4. Consider the surface defined by the portion of the cone z = -Vx2 + y? that lies below the rectangle (x, y) E [–1, 1] x [2, 4]. (a) Find parametric equations for this surface. (b) Compute the area of this surface.

Q: Consider the rormal line to the surface z= z + y? (parabokoid) at the point P(1,1, 2) Find the point…

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Q: 1. Let F(x, y, z) = yi+ xj+ yzk and the surface S be the part of the paraboloid z = 4 – x2 – y? that…

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Q: (c) The cone r² = 4(y² + z?) for (0 <x < 4

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Q: 25) Show that the curve with parametric equations x = tcos(t) , y = tsin(t), lz = t ies on the cone…

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Q: An equation for the tangent plane to the surface : = (xty at the point P(1, 1, 4) is a) 8x + 8y -: =…

A: Equation of a tangent to the surface z=f(x,y) at point (a,b,f(a,b)) is given by…

Q: Find an equation of the surface of revolution by revolving the plane curve 3y = √(9- [(x-1)] ^2)…

A: We need to find the equation of the surface. Please find step by step solution below:

Q: Consider the surface x2 + y2 − 2xy − x + 3y − z = −4 and the point P(2, −3, 18). Determine the the…

A: Given surface is x2 +y2-2xy-x+3y-z=-4 and point is P(2,-3,18)

Q: Find the parametric equations for the curve of intersection between the cylindrical surface S: x +…

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Q: Consider the surface z = (x + 1)² + (y – 2)² – 1 - Select an equation of the tangent plane to the…

A: we have to find the equation of the tangent plane of the surface 2=(x+1)2+(y-2)2-1 atthe point…

Q: Consider the equation below. x² – y2 + z² – 4x – 2y – 4z + 7 = 0 - Reduce the equation to one of the…

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Q: B Suppose we want to find all the points on the surface gx, y) x+ 4y - 4 where the tangent plane is…

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Q: 14) Find the points on the surface xy- z2 = 1 that are closed to the origin. %3D

A: To find the point on the below surface that is closed to the origin. xy−z2=1

Q: The attached figure shows the surface S1: y + z= 2, limited in the first octant with surface S2: x =…

A: The area of a surface z=f(x,y) bounded by the region R is given by the integral expression:-…

Q: Are there any points on the hyperboloid r2 - y? – 22 = 1 where the tangent plane is parallel to the…

A: Given hyperboloid ;    x2-y2-z2=1  and equation of plne which is parallel to tangent plane  x+y-z=0

Q: Sketch the curve of intersection of the surfaces. The ellipsoid 2x + 2y² + z² 24 and the paraboloid…

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Q: z = In(x²y*) + / x² + y²

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Q: 2. Bounded by the paraboloid = 10 – 322 – 3y? and the plane z = 4 (using polar coordinates)

A: The paraboloid is bounded by z=10-3x2-3y2 . Now, converting in polar coordinates, z=10-3r2.

Q: What is the equation of the plane to the surface z = e3y sin(3x) +2 at the point P ,0, 3 O x - y -…

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Q: Consider the surface x2 + y2 − 2xy − x + 4y − z2 + 4 = 0 at the point Po (2, −2, 5). Determine the…

A: This is a multi-variable calculus.

Q: 1. Find the closest point of the surface ry + xz + yz = 1 to the origin. %3D

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Q: The parametric representation of the surface y2 - 4y + z2 = 0,0 < x < 3 is a) 7(u, v) = (v, 2 + 2…

A: solution:y2-4y+z2=0 , 0≤x≤3considery2-4y+z2=0∴y2-4y+4+z2=4∴y-22+z2=22

Q: 4. The surface described by the equation x2 + y + z² = 0 is a(an) (a) elliptic paraboloid symmetric…

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Q: 3. Find the exact area of the surface obtained by rotating the curve described by the parametric…

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Q: Given the surface z = In(5x2 + 2xy - 3y2 – 12). (a) Find the equation of the tangent plane and the…

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Q: 3. Use polar coordinates to find the surface area of the paraboloid z = x² +y² – 5 that lies below…

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Q: Consider the parametric surface S given by the portion of the sphere x +y² +z² = 4 where z 2 1.

