By Jerrold Marsden, Alan Weinstein
Undergraduate Texts in arithmetic
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Additional info for Calculus I (Undergraduate Texts in Mathematics)
The calculus we will develop in Chapters 1 to 3 of this book can tell us exactly how many wiggles the graph of a function can have and so will greatly facilitate plotting. A Graphs of functions can have various shapes, but not every set of points in the plane is the graph of a function. Consider, for example, the circle x2 y2 = 5 (Fig. 9). If this circle were the graph of a function f, what would f(2) be? Since (2,l) lies on the circle, we must have f(2) = 1. Since (2, - 1) also lies on the circle, f(2) should also be equal to - 1.
R 2 ; the problem is to The equation must be of the form ( x - 2)2 + ( y determine r2. Since the point ( 5 , 6 ) lies on the circle, it must satisfy the equation. That is, so the correct equation is ( x - 2)' Example 3 Solution + ( y - 1)' Show that the graph of x 2 + y 2 - 6 x - 16y = 34. A + 8 = 0 is a circle. Find its center. Complete the squares: = ( X - 312+ ( y - 812- 65. 06. ~ i ~ 8 . 5 . 2~h~ , parabola y = x2. 53. Parabolas y = ax2 for various values of a. Consider next the equation y = x2.
Let p = m/n and q = k / l . Then = ( b"Ib kn ) I / n 1 (by Property 1 for integer powers) = (bm~)l/nl(bkn)l/nl (by the law (be)'/" = b'l "c I/" just proved) - b ml/nlbk n / n l (by the definition of b "/" ) = bPb4. The other properties are checked in a similar way (Exercises 33 and 34). Rational Powers Rational powers are defined by: b n = b - . . b (n times); b0 = 1 b-" = l / b " bl/" = if b > 0 and n is a natural number b"/" = @")I/" "6 I f b, c > 0 and p, q are rational, then: Example 3 Solution Example 5 Solullon Find g V 2 I 3 and g3l2.
Calculus I (Undergraduate Texts in Mathematics) by Jerrold Marsden, Alan Weinstein