18. Prove that b b ²1. I 2 a 2 F(x) f(y) dy dx = ( [° F(x) dx) ². [HINT: Notice that 2 b (11(x) dx)² = SAMMAM a [a, b] x [a, b] f(x) f(y) dx dy.]

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A: While finding partial derivative in multivariable function, if we have to find the partial…

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A: Hello. Since your question has multiple parts, we will solve first question  for you. If you want…

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A: Derivative

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A: Just I have used the definition of composition of two functions, and using given functions calculate…

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A: Given  ∫-22∫04-x2(x2+y2) dydx

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A: a) We divide the integral into three subintegrals

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A: using property of definite integration

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A: For the relation , to find its first derivative at the point where  .

Q: (x³ + y² + z) dx dy dz. 2 Evaluate

A: To evaluate the integral

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Q: Given y = f(u) and u = g(x), find y=8u+1, u = dx (글)x 2 || 3 + dy ᆞ = f'(g(x))g'(x). dx

A: According to Bartleby guide line we only do first part of multiple questions.(A) y=8u+1,

Q: Differentiate the function and simplify the answer. y = 8/xe* Select one: dy 4+8V %3D dx VA et 4+8x…

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Q: Which of the following functions satisfy x²dy+ y(x + y)dx = 0 with y(1) = 1 ?

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Q: Let Find f(x)|x=7 = y = f(x) = ( 2 ) ²³ d dæ f(x)\x=7² sin x.

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A: using the given value of the function and definite integral we need to evaluate the unknown value of…

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A: ddxxn=n xn-1

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A: Given that, f, g∈Rx on [a, b]We need to prove that∫abf2dx+∫abg2dx≥2∫ab|fg|dx