Multivariable Calculus: Early Transcendentals, 6th Edition by James Stewart

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 packed with correct, real-world examples. With CALCULUS: EARLY TRANCENDENTALS, 6th variation, Stewart conveys not just the software of calculus that will help you advance technical competence, but additionally promises an appreciation for the intrinsic great thing about the topic. His sufferer examples and integrated studying aids may also help you construct your mathematical self belief and accomplish your pursuits within the path!

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Find the area available for grazing by the cow. 3 POLAR COORDINATES |||| 639 ; BE´ZIER CURVES L A B O R AT O R Y PROJECT The Bézier curves are used in computer-aided design and are named after the French mathematician Pierre Bézier (1910–1999), who worked in the automotive industry. A cubic Bézier curve is determined by four control points, P0͑x 0 , y0 ͒, P1͑x 1, y1 ͒, P2͑x 2 , y 2 ͒, and P3͑x 3 , y 3 ͒, and is defined by the parametric equations x ෇ x0 ͑1 Ϫ t͒3 ϩ 3x1 t͑1 Ϫ t͒2 ϩ 3x 2 t 2͑1 Ϫ t͒ ϩ x 3 t 3 y ෇ y0 ͑1 Ϫ t͒3 ϩ 3y1 t͑1 Ϫ t͒2 ϩ 3y 2 t 2͑1 Ϫ t͒ ϩ y 3 t 3 where 0 ഛ t ഛ 1.

5. If the circle C rolls on the outside of the fixed circle, the curve traced out by P is called an epicycloid. Find parametric equations for the epicycloid. 6. Investigate the possible shapes for epicycloids. Use methods similar to Problems 2–4. 2 CALCULUS WITH PARAMETRIC CURVES Having seen how to represent curves by parametric equations, we now apply the methods of calculus to these parametric curves. In particular, we solve problems involving tangents, area, arc length, and surface area. TANGENTS In the preceding section we saw that some curves defined by parametric equations x ෇ f ͑t͒ and y ෇ t͑t͒ can also be expressed, by eliminating the parameter, in the form y ෇ F͑x͒.

In particular, you should identify the transitional values of a for which the basic shape of the curve changes. ; 82. The astronomer Giovanni Cassini (1625–1712) studied the family of curves with polar equations r 4 Ϫ 2c 2 r 2 cos 2␪ ϩ c 4 Ϫ a 4 ෇ 0 69. Show that the polar equation r ෇ a sin ␪ ϩ b cos ␪, where ab where a and c are positive real numbers. These curves are called the ovals of Cassini even though they are oval shaped only for certain values of a and c. ) Investigate the variety of shapes that these curves may have.

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