By Robert Carl Yates

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**Additional info for Analytic geometry with calculus**

**Example text**

X + 1)2 [Form: y = u°] 3x2 D=y = '(x' + 1)-12(3x2) _ 2(x' + 1)1/2 ' (D) r = sin2O - cos 0. Here sine 0 is in the form u2, so that Der = 2( sin 9) cos B + sin B. 6. Differentiation of Implicit Functions The differentiation formulas just developed are especially useful in the differentiation of functions given implicitly. For example, consider x2+xy+y2-y- 1 =0. Here y represents values of two functions-those obtained by solving the quadratic for y in terms of x. We shall regard y as representing the value of either of the two functions.

4. Limits. The Symbol -> We wish at times to let x represent any number in a set near a specified number a such that the difference* I x - a I can be made as near 0 as we please for every x in the set. This is indicated by the symbol x ->a. " It signifies the signless number. Thus 3 - 5 1 = 2. 35 VARIABLES, FUNCTIONS, LIMITS, CONTINUITY Sec. 4 that the function value f (x) approaches a constant L. That is, f (x) -> L, then x ---, a, as so that the value of the difference If(x) - LI becomes and remains arbitrarily near 0.

X xZ Fig. 2 The student should review here the material given previously in Par. 4. t Or at which Dsy fails to exist. Sec. 3 THE DERIVATIVE 48 A critical point is a "high" point if the tangent turns through it from positive to negative slope; a "low" pointif from negative topositive. Such critical points locate relative maxima and minima values of y and offer valuable information in curve sketching. For example, y= x3-3x2+2 (2,-2) with Fig. , at x = 0, 2. These points are (0, 2) and (2, -2). The accompanying table helps disclose the nature of these critical points.

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