca Stewart calculus concepts and contexts alculus is less static and more dynamic. Il is concerned with change and motion; it deals with quantities that approach other quantities. For that reason it may beuseful to have an overview of the subject before beginning its intensive study. Here we give a glimpse of some of the main ideas of calcu Stewart calculus concepts and contexts lus by showing how the concept of a limit arises when we attempt to solve a variety of problems.Fl Tho Area Problem/t = A1+A; + A, + /t4 + A,The origiStewart calculus concepts and contexts
ns of calculus go back at least 2500 years to the ancient Greeks, who found areas using the "method of exhaustion." They knew how to find the area 2$ CODCtPIS MilCUBĨÍXĨSA Preview of Calculuswww.pdfgrip.coniCalculus is fundamentally different from the mathematics that you have studied previously. Ca Stewart calculus concepts and contexts ea of a curved figure. The Greek method of exhaustion was to inscribe polygons in the figure and circumscribe polygons about the figure and then let the number of sides of the polygons increase. Figure 2 illustrates this process for the special case of a circle with inscribed regular polygons.FIGURE Stewart calculus concepts and contexts IFIGURE 2TEC The Preview Module is a numerical and pictorial Invesligarcn the app-oxirroNco cf th? area ol a circle by inscribed and circumscribed poStewart calculus concepts and contexts
lygons.Let .4. be the area of the inscribed polygon with /1 sides. As /1 increases, it appears that 4,, becomes closer and closer to the area of the cCODCtPIS MilCUBĨÍXĨSA Preview of Calculuswww.pdfgrip.coniCalculus is fundamentally different from the mathematics that you have studied previously. Ca Stewart calculus concepts and contexts limits explicitly. However, by indirect reasoning. Eudoxus (fifth century B.c.) used exhaustion to prove the familiar formula for the area of a circle: A = nr.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 ar Stewart calculus concepts and contexts eas 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.VStewart calculus concepts and contexts
/VAV. pdfgrip.com4■ A PREVIEW Of CALCULUSIs it possible to fill a circle with rectangles? Try it tor yourselt.Rsscurces / Mobile 1JTCW" I Rectanựeí h CODCtPIS MilCUBĨÍXĨSA Preview of Calculuswww.pdfgrip.coniCalculus is fundamentally different from the mathematics that you have studied previously. Ca Stewart calculus concepts and contexts r 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.FIGURE 5The tangent line at pX - aFIGURE 6The secant line PỌNow imagine that Q move Stewart calculus concepts and contexts s along the curve toward p as in Figure 7. You can see that the secant line rotates and approaches the tangent line as its limiting position. This meaStewart calculus concepts and contexts
ns that the slope m,v of the secant line becomes closer and closer to the slope m of the tangent line. We writem — lim mfu Ọ ->rand we say that m is tCODCtPIS MilCUBĨÍXĨSA Preview of Calculuswww.pdfgrip.coniCalculus is fundamentally different from the mathematics that you have studied previously. Ca Stewart calculus concepts and contexts -zĩ—-—---X — aFIGURE 7CODCtPIS MilCUBĨÍXĨSA Preview of Calculuswww.pdfgrip.coniCalculus is fundamentally different from the mathematics that you have studied previously. CaGọi ngay
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