Elementary Calculus: An Infinitesimal Approach, 2nd Edition by H. Jerome Keisler PDF

By H. Jerome Keisler

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Extra info for Elementary Calculus: An Infinitesimal Approach, 2nd Edition

Example text

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 x෇r ͩ ␲ ␲ Ϫ sin 3 3 ͪ ͩ ෇r ␲ s3 Ϫ 3 2 ͪ ͩ y ෇ r 1 Ϫ cos ␲ 3 ͪ ෇ r 2 dy sin͑␲͞3͒ s3 ͞2 ෇ ෇ ෇ s3 dx 1 Ϫ cos͑␲͞3͒ 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.

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Elementary Calculus: An Infinitesimal Approach, 2nd Edition by H. Jerome Keisler


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