The joint function of X and Y is given by 0

Q: Find a function f € span(1, cos x, sin x) which minimizes the integral 2πT √²* |x +1 − f(x)| ²dx.

A: Given a function                     f∈span1, cosx, sinx

Q: De min Jx when f(x, y) = cos(y – x) – x sin(y – x). 1. fr = -x sin(y – x) | 2. fa = x sin(y – x) 3.…

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Q: Find the local linear approximation to f(x) = Vx at x = 1 and use it to approximate both 1.1 and…

A: To know which one is better we have to compare with calculator values of sqrt(1.1)  and sqrt(0.1)

Q: Which of the following functions is homogeneous of degree 2/3. option I: 3 f(x, y) = /x² + y2 option…

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Q: If r sin(x) f(x)= Jtan(x) V 5+t2 dt and g(y)= f(x) dx 9" Then is equal to 16 -2 võ 2 23 - 2 V6 4. V-…

A: Here, we need to find g''π4. We are given that g(y)=∫1yf(x)dx and f(x)=∫tan(x)sin(x)5+t2dt. By…

Q: Prove that = va² – x* if y =(f)sin~+ () + (9)v&* - x²

A: Differentiation wrt x

Q: f(x.y)=xy'+ In x-5, then the approximation f at x=3, y=4,dx= .01, dy = 01 is (up to 4 decimal digit)

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Q: Let z(x, y) be the function implicitly defined as the solution to 2x + y + z + sin(xyz) = 4 + that…

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Q: Show that 1, x, and x2 - ½ are orthogonal, when the integration is from x = -1 to x = 1. Write f (…

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Q: A function (f) is homogeneous of degree • if and only if : ôf + y = Af (x, y). Does the function…

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Q: 2. Let f(x, y) = In(x² + y²). Show that fre + fyy = 0. %3D

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Q: Use the Mean Value Theorem to show that f(r) = 2x sinh(r – 1) +2" – r has a critical number between…

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Q: Solve using product rule d(uv) = u dv + v du: f(x) = (3x^2 – x + 4) (2x^2 – 1)

A: Our Aim is to solve the given function using the Product Rule where f(x)=(3x2-x+4)(2x2-1)

Q: Q2: Expand f(x,y) = [sin (x y³)]² about (5, 1).

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Q: Determine (x, y) and 2(x, y), where f(x, y) = X sen(xy) %3D y

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Q: Suppose that f(x, y) : at which {(x, y) | 0 < x < 4, – x < ys va}. 1+ x D

A: Consider a given function:  The objective is to evaluate the following double integral of fx, y…

Q: a) For 0 ≤ x ≤ 1, show that < x². √2 + x b) Use the estimate in a) and integrate it to prove that…

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Q: O Find all second order derivati ves for f(x,y) = Cos(2x) - x²e5Y + + 3y2

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Q: If f(x,y,z) = z In(x²ycosz): then fz is

A: Here, the given function is: f(x,y,z)=z ln(x2ycos z) Using the properties of the natural logarithm:…

Q: In integrating the function f(3x + 5)(cos) dv using integration by parts, what would you choose for…

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Q: 5. Given F(x) = cos (t) dt, determine F'(x). 2

A: Using leibniz rule  Explanation is given below....

Q: The linear approximation at x = and where B is: 0 to 1 X is A + Bx where A is:

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Q: If f =x-e" and F=x'yi-2y'zj + xy'z'k, find: div(F x grad(yf )) at (4,1,-2)

A: The function                     Then, we have to find

Q: Use the Mean Value Theorem to show that f(r) = 2x sinh(x – 1) + 2" – x %3D has a critical number…

A: Given: We can see that the function is continuous in [0, 1] and differentiable in (0, 1). So, the…

Q: (b) Show that the roots of y'(x) = 0 are x = 0, x = L, and x = L/2, so it follows (why?) that the…

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Q: Let z = f(x, y), x = rcos 0 and y = r sin 0. Evaluate the value of zg at the point (x, y) = (/3,1)…

A: Chain rule: If is function of and and are functions of and then, Given:

Q: Find the integration inner product on C[–4, 1] of the given functions. f(x) = 15x g(z) = 3r* (f, 9)…

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Q: Compute the first-order central difference approximations ofO(h^4) of t function of y at x = 1h =…

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Q: Find the critical numbers of the function. b) g(x) = |2x + 3|| c) f(e) = 2 cos 0 + sin2 0 d) f(x) =…

A: gx=2x+3 fθ=2cosθ+sin2θ fx=x lnx

Q: Find the best linear approximation, g, to f on the interval [0, 1]. (x) = 2/x g(x) %3D

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Q: Find He valve of fe2) when fg functions such that are dilferentrable %3D g(x) >o 1 for all x, while…

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Q: If f:R⇒Rf:R⇒R is defined by f(x)=e−cosxsin(x)f(x)=e−cos⁡xsin⁡(x) for every x∈Rx∈R then…

A: The given function is: f(x)=e-cos x sin x To find the derivative first. f'(x)=ddxe-cos x sin x…

Q: Suppose that the joint p.m.f. of Y and Y2 is defined by: 2x + y for x = 1, 2,3; y = 1,2,3 54 f(x, y)…

A: The probability mass function of Y1 and Y2 is,

Q: 10.Show that F(x, y) = (1 + xy)e*yi + (ev + x?e*y)j is conservative and find a function "f" such…

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Q: 14. Give the linear near-minimax approximation for f (x) = sin a on [–1, 1].

A: To determine linear near-minimax approximation for fx=sinx on -1,1 Let the approximation be px=a+bx…

Q: In x)2 To evaluate aX, apply integration by parts twice. For the second application of integration…

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Q: 4. If f is a differentiable function of x and y with w = f(x, y), x = r cos 0 and y = r sin 0, show…

A: Given, w=f(x,y) x=rcosθ and y=rsinθ

Q: (ii)If y=(x+Jx* +1. prove that y'= Vx +1

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Q: Find the integration inner product on C[–4, 2] of the given functions. f(x) = 15x g(x) = 2x³ %3D…

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Q: The function f(x, y) = 2xy + yx + 11 has two critical points. Select one: True O False

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Q: if (2,y) # (0,0) Let f(x, y) = aª + yª Show that D1f(0,0) and if (2,y) = (0,0) D2f(0,0) exist but f…

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Q: If z=f(x,y), x= r cos θ and y= sin θ, where f is a differentiable function expressing az/ax and…

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Q: Show that the p.d.f. of Z is given by g1(r) = e-1(+A)cosh(vAx), x > 0, where cosh(2) = }(e² +e=*) is…

A: Note: Hi there! Thank you for posting the question. As you have posted multiple questions, as per…

Q: 5 The function f(x, y) = | x' cos is homogenous inx and y of degree

A: Given that: f(x, y)=x2cos2yx-y2+x3y

Q: If the functions f(x², y²) and g(x, y) are homogeneous of degree 2 and 4, f (x, y) g(x, y) 1…

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Q: Consider the function f(x, y) = 3 x² y + cos(e*"). Which one of the following is correct- Oyfx-x…

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Q: prove using the IVT. e^x + ln x = 0 has a solution.

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Q: (a) Prove and verify that if (x) = vx , then f (1) is 8: Vx, then f³(1) is 3/8- (b) Integrate the…

A: Given

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.