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

SECTION 10.2 CALCULUS WITH PARAMETRIC CURVES 0 9
21. From the graph, it appears that the rightmost point on the curve :J: = t - t 6 , y = et
is about (0.6, 2). To find the exact coordinates, we find the value oft for which the
graph has a vertical tangent, that is, 0 = dxl dt = 1 - 6t 5 {::} t = 11 W.
Hence, the rightmost point is
23. We graph the curve x = t 4 - 2t 3 - 2e, y = t 3 - tin ~he viewing rectangle [-2, 1.1] by [- 0.5, 0.5]. This rectangle
corresponds approximately tot E [ -1, 0.8] .
o.s
- 0.5
We estimate that the curve has horizontal tangents at about (- 1, - 0.4) and (- 0:11, 0.39) and vertica·L tangents at
dy dyl dt· 3t2- 1 .
about (0, 0) and (-0.19, 0.37). We calculate -d = dxl d = 4 3 6 2 . The honzontal tangents occur when
X t t - t - 4 t ·
dy I dt = 3t2 - 1 = 0 1=} t = ± ~ , so both horizontal tangents are shown in our graph. The vertical tange~ts occur when
dxldt = 2t(2t2 - 3t-2) = 0 {::} 2t(2t + 1)(t - . 2) = 0 {::} t = 0, - ~or 2. It seems that we have missed one vertical
tangent, and indeed if we plot the curve on the t-interval [-1.2, 2.2] we see that there is another vertical tangent at (-8, 6).
25. x = cos t, y = sin tcost. dxl dt = - sint, dyl dt = - sin 2 t + cos 2 t = cos 2t.
(x,y)= (O, O) {::} cost = .O. {::} tisanoddmu lt ipleoflf.Whent=~ ,
dxl dt = - 1 and dyl dt = - 1, so dyl dx = 1. When t = 3 ;
, dxl dt = 1 and
dy I dt = - 1. So dy I dx = - 1. Thus, y = x andy = - x are both tangent to the
curve at (0, 0).
27. x = rB - d sinB, y = r - dcosB.
29. X
dx dy , . dy d sinB
(a) dB = r- dcosll, dB = dsme, so dx = r _ dcosB '
(b) IfO < d < r, then jdcosllj:::; d 0. This shows that dxldll never vanishes,
·so the trochoid can have no vertical tangent if d < r .
= 2t3' y = 1 + 4t - t2 ---'- dy - dy I dt = 4 - 2t N 1 . dy = 1 ~ 4 - 2t = 1. ......,._
_,.. ·dx - dxl dt 6t 2 • ow so ve dx .,_,. 6t2 .....,.
6t 2 + 2t - 4 = 0 {::} 2(3t - 2)(t + 1) = 0 {::} t = ~ or t := - 1. 1ft = ~ . the point is(~~ '
the point is ( -2, -4).
2 :), and ift = -1,
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