A: In this question, we have to parameterize the surface in cartesian coordinates and then find the…

Q: 1 Find the surface area of the part of the hyperbolic paraboloid z = (x² - y²) that lies inside the…

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Q: 9. Show that the sum of the T-, y-, and z- intercepts of any tangent plane to the surface VI + Vỹ +…

A: Given, Gradient of the surface at any point (x0,y0,z0) is given by,

Q: 3. Consider the surface z = xln y – x²y. (a) Show that Q = (1,1, –1) lies on this surface. (b) Find…

A: The given surface is z=xlny-x2y (a) Substitute (x,y,z)=(1,1,−1) in z=xlny-x2y.…

Q: Find all points on the surface xy - z2 = 1 which are closest to that origin. Let x,y,z be the…

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Q: 14. Under the paraboloid z=x² +y2 and above the disk x +y² <9 for (x,y) in the first quadrant.

A: we have to find the volume

Q: Show that the surfaces x2 + 2y2 + 3z2 = 3,   x2 + y2 + z2 + 6x − 10y + 14 = 0, are tangent to each…

A: Concept:   The calculus helps in understanding the changes between values that are related by a…

Q: Sketch the trace curves of x^2 − y^2 + z^2 − 4x + 2z = 4, and use these trace curves to sketch the…

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Q: Given the surface of z =10-4x² – y² : Sketch the surface of z=10-4x² – y² where z>0 and show the…

A: Given:-               z = 10- 4x2 - y2z = 10-- 4x2 + y2

Q: 1. Find a parametric representation for the surface: the part of the sphere x2 + y? + z2 = 4 that…

A: According to the given information, It is required to calculate the parametric representation for…

Q: Let C be the curve of intersection of the spheres x 2 + y 2 + z2 = 3 and(x - 2)2 + (y - 2)2 + z2 =…

A: Let fx=x2+y2+z2-3=0 &amp; gx=x-22+y-22+z2-3=0 At P=(1, 1, 1) ∇f=2x, 2y, 2z∇fP1, 1, 1=2, 2, 2…

Q: 1-) Given the surface z = In(-2x2 + 5y2 – xy). Find the equation of the tangent plane and the…

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Q: Find the equation for the tangent plane to the surface z=In(9x2 + 5y2+1 O x+y = 0 z = 0 O x-y = 0 O…

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Q: 1. Identify the parametric surface S given by: r(u, v) = 3 cos ui+ vj + 3 sinu k where 0 <u < 2n and…

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Q: Find a set of parametric equalions for the surface formed when the curve y(1+z²) – 1 = 0 is revolved…

A: Find a set of parametric equations for the surface formed if the curve y1+z2-1=0 is revolved about…

Q: Find the parametric equations for the curve of intersection between the cylindrical surface S : x +…

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Q: 3. Find the surface area of the surface generated by revolving the graph of the "curve" given…

A: The given parametric curve is: x=t,     y=t2,         0, 2

Q: Classify and Sketch the surface X2+ Zy² + Z? -4x+4y-2z+3:0

A: The given equation of the surface is x2 + 2y2 + z2 - 4x + 4y - 2z + 3 = 0.The general equation,…

Q: The parametric representation of the surface y2 - 4y + z2 = 0,0 <x< 3 %3D is a) F(u, v) = (v, 2 + 2…

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Q: - - Consider the surface x² + y² + 2xy − 3x + 2y – 2z = 2. Determine the following: (a) the general…

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Q: Identify the surface y-3=x²+z² paraboloid cylinder cone hyperboloid

A: y-3=x²+z²

Q: Consider the surface z = 4x2² + y² + 1. An equation for the tangent plane to this surface at the…

A: Given: z=4x2+y2+1

Q: 7. Sketch the surface 1≤z≤2-x² - y²

A: