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Supplementary Problems
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2 22.5 Find the second derivative Dx y of the following functions y: 1 (b) y = x 3 7x (c) y = x + 5 (a) y = x x x 4 (f ) y = (x + 1)2 (x 3)3 (g) y = (e) y = x 2 + 1 (1 x)2
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(d) y =
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22.6 Use implicit differentiation to nd the second derivative y in the following cases: (a) x 2 + y2 = 1 (b) x 2 y2 = 1 (c) x 3 y3 = 1 (d) xy + y2 = 1
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22.7 If x + y = 1, calculate y : (a) by explicitly solving for y and then differentiating twice; (b) by implicit differentiation. (c) Which of methods (a) and (b) is the simpler one 22.8 Find all derivatives ( rst, second, etc.) of y: (a) y = 4x 4 2x 2 + 1 x+1 (d) y = x 1 1 (b) y = 2x 2 + x 1 + x 1 (e) y = 3+x x 1 (f ) y = 2 x (c) y =
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22.9 Find the velocity the rst time that the acceleration is 0 if the equation of motion is: (a) s = t 2 5t + 7 (b) s = t 3 3t + 2 (c) s = t 4 4t 3 + 6t 2 4t + 3
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22.10 At the point (1, 2) of the curve x 2 xy + y2 = 3, nd the rate of change with respect to x of the slope of the tangent line to the curve. 22.11 If x 2 + 2xy + 3y2 = 2, nd the values of dy/dx and d 2 y/dx 2 when y = 1. 22.12 Let f (x) = (a) 1 + 3K(x 2) + (x 2)2 Lx + K if x 2 where L and K are constants. if x > 2 (b) With L and K as found in part (a), is f (x) continuous for all x
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If f (x) is differentiable at x = 2, nd L and K.
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22.13 Let h(x) = f (x)g(x) and assume that f and g have derivatives of all orders. (a) Find formulas for h (x), h (x), and h(4) (x). (b) Find a general formula for h(n) (x). 22.14 Let H(x) = f (x)/g(x), where f and g have rst and second derivatives. Find a formula for H (x).
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CHAP. 22]
HIGHER-ORDER DERIVATIVES
22.15 The height s of an object in free fall on the moon is approximately given by s = s0 + v0 t 27 t 2 , where s is measured in 10 feet and t in seconds. (a) What is the acceleration due to gravity on the moon (b) If an object is thrown upward from the surface of the moon with an initial velocity of 54 feet per second, what is the maximum height it will reach, and when will it reach that height 22.16 Two objects move on the x-axis. Their positions f (t) and g(t) are given by f (t) = 6t t 2 and g(t) = t 2 4t. (a) When do they have the same position (b) When do they have the same velocity (c) When they have the same position, are they moving in the same direction
Applications of the Second Derivative and Graph Sketching
23.1 CONCAVITY If a curve has the shape of a cup or part of a cup (like the curves in Fig. 23-1), then we say that it is concave upward. (To remember the sense of concave upward, notice that the letters c and up form the word cup.) A mathematical description of this notion can be given. A curve is concave upward if the curve lies above the tangent line at any given point of the curve. Thus, in Fig. 23-1(a) the curve lies above all three tangent lines.