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Home Differentiation Differentials and Tangent Lines Examples Example 1  
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Example 1
For the curve y = x^{3}, find the tangent lines at the points (0,0), (1,1), and (½, ⅛) (Figure 2.2.2). The slope is given by f'(x) = 3x^{2}. Figure 2.2.2 At x = 0, f'(0) = 3 · 0^{2} = 0. The tangent line has the equation y = 0(x  0) + 0, or y = 0. At x = 1, f'(1) = 3, whence the tangent line is y = 3(x  1) + 1, or y = 3x  2. At x = ½, f'(½) = 3 (½)^{2} = ¾, so the tangent line is y = ¾(x  (½)) + (⅛), or y = ¾x + ¼


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