EXERCISES 4.4 Analyzing Functions from Graphs 3. y = x + sin 2x, - 20 Sxs 47 0. y = tan x - 4x, - -# 0 27 . y = sin x cost OSXS T 80. y' = x x50 * > 0 28 . y = cos x + V3 sin x, 05 x 5 2r 29. y = x1/5 30. y = x2/5 Sketching y from Graphs of y' and y" Each of Exercises 81-84 shows the graphs of the first and second 32. y = VI - x2 31. y = - Vx2 + 1 2x + 1 derivatives of a function y = f(x). Copy the picture and add to it a sketch of the approximate graph of f, given that the graph passes 33. y = 2x - 3x2/3 34. y = 5x2/5 - 2x through the point P. 81. 35. y = x2/3 (5-x) 36. y = x2/3(x - 5) 37. y = xV8 - x2 38. y = (2 -x2)3/2 39. y = V16 - x2 40. y = x2+- 41. y = 1 - 3 x - 2 42. y = Vx3+ 1 8x 83.* 43. y =2+ 4 44. y = * 4 + 5 45. y = 1x2 - 1/ 46. y = 1x2 - 2x1 SV-x, x 0. y 0, y" >0 100. y = 7 -1 0 y' > 0, y" = 0 x (x - 2) -1 0 , y" 0 103. The accompanying figure shows a portion of the graph of a twice-differentiable function y = f(x). At each of the five 2 0 y' = 0, y" > o labeled points, classify y' and y" as positive, negative, or zero. x > 2 y' > 0 , y " > 0 108. Sketch the graph of a twice-differentiable function y = f(x) that passes through the points (-3, -2), (-2, 0), (0, 1), (1, 2), and (2, 3) and whose first two derivatives have the following sign patterns. 104. Sketch a smooth connected curve y = f(x) with f (- 2) = 8, f' (2) = f'(-2) = 0, y " : f(0) = 4, f'(x) 0 for | x/ > 2, f" (x) > 0 for x > 0. In Exercises 109 and 110, the graph of f' is given. Determine x-values corresponding to inflection points for the graph of f. 105. Sketch the graph of a twice-differentiable-function y = f(x) with the following properties. Label coordinates where possible. 109 y Derivatives * 0 1 y' = 0, y" > 0 2 0, y" > 0 - 2 2 4 y' > 0, y" = 0 -2 4 0 , y" 6 y' 0, sketch a curve y = f(x) that has f(1) = 0 and Displacement s = f(t) f'(x) = 1/x. Can anything be said about the concavity of such a curve? Give reasons for your answer. 120. Can anything be said about the graph of a function y = f(x) that 5 10 15 has a continuous second derivative that is never zero? Give rea- Time (sec) sons for your answer. 114. s 121. If b, c, and d are constants, for what value of b will the curve y = x3 + bx2 + ox + d have a point of inflection at x = 1? Give reasons for your answer. 122. Parabolas Displacement s = f(1) a. Find the coordinates of the vertex of the parabola y = ax + bx + c, a # 0. 5 15 b. When is the parabola concave up? Concave down? Give reasons for your answers. Time (sec) 123. Quadratic curves What can you say about the inflection points of a quadratic curve y = ax + bx + c, a + 0? Give 115. Marginal cost The accompanying graph shows the hypotheti reasons for your answer. al cost c = f(x) of manufacturing x items. At approximately 124. Cubic curves What can you say about the inflection points of what production level does the marginal cost change from a cubic curve y = ax + bx2 + cx + d, a * 0? Give reasons decreasing to increasing? for your