27.70
If cos~
1 xy = e
2y , find/.
INVERSE TRIGONOMETRIC FUNCTIONS
27.71
Sketch the graph of y = tan~
1 x - In Vl + x
2
.
Fig. 27-7
27.73
Use implicit differentiation.
See Fig. 27-7.
For the only critical number, x = l, y" = -| <0, and, therefore, there is a relative maximum at * = 1,
y= IT/4- \ In 2 = 0.4. Note that y(0) = 0. Also, as x->±°°, y->-<». Setting y" = 0, we find two
inflection points at x = 1 ± V2.
27.72
Find the derivative of y = sin (sin" x ).
Since sin (sin ' x
2 ) = x
2 , y' - 2x. Note that y is defined only for -1 < x =£ 1 (and y' only for -1 <
x If y = tan-'(esinj:), find/.
27.74
27.75
Refer to Fig. 27-8. In a circular arena of radius r, there is a light at L. A boy starting from B runs toward the
center O at the rate of 10 ft/s. At what rate will his shadow be moving along the side when he is halfway from B
tn Df
Let P be the boy's position, x the distance of P from B, B the angle OLP, and s the arc intercepted by We
shadow is moving at the rate of 16 ft/s.
are given that
DJC = 10. Then
s = r(20),
0 = tan (r-^:)/r.
D,s = 2rD,0 = 2r
When
Hence, the
(xy' + y) = e
2y -2y', xy'+ y =-2e
2y y'^1-x
2 y
2 ,
229
y'(* + 2e
2 'Vl-*
2 y
2 ) = -;y,
find _y'.
IF