3. Find the extreme values of the given f(x, y) on the indicated surface. f (x, y) = 8x² – 2y; x² + y² = 1

Q: 10. (1 – y³) dx + (x³ + e*) dy, Č is the boundary of the region between the circles x² + y² = 4 and…

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Q: 2.) Find the acute angle between the given curves y² = 2x and 4x²+ 4y² +5y=0

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Q: Suppose f(x, y, z) = In(7x + 9y+7z). Find an equation for the level surface of f that passes through…

A: Given Data: The level of the curve for fx,y,z=In 7x2+9y+7z2 is of the form,…

Q: 2. Evaluate f (3x + 4y2) dA, where R is the region in the upper half-plane 15 bounded by the circles…

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Q: The temperature in degrees centigrade (°C) at each point (x, y) on a curve 2x + y = 3 is given by…

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Q: Find the point in the plane x – 2y + z = 4 that is closest to the origin.

A: Solution)                   Given                           plane  x-2y+z =4

Q: 2. Use the method of Lagrange Multipliers to determine which point on the surface !++ = 1, æ, y, z >…

A: According to the Legrange multipliers method to find closest point on surface to origin, solve for…

Q: Find the points on the surface x² + y² + z² = 4 farthest from the point (1,–1,1), 2 %D

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Q: The numerical value of 34 3dA (where D is the region bounded by lines y=0 and x = 1, D and the…

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Q: 8. (x – y) dV, where E is enclosed by the surfaces z = x² – 1, z = 1 – x² , y = 0, and y = 2 E

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Q: 5. D is the region in the xy-plane bounded by the curves x = cos(y), 3y 27Tx, Зу 3 — 2тх. Evaluate x…

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Q: The surface of many airfoils can be described with an equation of the form Thickness max y = =…

A: As per the question we are given the general form of the equation y = f(x) of the surface of a…

Q: Q2: f(x. y)- x/y over the region in the first quad- rant bounded by the lines y x,y 2x,x 1,x-2

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Q: The area bounded by the curve y3a and the lines z 4 and a-9 above the z-azis is aquare units O 27v0…

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Q: The surface area of the surface generated by revolving y= 1 – x² for 0<x<1 about the x – axis is s=…

A: The solution is as follows

Q: 3. Find the area of the band cut from the paraboloid x² + y² -z = 0 by the planes z = 2 and z = 6.…

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Q: 4. Let C be the unit circle in the xy-plane, and S be the boundary of the square with side 1,…

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Q: A surface in R³ is defined by the equation 4z = x² + 4y“.

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Q: Find the absolute maximum and minimum values of f(x, y)= xy on the quarter-disk D={(x,y)l x20, y20,…

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Q: Find the absolute extrema for f(x,y)=4-x^2-y^4+1/2y^2 over the closed disk D:x^2+y^2 is less than or…

A: If f(x,y) is function of two variables then the points which satisfyfx=0fy=0 are known as a…

Q: 9. Find the point (x, y) on the line y = 2x + 3 such that the distance from (x, y) to (1, 1) is as…

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Q: Find the absolute maximum and minimum values of f(x, y) = 3xy−6x−3y + 7 on the closed triangular…

A: Find out the absolute maximum and minimum values of  f(x,y)=3xy-6x-3y+7 for the maximum and minimum…

Q: 16. Let h(x, y) = xy – 2x². Find the minimum and maximum values of h on the rectangle where 0 < x <…

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Q: 11. What is the equation of the surface revolved about the x-axis for 2² - xz = ln x + 3x? A. y²…

A: Introduction: When a surface is revolved around the coordinate axis, another surface is obtained. In…

Q: (iii) Sf xy²e*dA, Dis bounded by y = x,y = 2,x = 0.

