Problems - Derivatives of Trigonometric Functions

In Problems 1-14, find the derivative.

In Problems 25-34, find the maxima, minima, inflection points, and limits when necessary, and sketch the curve for 0 ≤ x ≤ 2π.

35            Show that at lim_x→0 sin (1/x) does not exist.

36            Let f(x) = x sin (l/x), with f(0) = 0. Show that f is continuous but not differentiate at x = 0.

In Problems 37-53, evaluate the integral.

54            A revolving light one mile from shore sweeps out eight revolutions per minute. Find the velocity of the beam of light along the shore at the instant when it makes an angle of 45° with the shoreline.

55            A ball is thrown vertically upward from a point P so that its height at time t is y = l00t - 16t² feet. Q is another point on the surface 100 ft from P. At time t = 5 find the rate of change of the angle between the horizontal line QP and the line from Q to the ball.

56            Two hallways of width a and b meet at right angles. Find the length of the longest rod which can be slid on the floor around the corner.

57            Find the area under one arch of the curve y = 3 sin x.

58            Find the area under one arch of the curve y = sin (3x).

59            Find the area of the region between the curves y = sin x cos x and y = sin x, 0 ≤ x ≤ π/2.

60            The region between the x-axis and the curve y = tan x, 0 ≤ x ≤ π/4, is rotated about the x-axis. Find the volume of the solid of revolution.

61            The region between the x-axis and the curve y = (sin x)/x, π/2 ≤ x ≤ π, is rotated about the y-axis. Find the volume of the solid of revolution.

62            Find the length of the parametric curve x = 2 cos (3t), y = 2 sin (3t), 0 ≤ t ≤ 1.

63            Find the length of the parametric curve x = cos² t, y = sin² t, 0 ≤ t ≤ π/2.

64            Find the length of the parametric curve x = cos³ t, y = sin³ t, 0 ≤ t ≤ π/2.

65            Find the area of the surface generated by rotating the curve in Problem 63 about the x-axis.

66            Find the area of the surface generated by rotating the curve in Problem 64 about the y-axis.