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Extra resources for Calculus: Single Variable (5th Edition)
1 = 8 cos(2x + 1) − 3 30. 8 = 4 tan(5x) 31. 1 = 8 tan(2x + 1) − 3 32. 8 = 4 sin(5x) Problems 33. What is the difference between sin x2 , sin2 x, and sin(sin x)? Express each of the three as a composition. ) 2 y ✛ ✛ t 2π 34. 57. 5 TRIGONOMETRIC FUNCTIONS 35. 5 + sin t (b) Based on the table, what is the relationship between the values of g(t) and k(t)? Explain this relationship using the formulas you chose for g and k. (c) Using the formulas you chose for g and h, explain why all the values of g are positive, whereas all the values of h are negative.
V(u(40)) For Problems 55–58, determine functions f and g such that h(x) = f (g(x)). [Note: There is more than one correct answer. ] 55. h(x) = (x + 1)3 √ 57. h(x) = x2 + 4 56. h(x) = x3 + 1 58. h(x) = e2x 59. A spherical balloon is growing with radius r = 3t + 1, in centimeters, for time t in seconds. Find the volume of the balloon at 3 seconds. 42. f (t) is the number of customers in Macy’s department store at t minutes past noon on December 18, 2008. 60. A tree of height y meters has, on average, B branches, where B = y − 1.
507t). 34 Chapter One A LIBRARY OF FUNCTIONS Example 5 Use the information from Example 4 to write a formula for the water level in Boston on a day when the high tide is at 2 pm. 507t). Since 2 pm is 14 hours after midnight, we replace t by (t − 14). 507(t − 14)). The Tangent Function If t is any number with cos t = 0, we define the tangent function as follows tan t = sin t . 46 on page 31 shows the geometrical meaning of the tangent function: tan t is the slope of the line through the origin (0, 0) and the point P = (cos t, sin t) on the unit circle.