By H. Jerome Keisler
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Extra info for Elementary Calculus: An Infinitesimal Approach, 2nd Edition
The four-leaved rose is also symmetric about the pole. These symmetry properties could have been used in sketching the curves. For instance, in Example 6 we need only have plotted points for 0 ഛ ഛ ͞2 and then reflected about the polar axis to obtain the complete circle. 2) and the Product Rule, we have dy dr sin ϩ r cos dy d d 3 dx dx dr cos Ϫ r sin d d We locate horizontal tangents by finding the points where dy͞d 0 (provided that dx͞d 0 ). Likewise, we locate vertical tangents at the points where dx͞d 0 (provided that dy͞d 0).
V EXAMPLE 2 (a) Find the tangent to the cycloid x r͑ Ϫ sin ͒, y r͑1 Ϫ cos ͒ at the point where ͞3. ) (b) At what points is the tangent horizontal? When is it vertical? SOLUTION (a) The slope of the tangent line is dy dy͞d r sin sin dx dx͞d r͑1 Ϫ cos ͒ 1 Ϫ cos M 632 |||| CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES When ͞3, we have xr ͩ Ϫ sin 3 3 ͪ ͩ r s3 Ϫ 3 2 ͪ ͩ y r 1 Ϫ cos 3 ͪ r 2 dy sin͑͞3͒ s3 ͞2 s3 dx 1 Ϫ cos͑͞3͒ 1 Ϫ 12 and Therefore the slope of the tangent is s3 and its equation is yϪ r s3 2 ͩ xϪ r rs3 ϩ 3 2 ͪ s3 x Ϫ y r or ͩ ͪ Ϫ2 s3 The tangent is sketched in Figure 2.
X sin 2 t, y cos 2 t, 0 ഛ t ഛ 3 52. x cos 2t, y cos t, 0 ഛ t ഛ 4 53. Show that the total length of the ellipse x a sin , _a y b cos , a Ͼ b Ͼ 0, is 35. 1 for the case d Ͻ r. ͞2 0 36. Let be the region enclosed by the loop of the curve in Example 1. (a) Find the area of . (b) If is rotated about the x-axis, find the volume of the resulting solid. (c) Find the centroid of . s1 Ϫ e 2 sin 2 d where e is the eccentricity of the ellipse (e c͞a, where c sa 2 Ϫ b 2 ) . 54. Find the total length of the astroid x a cos 3, y a sin 3, where a Ͼ 0.
Elementary Calculus: An Infinitesimal Approach, 2nd Edition by H. Jerome Keisler