Find the points on the surface xy – z2 = 1 that are closed to the origin.

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

A: The solution is given in the next step.

Q: Show that the curve r= Vti + Vtj + (2t – 1)k is tangent to the surface x +y² – z = 1 when t=1

A: Given that r=ti+tj+2t-1k and f=x2+y2-z-1 Take the derivative for f, ∇f=ddx,ddy,ddzf=2x,2y,-1…

Q: Evaluate the surface SS₂ xz ds S is the part of the 2x+2y+z=4 that li- the first octant.

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Q: Use Green's Theorem to compute y dx+(x³+3xy²) dy where C is the closed curve around the triangle…

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Q: At what points does the curve r(t)=(t,t,14t) intersect the surface z=x2+3y2?

A: Given: rt=t,t,14t, z=x2+3y2 for finding intersection point of curve and surface we put point t,t,14t…

Q: Find the area of the portion of the cylindrical surface Y = 4 – x² in the first octant that is…

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Q: Find an equation for the tangent plane to the surface z = 6x3y2 + 3y at the point P(1, -3). Hint:…

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Q: Find the area of the surface S where S is the cap cut from the paraboloid z = 2– x – y by the cone z…

A: Given that : z=2-x2-y2z=x2+y2 Now, to find the point of intersection , 2-x2-y2=x2+y2Put…

Q: Find the area of the portion of the cylindrical surface y = 4 – x² in the first octant that is…

A: Find your answer below below

Q: the surface z = &x + 4y th=

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Q: 4. Where does the curve r(t) = ti + t²j+ (3 t2)k intersect the paraboloid z = x2 +y² ?

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Q: The volume of the solid generated by the curve y = e, between z = 2 and z = 6 about the z- azis is

A: Given: The curve is y=e-x   To determine: The volume of the solid generated by the curve y=e-x…

Q: Evaluate ſ F . n dS if F = (x + y² ™)ī − (2x)j + (2yz)k and $ is the surface of the plane 2x + y +…

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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: a) Evaluate x²y? ds, where the curve C is the intersection between the surface z = := V1– y? and the…

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Q: Consider the normal line to the surface z = x2 + y² (paraboloid) at the point P(1, 1, 2). Find the…

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Q: Evaluate the surface integral I F. dS when F(x, y, 2) = y²i+ 3x² j and S is the graph of - ža (1+æ²…

A: Here, Fx,y,z=y2i+3x2jz=121+x2-y2

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: Determine the projection of the following surface on the xy plane. (xy trace) 4z-목 + =0

A: here first solve the given surface and then we trace it

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: If (a,B.y) is a point at which the surface x2+y2-z²-2x+12=0 has a horizontal tangent plane, then y|…

A: As per guidelines, we will solve first question only. Let α,β,γ be the point on the surface…

Q: Find the area of the portion of the cylindrical surface y = 4 - x2 in the first octant that is…

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Q: 2. Evaluate f, z³ /x² + y² + z² dV where E is the solid hemisphere that lies above the xy-plane and…

A: Solution:- equation of sphere having centre origin (0,0) and radius=1 x2+y2+z2=1…

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: Find the point closest to the origin on the curve of intersection of the plane x + y + z = 1 and the…

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Q: Evaluate the integral | (3x – 2 y + :)ds where C is the curve, C, and C, as shown. 3- y² + z² 2. 3…

A: Here, Curve C is equivalent to c1+c2. The point of intersections of c1 and c2 is at 0,1,0.

Q: Sketch the surface z = 3x? + y? +1 and find its linear approximation at P-(0, -1)

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Q: 2. Show that the surface area of the solid of revolution obtained by revolving the curve y = 1/x, x…

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Q: Find the point on the surface f(x,y)=x2+y²+xy+21x+7y at which the tangent plane is horizontal.

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

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Q: Find all points on the surface at which the tangent plane is horizontal. (a) z = x³y² (b) z = x² −xy…

A: (a). Given the surface z=x3y2,the tangent lane at a given point is given by :…

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

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Q: Find the surface area of the paraboloid z =1-x² - y? that is above the xy-plane.

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Q: Find the area of the surface defined by z = xy and x2 + y2 < 7.

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Q: 4. Compute |F- dš, where F = (y, –x,2z) and S is the surface of the sphere of radius 2, centered at…

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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: What is the equation of the plane to the surface z = e3Y sin(3x) +2 at the point P,0, 3) O x- y + ez…

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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: 2. Find the area of the surface generated by revolving the given curve about the y -axis. (a)…

A: Given b x=y3, 0≤y≤2c x=9-y2, -2≤y≤2

Q: Let f(x,y)=/4– 4x² – y² . Find its domain and give a rough sketch of its surface.

A: The given function is:fx, y=4-4x2-y2To find domain and sketch the surface

Q: Evaluate the integral // Fn ds where F = (2r+32)i-(rz+y)3+(y² +2)& and S is the surface of the…

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Q: Find all points where the surface z = xy e-* + 2y-) has a horizontal tangent plane. %3D O A. O B.…

A: Given the surface

Q: Find the minimum of f(x,y) = 4x + 3y on the ellipsoid 4x2 + 2y2 = 6. %3D Leave your answer correct…

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Q: Find the points on the surface xy + yz + zx - x- z =0 where the ant pla

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Q: Let C be a curve given by the intersection of the surfaces z = x2 +y2;z = 3−2x . The value of the…

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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 f[fp Vx2 + y2 + z² dV, where D is the sphere of radius a centered at the origin.

A: We need to find the integral:∫∫∫Dx2+y2+z2dVwhere D is the sphere of radius a centered at the…

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.

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.