Let f(x, y, z) = x² + 2y? + 4z2. Find the point on the plane x + y + z = 14 at which f has its smallest value.

Q: Find the plane tangent to the surface z=x x+ y° at the point (3,4,25). a. 6x+8y-z=25 O b. 6x-8y-z=25…

A: As we know that; The equation of the tangent plane: nxx-x0+nyy-y0+nzz-z0=0 And The normal vector at…

Q: Find the points on the surface 1x? + 3y2 + 5z2 = 1 at which the tangent plane is parallel to the…

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Q: Find the area of the surface 2x^(3/2) + 2y^(3/2) - 3z = 0 above the square R: 0<= x<= 1,…

A: find z.

Q: Find the equation of the plane tangent to the surface G(x,y, z) = 2x² - y² – 3z2 = -4 at P = (-2,…

A: Given surface is                                                G(x, y, z): 2x2-y2-3z2=-4 we have to…

Q: Integrate G(x, y, z) = xyz over the surface of the cube cut from the first octant by the planes x =…

A: Here, the surface is in the first octant. Therefore, the integration is

Q: Verify Green's theorem in a plane with respect to S. [(x2 - y²)dx + 2xy dy], where C is the boundary…

A: Using greens theorem

Q: Find the highest and lowest point of the surface z2 = x2 + y2 ,on the plane 4x – 3y + 8z = 10 %3D…

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Q: Find a vector tangent to the curve of intersection of the surfaces given by x² + y² + z² = 1 and 2²…

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Q: Find the points on the surface ( y + z)^2 + (z - x)^2 = 16 where the normal line is parallel to the…

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Q: Find a vector which is tangent to the curve of intersection of the surfaces z = x² – y? and xyz + 30…

A: We will find out the required tangent vector.

Q: Consider the surfaces  z = x^2 +y^2   and z+ 2x+ 2y = 0 Find the projection of the curve on…

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Q: Let D be the half-disc in the xy-plane given by x +y <9. Calculate e-(x2+v)dA

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Q: Compute x dS where S is the portion of the plane z=1+x+2y that lies over the square [0, 1] × [0, 1].…

A: We have to find the surface integral ∫sxds.

Q: A point moves along the intersection of the elliptic paraboloid z = a2 + 3y? and the plane y = 1. At…

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Q: F = (x² – y²,x, 2yz); C is the boundary of the plane z = 6 – 2x – y in the first octant.

A: Given, F=x2-y2,x,2yz Find the line integral ∮CF·dr in the following. Here C is the boundary of the…

Q: Let F(x, y, z) =7x2+8y2+13z2. Find the equation to the tangent plane to the level of surface F(x,…

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Q: The vector (x, 2y, –3z) is normal to the surface x2 + 2y? – 32 – 4 = 0 at any point (x, y, z). - - -

A: Let's find.

Q: Find the orthogonal trajectories on the sphere x2 + y2 + z2 = a2 of its intersection with the…

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Q: Find the points on the surface 6x² + y² – 3z² = 4 where the tangent plane to the surface is parallel…

A: We need to find points of intersection.

Q: Suppose that the plane tangent to the surface z = f(x, y) at (-4, 2, 5) has equation z + 5x + 4y =…

A: From the equation of tangent, we can derive: z = -(7 + 5x + 4y) Hence,

Q: /Sketch the Surface Z=V9-X+y- and verify that fxg = Exx %31

A: Second order Partial Differentiation

Q: Integrate F = 3x2yi + (x3 + 1)j + 9z2k around the circle cut from the sphere x2 + y2 + z2 = 9 by the…

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Q: Find vohume fome above ZVy-xLy, below Z0(xy plane) inside xy/ 3.

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Q: Compute the equation of the plane tangent to the surface x^2+3 y^2+z^3 = 8 at the point (2,1,1).

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Q: = xyi + yzĵ + de is a in x-y plane.

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Q: Find the equation for the tangent plane to the surface z= 7x2. 10y² at the point (2, 1, 18). O 2x +…

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

Q: (a) Between 1 and the disk x²+ (1+x² +y*) (b) Between % = 1 and the entire xy-plane. (1+x* + y?)*

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Q: Find an equation of the tangent plane to the surface zx + 2xy + 3yz3 = 6 at a point in the…

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Q: 2. For the surface z = In(y+ Vx² + y²). (b) Find the linear approximation at the point (3,-4).

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Q: Calculate fc((3x + 5y) i + (3x + 4y) j) - dr where C is the circular path with center (a, b) and…

A: According to the greens theorem:

Q: Find an equation of the plane tangent to the surface z = x2 - y at the point (2, 1, 3). O A. 4x - 2y…

A: Find an equation of the plane tangent to the surface z= x2 - y2 at the point (2,1,3). Here i…

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

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Q: Find an equation of the plane tangent to the surface xy z2 = 2 at the point (2, 1, -1). О А.х+ бу +…

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Q: Let F(r, y, z) = 2x² + 14y? + 722. Find the equation to the tangent plane to the level of surface…

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Q: At what point on the curve x = t³, y = 6t, z = t4 is the normal plane parallel to the plane 6x + 12y…

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Q: Calculate F- dS if F(x, y, z) = x²ye²i+ æy?j – 2xye*k, where S is the surface of the box bounded by…

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Q: Find the center of mass of a thin triangular plate bounded by the y-axis and the lines y = x and y =…

A: Consider M is the mass of the triangular plate bounded by the y-axis and both the lines. To find…

Q: Find the area of the surface cut from the paraboloid x2 + y2 - z = 0 by the plane z = 2.

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Q: Find the equation of tangent plane to the surface x² + 2y? + 3z2 = 21 which is parallel to the plane…

A: Topic- equation of tangent to surface

Q: Find the orthogonal trajectories of the conicoid (x + y)z = 1 of the conics in which it is cut by…

A: Given:The equation of conicoid is (x+y)z=1The system of the plane is x-y+z=K, where k is…

Q: Find the average height of the hemispherical surface z = √(a2 - x2 - y2) above the disk x2 + y2 ≤ a2…

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Q: (b) Integrate (x, y,z)=(y+z)+1 the paraboloid 4x - y-z -0 by the plane x 1. over the surface cut…

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Q: Find the points on the surface 1x^2+2y^2+5z^2=1 at which the tangent plane is parallel to the plane…

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Q: Find a parametrization of the curve of intersection of the surfaces z = x² – y² and z = x² + xy + 1.

A: The given curves are z=x2-y2 and z=x2+xy+1 To find the intersecting curve : Let us equate both the…

Q: find ⌠⌠F*dS if F(x,y,z)=2xi+2yj+2zk and S is the paraboloid z=8-x^2-y^2 lying above the rectangle…

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Q: Find a vector function that represents the curve of intersection of the surface 4x+2y-8z^2=16 and…

A: To find a vector function that represents the curve of intersection of the surface 4x+2y-8z2=16 and…

Q: Use t = y to parametrize the part of the line of intersection of the two planes y = 2x – 4 and z =…

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Q: Find a parametrization for the intersection of the surfaces: z = x2 + y2 and  x2 + 4y2 = 1

A: Find the parametric representation for the intersection of the given surfaces.

Q: Calculate / F . dS where F = (e* sin y, e cos Y, yz²) S is the surface of the box bounded by the…

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