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Q: Find the length of the segment of the curve y = f (x) = e ^ x between the abscissa points x = 0 and…

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Q: What are the control points of the splines Bezier curve from the two equations given below: a(x) =…

A: Given, ax=1+5x2+2x3...1bx=1-x+x3...2 The control points are lie on both the curves. So, to get…

Q: A surface in R³ is defined by the equation 4z = x² + 4y².

A: Given:-                 The surface in  is given by  . Sketch the intersections (traces) of this…

Q: 1. Find the absolute minimum value of 0 < x < 4 f (x, y) = 4x² + 2xy – 2y? – 18x on the rectangular…

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Q: (ii) Find the extremum of Z= x2 +y2 subject to x + 4y 2 and examine whether the %3D %3D stationary…

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Q: 2 Find the point onthe curve y=Jx that is. the point (4,0). minimum distance from a

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Q: 2. Sketch the level surface of the function w = f(x, y, z) = 25 – x² – (z – 2)² for w = 9.

A: 2. Sketch the level surface of the function w=fx,y,z=25-x2-z-22 for w=9.

Q: Suppose that a thin metal wire in the shape of a semicircle occupies the curve C, where C is the top…

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Q: The length of the curve y = Ey3/2 between x = 0 and X 1 is %3D

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Q: 川- Q1 Verify that [(y² – 2x )dx +(2xy +x² )dy = [[] dA where C is the boundary of a regi ôy R R…

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Q: The surface area of the surface generated by revolving y= 1-x2 for 0<x<1 about the x - axis is S = […

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Q: Find the length of the curve y = 4 1 between x 1 to x 2. %3D &x2

A: It is length of curve.

Q: Given w = f (x, y, z) = 4.x² + y² + 4z². Sketch the level surface of the function for w = 1.

A: We have to sketch the level surface of the following function for w=1, w=f(x,y,z)=4x2+y2+4z2

Q: Find the point on the graph of z = x2 + y2 + 10 nearest the plane x + 2y - z = 0.

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Q: Find an equation for the level surface of the function f(x,y,z) = Vx+ y - Inz through the given…

A: Solve the given

Q: 4. Evaluate ||| x y dV , where E is bounded by the parabolic cylinders y = x², y' = x and the planes…

A: The region E is bounded by the parabolic cylinders y = x2, x = y2 and the planes z = 0 and z = x +…

Q: 4. (a) Use the trapezoidal rule to numerically integrate from a = 0 to b= 2 /(z)-0.2+25 +3r

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Q: The numerical value of л за 3dA (where D is the region bounded by lines y=0 and x = 1, D and the…

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Q: 2. Suppose that the edge lengths r, y, and z of a closed rectangular box are changing at the…

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Q: (b) Construct the Green's function of the problem: y" +y' - 2y r(x), 0Sxs1 %3D y(0) = y'(1) = 0 %3D…

A: Given problem: y''+y'-2y=r(x),  0≤x≤1y(0)=y'(1)=0 Note: 1. Green function exists iff solution of the…

Q: Below is a curve C, which is the intersection between the surfaces  S1 : z = 4 − x2 S2 : z + 2x −…

A: Given that,Below is a curve C, which is the intersection between the surfaces   S1 : z=4−x2  S2 :…

Q: Solve the boundary-value problem by method of separation of variables: du Su +3 Sy = 2u, given u (x,…

A: A

Q: Let h(x, y) = 2x² – 2x²y + y². Find the absolute minimum and maximum values of the surface on the…

A: We use partial derivative to find critical points to check maximum and minimum value

Q: Given w = f(x, y, z) = 4r + y + 4z². Sketch the level surface of the function for w = 1.

A: For the function of the form: w=fx, y, z, the level surfaces are defined using the equation, f(x, y,…

Q: *6. Let f: R' →R be given by =+4x'z-6xyz+4y-3x'y. Is M=f-((0}) a smooth surface (2-dimensional…

A: Given, fxyz=z2+4x3z-6xyz+4y3-3x2y2 Now, writing the given function in matrix form.…