APPLICATIONS OF THE SECOND DERIVATIVE AND GRAPH SKETCHING in .NET framework

Generation QR Code JIS X 0510 in .NET framework APPLICATIONS OF THE SECOND DERIVATIVE AND GRAPH SKETCHING

APPLICATIONS OF THE SECOND DERIVATIVE AND GRAPH SKETCHING
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[CHAP. 23
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Fig. 23-14
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23.12 Sketch the graph of a continuous function f such that: (a) f (1) = 2, f (1) = 0, f (x) > 0 for all x (b) f (2) = 3, f (2) = 0, f (x) < 0 for all x (c) f (1) = 1, f (x) < 0 for x > 1, f (x) > 0 for x < 1, lim f (x) = + , lim f (x) =
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(d) f (0) = 0, f (x) < 0 for x > 0, f (x) > 0 for x < 0, lim f (x) = 1, lim f (x) = 1
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(e) f (0) = 1, f (x) < 0 for x = 0, lim f (x) = + , lim f (x) =
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(f) f (0) = 0, f (x) > 0 for x < 0, f (x) < 0 for x > 0, lim f (x) = + , lim f (x) = +
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(g) f (0) = 1, f (x) < 0 if x = 0, lim f (x) = 0, lim f (x) =
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23.13 Let f (x) = x|x 1| for x in [ 1, 2]. (a) At what values of x is f continuous (b) At what values of x is f differentiable Calculate f (x). [Hint: Distinguish the cases x > 1 and x < 1.] (c) Where is f an increasing function (d) Calculate f (x). (e) Where is the graph of f concave upward, and where concave downward (f ) Sketch the graph of f . 23.14 Given functions f and g such that, for all x, (i) (g(x))2 (f (x))2 = 1; (ii) f (x) = (g(x))2 ; (iii) f (x) and g (x) exist; (iv) g(x) < 0; (v) f (0) = 0. Show that: (a) g (x) = f (x)g(x); (b) g has a relative maximum at x = 0; (c) f has a point of in ection at x = 0. 23.15 For what value of k will x kx 1 have a relative maximum at x = 2 23.16 Let f (x) = x 4 + Ax 3 + Bx 2 + Cx + D. Assume that the graph of y = f (x) is symmetric with respect to the y-axis, has a relative maximum at (0, 1), and has an absolute minimum at (k, 3). Find A, B, C, and D, as well as the possible value(s) of k. 23.17 Prove Theorem 23.1. [Hint: Assume that f (x) > 0 on (a, b), and let c be in (a, b). The equation of the tangent line at x = c is y = f (c)(x c) + f (c). It must be shown that f (x) > f (c)(x c) + f (c). But the mean-value theorem gives f (x) = f (x )(x c) + f (c) where x is between x and c, and since f (x) > 0 on (a, b) , f is increasing.] 23.18 Give a rigorous proof of the second-derivative test (Theorem 23.3). [Hint: Assume f (c) = 0 and f (c) < 0. Since f (c) < 0, f (c + h) f (c) f (c + h) f (c) < 0. So, there exists > 0 such that, for |h| < , < 0, and since f (c) = 0, lim h h h 0 f (c + h) f (c) = f (c + h1 ) f (c + h) < 0 for h > 0 and f (c + h) > 0 for h < 0. By the mean-value theorem, if |h| < , h
CHAP. 23]
APPLICATIONS OF THE SECOND DERIVATIVE AND GRAPH SKETCHING
for some c + h1 between c and c + h. So, |h1 | < |h|, and whether h > 0 or h < 0, we can deduce that f (c + h) f (c) < 0; that is, f (c + h) < f (c) . Thus, f has a relative maximum at c. The case when f (c) > 0 is reduced to the rst case by considering f .] 23.19 Consider f (x) = 3(x 2 1) . x2 + 3
(a) Find all open intervals where f is increasing. (b) Find all critical points and determine whether they correspond to relative maxima, relative minima, or neither. (c) Describe the concavity of the graph of f and nd all in ection points (if any). (d) Sketch the graph of f . Show any horizontal or vertical asymptotes. 23.20 In the graph of y = f (x) in Fig. 23-15: (a) nd all x such that f (x) > 0; (b) nd all x such that f (x) > 0.
Fig. 23-15
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