6. Sketch the surface described by the parameterization r(u, v) = (u, v cos(u), v sin(u)) for 0 ≤ u≤ 8π and 1 ≤ v ≤ 3. Find the area of this surface. Hint: you may use that the function =x√1 + x² +ln(x + √√1+x²) satisfies f'(x) = 2√1+x². f(x) =

Q: 5. Find an equation plane to the surface (u, v) = (3 cos(u) sin(v), 2 sin(u) sin(v), cos(u)) at the…

A: First, we find the point on the surface where the tangent plane would be drawn. Next, we calculate…

Q: = 21 which is parallel to the plane Find the equation of tangent plane to the surface x? + 2y? + 3z2…

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Q: 2. Find the equation of the tangent plane to the surface x²+3y²+4z² = 20 (2,2, 1). at the point

A: Step 1:To find the equation of the tangent plane to the surface x2+3y2+4z2=20 at the point (2,2,1),…

Q: 'ind parametrizations (a) The line from (a, b) to (c,d). (b) A path from (2,0) to (0, −2) along a…

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Q: = . Find an equation of the tangent plane to the parametric surface R(u, v) 0≤u≤, at the point…

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Q: n exact form. Use symbolic notation and fractions where needed.)

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Q: B) r(t) = (4 cos t, 4 sin t, 16)

A: We consider x=rcos(theta) and  y=rsin(theta)  Therefore r^2=x^2+y^2

Q: Let S be the part of the surface x2 + y2 +2²=4 that lies in the first octant above the cone ZU √x² +…

A: Given that S is the part of the surface  that lies above the cone  in the first octant.One has to…

Q: Use hyperbolic functions to parametrize the intersection of the surfaces x' - y 49 and z= (Use…

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Q: Describe and sketch the following surfaces z = f(x, y): (a) * (c) 2= /1-x² - y² z = 6 - 2x - 3y (b)…

A: We have to sketch and describe the following curve

Q: I, Use Stoke's theorem to evalueate S.Fidr, where F Cau2)= - it +x3 + 2î aind C is the Ceurve of…

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Q: The tangent plane to the surface 2x² + 3y² = 5z² at (1, 1, 1) has an equation ax +by+cz = d.What is…

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Q: (b) Find the two points on the surface x² + y² + 24 = 2 where the tangent plane is parallel to the…

A: The two points on the surface x4+y4+z4=2 where the tangent plane is parallel to the plane x-z=5.

Q: Q1.6 Find the point on the surface given by the equation x^2 + 5y^2 + 10z^2 = 18016 that has a…

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Q: Find an equation of the tangent plane to the surface given implicitly by z= sin(zy) at the point (,…

A: A tangent plane to a surface is a plane that just touches the surface at a single point. The…

Q: 1. The equation r²y³ +2ln(yz) + ye" - 1 determines a surface. Note that you cannot easily write z as…

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Q: Sketch the surface 1<z</9-x² - y² and x²+y² <1

A: To sketch the given surfaces.

Q: Please help with number 24

A: First, we will find the magnitude of the vector field. We are given, F = (x-y)i + (2x+y+3)j. So, the…