Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

CHAPTER 10 REVIEW 0 37
EXERCISES
1. X = t 2 + 4t, y = 2 - t, - 4 ~ t ~ 1. ~ = 2 - y, SO
y
X = {2 - y) 2 + 4{2- y) = 4- 4y + y 2 + B- 4y = y 2 - By+ 12 {::}
x + 4 = y 2 -
By+ 16 = (y - 4) 2 . This is part of a parabola with vertex
( -4, 4), opening to the right.
X
3. y = sec() = ~() = .! . Since 0 ~ () ~ 1r /2, 0 < x ~ 1 and y ;::: 1.
COS X
This is part of the hyperbola y = 1/x.
y
\ ,1,1),0-0
0
5. Three different sets of parametric equations for the curve y = -IX are
(i) X = t, y = Vt
(ii) X = t 4 , y = t 2
(iii) x = tan 2 t, y = tan t, 0 ~ t < 1r /2
There are many other sets of equations that also give this curve.
7. (a) The Cartesian coordinates are x = 4 cos 2 ; = 4 (- ~) = -2 and
y = 4sin ;,. = 4( 4) = 2 J3, that is, the point ( - 2,2 J3).
(b) Give~ x = - 3 and y= 3, we haver = J( -3) 2 + 3 2 = JIB = 3 v'2. Also, tan() = ~ '* tan() = ~ , and since
3
( -3, 3) is in the second quadrant,(}= ' 3 4 ". Thus, one set of polar coordinates for ( -3, 3) is (3 V2, 3 ,;'"), and two others are
(3 '2 1 111") and (- 3 '2 7 ,.)
y ~, 4 v. .L.i, 4 .
9. r = 1 - cosO. This cardioid is
symmetric about the polar axis.
r
2
(2, 7T)
7T 27T 0
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