Solução_Calculo_Stewart_6e

F.
TX.10
32 ¤ CHAPTER 1 FUNCTIONS AND MODELS
63. (a) In general, tan(arctan x) =x for any real number x. Thus,tan(arctan 10) = 10.
(b) sin −1 sin 7π 3
=sin
−1 sin π 3
=sin
−1 √ 3
2 = π 3 since sin π 3 = √ 3
2 and π 3 is in − π 2 , π 2
.
[Recall that 7π 3
= π 3
+2π and the sine function is periodic with period 2π.]
65. Let y =sin −1 x.Then− π 2 ≤ y ≤ π 2
⇒ cos y ≥ 0,socos(sin −1 x)=cosy = 1 − sin 2 y = √ 1 − x 2 .
67. Let y =tan −1 x.Thentan y = x, so from the triangle we see that
sin(tan −1 x)=siny =
x
√
1+x
2 .
69. The graph of sin −1 x is the reflection of the graph of
sin x about the line y = x.
71. g(x) =sin −1 (3x +1).
Domain (g) ={x | −1 ≤ 3x +1≤ 1} = {x | −2 ≤ 3x ≤ 0} = x | − 2 3 ≤ x ≤ 0 = − 2 3 , 0 .
Range (g) = y | − π 2 ≤ y ≤ π 2
=
−
π
2 , π 2
.
73. (a) If the point (x, y) is on the graph of y = f(x), then the point (x − c, y) is that point shifted c units to the left. Since f is
1-1, the point (y, x) is on the graph of y = f −1 (x) and the point corresponding to (x − c, y) on the graph of f is
(y, x − c) on the graph of f −1 . Thus, the curve’s reflection is shifted down the same number of units as the curve itself is
shiftedtotheleft.Soanexpressionfortheinversefunctionisg −1 (x) =f −1 (x) − c.
(b) If we compress (or stretch) a curve horizontally, the curve’s reflection in the line y = x is compressed (or stretched)
vertically by the same factor. Using this geometric principle, we see that the inverse of h(x) =f(cx) canbeexpressedas
h −1 (x) =(1/c) f −1 (x).