If 0 ≤ f (x) ≤ g(x) for x in the interval [a, b], can the surface obtained by rotating the graph of y = g(x) around the x-axis over the interval have less surface area than the surface obtained by rotating the graph of y = f (x) around the x-axis over the same interval?

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

A: Given: f(x,y)=3xy-6x-3y+7   Find: The absolute maximum and minimum values.

Q: Find the value of c > 0 such that the regionbounded by the cubic y = x(x - c)2 and the x-axis on…

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Q: Find the absolute minimum and absolute maximum of the function f(x, y) = x^3 – 2y^3 + 3y + (x – 1)^2…

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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: Use Taylor's formula for f(x,y) at the origin to find quadratic and cubic approximations of f(x,y) =…

A: The function fx,y=11-3x-y Then the taylor's formula for fx,y is, fxx,y=31-3x-y2fyx,y=11-3x-y2 Then…

Q: Find an equation of the tangent plane to the given surface at the specified point. f(x, y) = tan xy.…

A: From the partial derivative with respect to x and with respect to y

Q: The area of the surface formed by revolving the graph of x = /ỹ on [2,9] about the x-axis is given…

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Q: what is the area of the rotational surface formed by rotating the part of the function y= x^3/3…

A: We apply the above formula to solve this

Q: Find absolute maximum and minimum values for f (x, y) = x² + 14xy + y, defined on the disc D = {(x,…

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Q: Use Green's theorem to evaluate S. (5xy + x² + y²) dx + (x² – y)dy where C is a closed curve that is…

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Q: (a) Find the solution of the limit using polar coordinate system. x² + y2 lim (x,y)-(0,0) x2 + y?…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: [39] Consider the heat conduction in a circular cylinder: x² + y² < a²,0 < z <b. Assume that the…

A: Given Data: Heat conduction in a circular cylinder  x2+y2&lt;a2, 0 &lt;z&lt;b To Find: (a) Boundary…

Q: If f′(0)=f′(2), find the difference c-2d for the following cubic spline S(x) with clamped boundary…

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Q: Find the absolute maximum and absolute minimum values of the function f(x, y) 2x3 + 3xy + 2y %3D…

A: We have to find the absolute maximum and absolute minimum values of the function fx,y=2x3+3xy+2y3…

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: Find the rates at which the volume, surface area, and interior diagonal are changing at the instant…

A: Volume of rectangular box (V) = xyz Take derivative with respect to t, we get…

Q: Consider a circle of radius 2 centered at the origin. Use the Lagrange multiplier method to find the…

A: It is given that circle is of radius 2 and centered at origin. The function isfx,y=3x-y+5.

Q: The numerical value of f3dA 3dA (where D is the region bounded by lines y=0 and x= 1 and the…

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Q: Use the method of Lagrange multipliers to find the absolute extrema of the f(x, y, z) = y + xz² x² +…

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Q: Determine the x-coordinate of the local maximum of the curve y = (2x - 1)2 (3x+4) in the interval…

A: Given: y=2x-123x+4

Q: Find the absolute maximum and minimum of f(x,y)= 4xy^2 - (x^2)(y^2) - xy^3 on the closed triangular…

A: f(x,y)=4xy2-x2y2-xy3fx=4y2-2xy2-y3fx=04y2-2xy2-y3=04-2x-y=0…

Q: Given (1 + y³)y' = x². (i) Determine a region of the xy-plane for which the given differentia…

A: Explanation of the solution is given below...

Q: For the function f(x, y) = x4 + yA - 4xy on the set D = {I(x, y)| 0<x< 2, 0sys 3}, calculate the…

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Q: A surface x(y, z) is defined implicitly xy + yz +x'z' = 3-xZ Calculate ду

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Q: Find the local minimum value of f(z, y) = 4 + x² + y7 – 7xy.

A: Explanation of the answer is as follows

Q: 4. Let f(x, y) = sin(x²y) + vx + e+y. Find the equation of the tangent plane to the surface z f(x,…

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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: 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: find the absolute maximum and minimum of the function f(x,y,z)=x+2y on the intersection curve…

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Q: Find the absolute maximum and minimum of the function f(x, y) = 3y – 2x2 , constrained to the region…

A: f(x, y) = 3y – 2x2 , constrained to the region bounded by the parabola y=x2+x-1 and the line y=x.To…

Q: Find the absolute extrema for f(x, y) = (x² + 2y²)e-x²-y' over the closed disk D: x° + y° < 4. 2.5 2…

A: Given a function of x and y. We need to find the absolute extrema ( maxima and minima) over a given…

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: If f′(0)=f′(2), find the difference 2c-d for the following cubic spline S(x) with clamped boundary…

A:

Q: Find absolute maximum and minimum values of the function f(x,y) = 4xy – x²y² – xy + 1 in the closed…

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Q: Find the absolute maximum and minimum values of D(x,y)=x²-xy + y² + 1 On the closed triangular plate…

A: Here I am using basic calculus we and also use short method.

Q: By minimizing the function ƒ(x, y, u, y) = (x - u)2 + (y - y)2 subject to the constraints y = x + 1…

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

A: The solution of the problem is given in the next step.

Q: Show that the g(x,y) has minimum of 0 in the region 0 ≤ x ≤ 5 and 0 ≤ y ≤ 5, occurring at x =3 and y…

A: It is given that ;                   gx, y = x2+y-112 + x+y2-72   ..............(I) We have to show…

Q: Find the surfaces on which the function u = 2 (x2 + y²)/z is constar

A: Since u is a constant so, x2+y2z=4a , where a is an arbitrary constant.

Q: For x,y,z > 0, use the Lagrange Multiplier Method to find the minimum value of 1 1 1 + + x y Subject…

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Q: Find an equation for the level surface of the function f (x, y) = ln (x² + y²+ z²) that passes…

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Q: Find the absolute maximum and the absolute minimum of f(x,y)= 3xy-3x-6y+1 on the closed triangular…

A: Given:- The function is, f(x,y)=3xy-3x-6y+1on the closed triangular region whose vertices are (0,0),…

Q: (1) Find the area of the curved surface of the upper half of the sphere x2 + y² + z² = a²(a > 0) on…

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Q: 4x Estimate the error in the quadratic approximation of f(x,y) = e sin y at the origin over the…

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Q: Find the absolute maximum and absolute minimum values of the function f(x,y)=x2+y3 on the region…

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Q: 18. Find the absolute maximum and absolute minimum values for the function f (x, y) = e-**-y* (x² +…

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Q: Show that the subtangent to y = x' at x = a is equal to a/3.

A: Definition used -                            Length of subtangent =ydxdy

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.…

Q: The volume of the solid generated by the curve y = e¬ª, between x = 3 and x = 7 about the |3| x –…

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Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.