Consider the surface that can be parameterized as x (u, v) y (u, v) z (u, v) for u, v € [0, 2π). = = = U ICOS U COS V cos u sin v
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- 2 Sketch the parametric surface x = Vu“ +v , y = u, z = v. luThe two surfaces x2 + y2 + z2 = 6 and 2x2 + 3y2 + z2 = 9 intersect at the point(1,1,2). Find the angle between the tangent planes at the point(1,1,2). Also, find the tangent vector to the curve in which surface intersect?Find the surface area of the parametric surface 7(u, v) = (2 cos(2u) + v cos(u), 2 sin(2u) + v cos(u), v sin(u)) where -T < uFind an equation of the normal line to the parametric surface given by R(u, v) = ((2+ cos v) cos Tu, U +(1 – u) sin v, (1+ cos v) sin Tu) at the point where (u, υ) (1, π).Consider the surface S parametrized by R(u, v) = (uv, sin u, cos v). Find an equation for the tangent plane to S at the point (, 1, ).Find an equation of the tangent plane to the parametric surface R(u, v) = ((u - sin u) cos v, (1 - cos u) sin v, u) at the point where u = v= FIN 2Match each parametrization with the corresponding surface. (i) (u, cos (u), sin (v)) Z (iv) (u, v³, v) (ii) Answer Bank (u, u + v, v) (cos (u) sin (v), 3 cos (u) sin (u), cos (v)) (v) (iii) (u, u (2+ cos (v)), u (2+ sin (u)))If the gradient of f is Vf = x³ } – zỉ+ 2yx k and the point P = ( surface f (x, y, z) = 0, find an equation for the tangent plane to the surface at the point P. (7, –7, 10) lies on the levelThe surface z = x² - y² and the surface ryz = -30 intersect along a curve in R³. Find a vector tangent to this curve at the point P = (-3,2,5). Hint: such a vector is parallel to tangent planes at P to both surfaces.Consider the surface S: z = 4ex^2 - 4xy, where P (0, 1, 1) is a point on S. An equation of the tangent line to S at P, in the direction of the vector v = (−2, −4) corresponds to: Answers in the picture:If a parametric surface given by r₁(u, v) = f(u, v)i + g(u, v)j + h(u, v)k and -4 ≤ u ≤ 4,-2 ≤ v ≤ 2, has surface area equal to 8, what is the surface area of the parametric surface given by r2(u, v) = 5r₁(u, v) with −4 ≤ u ≤ 4, −2 ≤ v ≤ 2? 27