47. For what value of the constant c is the function f continuous on-
12. If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height (in meters after t seconds) is given by H = 10t - 1.86t^2. a) Find the velocity of the rock after one second. b) Find the velocity of the rock when t = a. c) When will the rock hit the surface? d) With what velocity will the rock hit the surface?
f(x) = cx + 2x if x < 2, -cx if x >= 2
8. Find the values of a and b that make f continuous everywhere.
f(x) = x^2 + 4 if x < 2, x^2 - bx + 3 if 2x < 3, 2x - a + b if x >= 3
31. a) If F(x) = 5x/(1 + x^2), find F(2) and use it to find an equation of the tangent line to the curve y = 5x/(1 + x^2) at the point (2, 2).
73. Show that the function f(x) = xsin(1/x) is continuous on (0, infinity).
33. If an equation of the tangent line to the curve y = f(x) at the point where x = 2 is y = 4x - 5, find f'(2) and f''(2).
f(x) = xsin(1/x) if x > 0, 0 if x = 0
57-58. Determine whether f'(0) exists.
65. Let f(x) = 1/x sin(x) if x > 0, 0 if x = 0. Determine the intervals where f is continuous.
75. A woman leaves her house at 7:00 AM and takes her usual path to the top of a mountain, arriving at 7:00 PM. The following morning, she starts at 7:00 AM at the top and takes the same path back, arriving at her home at 7:00 PM. Use the Intermediate Value Theorem to show that there is a point on the path that the woman will cross at exactly the same time of day on both days.
f(x) = 1/x sin(x) if x > 0, 0 if x = 0
a) Find f(4) and f'(4). b) Sketch the graph of f. c) Where is f discontinuous? d) Where is f not differentiable?
