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The tangent plane at a point Po(f(uo, vo), g(uo, vo), h(uo, vo)) on a parametrized surface r(u, v) = f(u, v)i + g(u, v)j + h(u, v)k is the plane through Po normal to the vector r,(uo, vo) × r,(uo, vo), the cross product of the tangent vectors r,(uo, v) and r„(u0, vo) at Po. In Exercises 27–30, find an equation for the plane tangent to the surface at Po. Then find a Cartesian equation for the surface and sketch the surface and tangent plane together. 27. Cone The cone r(r, 0) = (r cos 0)i + (r sin 0)j + rk, r > 0, 0 < 0 < 27 at the point Po(V2, V2, 2) corresponding to (r, 0) = (2, 1/4) 28. Hemisphere The hemisphere surface r(, 0) = (4 sin øcos 0)i + (4 sin o sin 0)j + (4 cos 4)k, 0 < ¢ < T/2, 0

The tangent plane at a point Po(f(uo, vo), g(uo, vo), h(uo, vo)) on a parametrized surface r(u, v) = f(u, v)i + g(u, v)j + h(u, v)k is the plane through Po normal to the vector r,(uo, vo) × r,(uo, vo), the cross product of the tangent vectors r,(uo, v) and r„(u0, vo) at Po. In Exercises 27–30, find an equation for the plane tangent to the surface at Po. Then find a Cartesian equation for the surface and sketch the surface and tangent plane together. 27. Cone The cone r(r, 0) = (r cos 0)i + (r sin 0)j + rk, r > 0, 0 < 0 < 27 at the point Po(V2, V2, 2) corresponding to (r, 0) = (2, 1/4) 28. Hemisphere The hemisphere surface r(, 0) = (4 sin øcos 0)i + (4 sin o sin 0)j + (4 cos 4)k, 0 < ¢ < T/2, 0

Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter9: Vectors In Two And Three Dimensions
Section9.6: Equations Of Lines And Planes
Problem 2E
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The tangent plane at a point Po(f(uo, vo), g(uo, vo), h(uo, vo)) on a parametrized surface r(u, v) = f(u, v)i + g(u, v)j + h(u, v)k is the plane through Po normal to the vector r,(uo, vo) × r,(uo, vo), the cross product of the tangent vectors r,(uo, v) and r„(u0, vo) at Po. In Exercises 27–30, find an equation for the plane tangent to the surface at Po. Then find a Cartesian equation for the surface and sketch the surface and tangent plane together. 27. Cone The cone r(r, 0) = (r cos 0)i + (r sin 0)j + rk, r > 0, 0 < 0 < 27 at the point Po(V2, V2, 2) corresponding to (r, 0) = (2, 1/4) 28. Hemisphere The hemisphere surface r(, 0) = (4 sin øcos 0)i + (4 sin o sin 0)j + (4 cos 4)k, 0 < ¢ < T/2, 0 <o < 27, at the point Po(V2, V2, 2V3) corresponding to (4, 0) = (п/6, п/4) 29. Circular cylinder The circular cylinder r(0, z) = (3 sin 20)i + (6 sin²0)j + zk, 0 < 0 < , at the point Po(3V3/2, 9/2, 0) corresponding to (0, z) = (T/3, 0) (See Example 3.) 30. Parabolic cylinder The parabolic cylinder surface r(x, y): xi + yj – x*k, -0 < x < o, –∞ < y < ∞, at the point Po(1, 2, – 1) corresponding to (x, y) = (1, 2)
Transcribed Image Text:The tangent plane at a point Po(f(uo, vo), g(uo, vo), h(uo, vo)) on a parametrized surface r(u, v) = f(u, v)i + g(u, v)j + h(u, v)k is the plane through Po normal to the vector r,(uo, vo) × r,(uo, vo), the cross product of the tangent vectors r,(uo, v) and r„(u0, vo) at Po. In Exercises 27–30, find an equation for the plane tangent to the surface at Po. Then find a Cartesian equation for the surface and sketch the surface and tangent plane together. 27. Cone The cone r(r, 0) = (r cos 0)i + (r sin 0)j + rk, r > 0, 0 < 0 < 27 at the point Po(V2, V2, 2) corresponding to (r, 0) = (2, 1/4) 28. Hemisphere The hemisphere surface r(, 0) = (4 sin øcos 0)i + (4 sin o sin 0)j + (4 cos 4)k, 0 < ¢ < T/2, 0 <o < 27, at the point Po(V2, V2, 2V3) corresponding to (4, 0) = (п/6, п/4) 29. Circular cylinder The circular cylinder r(0, z) = (3 sin 20)i + (6 sin²0)j + zk, 0 < 0 < , at the point Po(3V3/2, 9/2, 0) corresponding to (0, z) = (T/3, 0) (See Example 3.) 30. Parabolic cylinder The parabolic cylinder surface r(x, y): xi + yj – x*k, -0 < x < o, –∞ < y < ∞, at the point Po(1, 2, – 1) corresponding to (x, y) = (1, 2)
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