INVERSE TRIGONOMETRIC FUNCTIONS
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Let w = x
2
, du = 2x dx. Then
Let w = e*, du = e* dx. Then
Let M = sin x, du = cos x dx. Then
Find an equation of the tangent line to the graph of y = sin
1
at the origin.
which is 5 when x = 0. Hence, the slope of the tangent line is 5, and, since it goes
through the origin, its equation is
A ladder that is 13 feet long leans against a wall. The bottom of the ladder is sliding away from the base of the
wall at the rate of 5 ft/s. How fast is the radian measure of the angle between the ladder and the ground changing
at the moment when the bottom of the ladder is 12 feet from the base of the wall?
Let x be the distance of the bottom of the ladder from the base of the wall, and let 6 be the angle between the ladder and the ground. We are told that D.x = 5, and
Hence, D,8 =
The beam from a lighthouse 3 miles from a straight coastline turns at the rate of 5 revolutions per minute. How
fast is the point P at which the beam hits the shore moving when that point is 4 miles from point A on the shore
directly opposite the lighthouse?
Let x be the distance from P to A, and let 0 be the angle between the beam and the line PA. We are
told D,e = IOTT rad/min. Clearly, 6 = tan
So 107r = D,0 = {l/[l + (x/3)
2 ]}
Hence, D,x = (2507T/3) mi/min = SOOOir mi/h « 15,708 mi/h.
D.x. When x = 4,
Find the area under the curve y = 1/(1 + x
2 ), above the x-axis, and between the lines x = 0 and x - 1.
Find the area under the curve y = 1 /V1 - x
2 , above the jr-axis, and between the lines x = 0 and
The region $ under the curve y = 1/Jc
2 Voc
2 - 1, above the *-axis and between the lines jc = 2/V3 and
A: = 2, is revolved around the y-axis. Find the volume of the resulting solid.
By the cylindrical shell formula,
When x = 12,