By James Stewart
Good fortune on your calculus path begins the following! James Stewart's CALCULUS texts are world-wide best-sellers for a cause: they're transparent, actual, and choked with appropriate, real-world examples. With MULTIVARIABLE CALCULUS, 7th version, Stewart conveys not just the application of calculus that can assist you boost technical competence, but additionally grants an appreciation for the intrinsic fantastic thing about the topic. His sufferer examples and integrated studying aids can assist you construct your mathematical self assurance and accomplish your objectives within the path!
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Luck on your calculus direction starts off the following! James Stewart's CALCULUS texts are world-wide best-sellers for a cause: they're transparent, exact, and jam-packed with appropriate, real-world examples. With MULTIVARIABLE CALCULUS, 7th variation, Stewart conveys not just the application of calculus that can assist you strengthen technical competence, but additionally supplies an appreciation for the intrinsic fantastic thing about the topic.
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Additional info for Multivariable Calculus (7th Edition)
3 ; 76. Investigate the family of polar curves r 1 ϩ cos n 61–64 Find the points on the given curve where the tangent line where n is a positive integer. How does the shape change as n increases? What happens as n becomes large? Explain the shape for large n by considering the graph of r as a function of in Cartesian coordinates. is horizontal or vertical. 61. r 3 cos 62. r 1 Ϫ sin 63. r 1 ϩ cos 64. r e 77. Let P be any point (except the origin) on the curve r f ͑ ͒.
It is the straight line that passes through O and makes an angle of 1 radian with the polar axis (see Figure 7). Notice that the points ͑r, 1͒ on the line with r Ͼ 0 are in the first quadrant, whereas those with r Ͻ 0 are in the third quadrant. (2, 1) (1, 1) O 681 EXAMPLE 5 Sketch the polar curve 1. (3, 1) ¨=1 POLAR COORDINATES 1 x (_1, 1) EXAMPLE 6 (a) Sketch the curve with polar equation r 2 cos . (b) Find a Cartesian equation for this curve. (_2, 1) SOLUTION (a) In Figure 8 we find the values of r for some convenient values of and plot the corresponding points ͑r, ͒.
3) and the circle are sketched in r=3 sin ¨ π 5π ¨= 6 ¨= 6 O FIGURE 5 Figure 5 and the desired region is shaded. The values of a and b in Formula 4 are determined by finding the points of intersection of the two curves. They intersect when 3 sin 1 ϩ sin , which gives sin 12 , so ͞6, 5͞6. The desired area can be found by subtracting the area inside the cardioid between ͞6 and 5͞6 from the area inside the circle from ͞6 to 5͞6. Thus A 12 y r=1+sin ¨ 5͞6 ͞6 ͑3 sin ͒2 d Ϫ 12 y 5͞6 ͞6 ͑1 ϩ sin ͒2 d Since the region is symmetric about the vertical axis ͞2, we can write ͫy A2 y 1 2 ͞2 ͞6 y ͞2 ͞6 ͞2 ͞6 9 sin 2 d Ϫ 12 y ͞2 ͞6 ͬ ͑1 ϩ 2 sin ϩ sin 2 ͒ d ͑8 sin 2 Ϫ 1 Ϫ 2 sin ͒ d ͑3 Ϫ 4 cos 2 Ϫ 2 sin ͒ d ͞2 ͞6 ] 3 Ϫ 2 sin 2 ϩ 2 cos [because sin 2 12 ͑1 Ϫ cos 2 ͒] Copyright 2010 Cengage Learning.