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(b) f (x) = 4x 3 7x 2 , at x = 3
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CHAP. 12]
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THE DERIVATIVE
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12.14 A function f , de ned for all real numbers, is such that: (i) f (1) = 2; (ii) f (2) = 8; (iii) f (u + v) f (u) = kuv 2v 2 for all u and v, where k is some constant. Find f (x) for arbitrary x. 12.15 Let f (x) = 2x 2 + 3x for all x.
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(a) Find the nonnegative value(s) of x for which the tangent line to the graph of f at (x, f (x)) is perpendicular to the tangent line to the graph at ( x, f ( x)). (b) Find the point of intersection of each pair of perpendicular lines found in part (a). 12.16 If the line 4x 9y = 0 is tangent in the rst quadrant to the graph of y = 1 x 3 + c, what is the value of c 3
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1 12.17 For what nonnegative value of b is the line y = 12 x + b normal to the graph of y = x 3 + 1 3
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12.18 Let f be differentiable (that is, f exists). De ne a function f # by the equation f # (x) = lim (a) Find f # (x) if f (x) = x 2 x. Hint: f (x + h) f (x h) h h 0
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(b) Find the relationship between f # and the derivative f .
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f (x + h) f (x) f (x + k) f (x) f (x + h) f (x h) = + where k = h. h h k
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12.19 Let f (x) = x 3 + x 2 9x 9. (a) Find the zeros of f . algebra If f (x) = an x n + an 1 x n 1 + + a1 x + a0 , where the ai s are integers, then any integer root k of f (x) must be a divisor of the constant term a0 .1 (b) Find the slope-intercept equation of the tangent line to the graph of f at the point where x = 1. (c) Find a point (x0 , y0 ) on the graph of f such that the tangent line to the graph at (x0 , y0 ) passes through the point (4, 1). 12.20 Let f (x) = 3x 3 11x 2 15x + 63. (a) Find the zeros of f . (b) Write an equation of the line normal to the graph of f at x = 0. (c) Find all points on the graph of f where the tangent line to the graph is horizontal. f (x + h) f (x) 12.21 De ne f+ (x), the right-hand derivative of f at x, to be lim , and f (x), the left-hand derivative of f at x, h h 0+ f (x + h) f (x) . Show that the derivative f (x) exists if and only if both f+ (x) and f (x) exist and are equal. to be lim h h 0 12.22 Determine whether the following functions are differentiable at the given argument. [Hint: Use Problem 12.21.] (a) The function f of problem 10.5(a) at x = 0. (b) g(x) = x3 0 if x 0 , at x = 0 if x < 0
1 If
0 = f (k) = an k n + an 1 k n 1 + + a1 k + a0 , then a0 = (an k n + an 1 k n 1 + + a1 k) = k(an k n 1 + an 1 k n 2 + + a1 ).
THE DERIVATIVE
[CHAP. 12
(c) h(x) = (d) G(x) =
x2 x
if x 1 , at x = 1 if x > 1
if x 1 x3 , at x = 1 4x 3 if x > 1 x2 Ax 1 if x > 1 . if x 3
12.23 Let f (x) =
(a) Find the value of A for which f is continuous at x = 3. (b) For the value of A found in (a), is f differentiable at x = 3 1 12.24 Use the original de nition of derivative to nd the derivative of the function f such that f (x) = . x 1 12.25 Assuming that f is differentiable, prove: (a) If f is even, f is odd; (b) If f is odd, f is even.
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