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13.6 Find the derivatives of the functions de ned by the following formulas: x2 3 (b) (a) (x 100 + 2x 50 3)(7x 8 + 20x + 5) x+4 4 2 3 (e) 8x 3 x 2 + 5 + 3 (d) 5 x x x (c) (f ) x5 x + 2 x3 + 7 3x 7 + x 5 2x 4 + x 3 x4
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Find the slope-intercept equation of the tangent line to the graph of the function at the indicated point: 1 x+2 , at x = 1 (a) f (x) = 2 , at x = 2 (b) f (x) = 3 x x 1 Let f (x) = x+2 for all x = 2. Find f ( 2). x 2
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13.8 13.9
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Determine the points at which the function f (x) = |x 3| is differentiable.
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13.10 The parabola in Fig. 13-2 is the graph of the function f (x) = x 2 4x. (a) Draw the graph of y = | f (x)|. (b) Where does the derivative of | f (x)| fail to exist
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Fig. 13-2
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13.11 GC Use a graphing calculator to nd the discontinuities of the derivatives of the following functions: (a) f (x) = x 2/3 (b) f (x) = 4 |x 2| + 3 (c) f (x) = 2 x 1 13.12 Evaluate lim 1 1 1 . (1 + h)8
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14.1 RELATIVE EXTREMA A function f is said to have a relative maximum at x = c if f (x) f (c) for all x near c. More precisely, f achieves a relative maximum at c if there exists > 0 such that |x c| < implies f (x) f (c). EXAMPLE For the function f whose graph is shown in Fig. 14-1, relative maxima occur at x = c1 and x = c2 . This is obvious,
since the point A is higher than nearby points on the graph, and the point B is higher than nearby points on the graph. The word relative is used to modify maximum because the value of a function at a relative maximum is not necessarily the greatest value of the function. Thus, in Fig. 14-1, the value f (c1 ) at c1 is smaller than many other values of f (x); in particular, f (c1 ) < f (c2 ). In this example, the value f (c2 ) is the greatest value of the function.
Fig. 14-1
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CHAP. 14]
MAXIMUM AND MINIMUM PROBLEMS
A function f is said to have a relative minimum at x = c if f (x) f (c) for x near c. In Fig. 14-1, f achieves a relative minimum at x = d, since point D is lower than nearby points on the graph. The value at a relative minimum need not be the smallest value of the function; for example, in Fig. 14-1, the value f (e) is smaller than f (d). By a relative extremum is meant either a relative maximum or a relative minimum. Points at which a relative extremum exists possess the following characteristic property. Theorem 14.1: If f has a relative extremum at x = c and if f (c) exists, then f (c) = 0. The theorem is intuitively obvious. If f (c) exists, then there is a well-de ned tangent line at the point on the graph of f where x = c. But at a relative maximum or relative minimum, the tangent line is horizontal (see Fig. 14-2), and so its slope f (c) is zero. For a rigorous proof, see Problem 14.28. The converse of Theorem 14.1 does not hold. If f (c) = 0, then f need not have a relative extremum at x = c.
Fig. 14-2
Fig. 14-3
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