clp_1_dc-pages-36-82.pdf - Chapter 1 L IMITS So very roughly speaking “Differential Calculus” is the study of how a function changes as its input

clp_1_dc-pages-36-82.pdf - Chapter 1 L IMITS So very...

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L IMITS Chapter 1 So very roughly speaking, “Differential Calculus” is the study of how a function changes as its input changes. The mathematical object we use to describe this is the “derivative” of a function. To properly describe what this thing is we need some machinery; in particular we need to define what we mean by “tangent” and “limit”. We’ll get back to defining the derivative in Chapter 2. 1.1 IJ Drawing Tangents and a First Limit Our motivation for developing “limit” — being the title and subject of this chapter — is going to be two related problems of drawing tangent lines and computing velocity. Now — our treatment of limits is not going to be completely mathematically rigorous, so we won’t have too many formal definitions. There will be a few mathematically precise definitions and theorems as we go, but we’ll make sure there is plenty of explanation around them. Let us start with the “tangent line” problem. Of course, we need to define “tangent”, but we won’t do this formally. Instead let us draw some pictures. Figure 1.1.1. Here we have drawn two very rough sketches of the curve y = x 2 for x ě 0. These are not very good sketches for a couple of reasons 29
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L IMITS 1.1 D RAWING T ANGENTS AND A F IRST L IMIT The curve in the figure does not pass through ( 0, 0 ) , even though ( 0, 0 ) lies on y = x 2 . The top-right end of the curve doubles back on itself and so fails the vertical line test that all functions must satisfy 1 — for each x -value there is exactly one y -value for which ( x , y ) lies on the curve y = x 2 . So let’s draw those more carefully. Tangent to the curve at this point Not a tangent line Sketches of the curve y = x 2 . (left) shows a tangent line, while (right) shows a line that is not a tangent. Figure 1.1.2. These are better. In both cases we have drawn y = x 2 (carefully) and then picked a point on the curve — call it P . Let us zoom in on the “good” example: We see that, the more we zoom in on the point P , the more the graph of the function (drawn in black) looks like a straight line — that line is the tangent line (drawn in blue). Figure 1.1.3. We see that as we zoom in on the point P , the graph of the function looks more and more like a straight line. If we kept on zooming in on P then the graph of the function would be indistinguishable from a straight line. That line is the tangent line (which we 1 Take a moment to go back and reread Definition 0.4.1 . 30
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L IMITS 1.1 D RAWING T ANGENTS AND A F IRST L IMIT have drawn in blue). A little more precisely, the blue line is “the tangent line to the func- tion at P ”. We have to be a little careful, because if we zoom in at a different point, then we will find a different tangent line. Now lets zoom in on the “bad” example we see that the blue line looks very different from the function; because of this, the blue line is not the tangent line at P .
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