Let R = [a, b] × [c, d] be a rectangle. Also suppose that f (x) is continuous on [a, b] and that g(y) is continuous on [c, d]. Show that qu //, f(2)g(y) dA = f (x) dx g(y) dy R a Use your answer to part (a) to evaluate xy + x + y + 1 dA, where R R= [0, 1] x [0, 1]. (Hint: Factor the integrand.)

Q: $. flx) =3=x*, glx)=x² -4 d) Domain A domain H) Domain

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Q: Suppose  f ( 200 , 300 ) = 1000and ∂ f ∂ x ( 200 , 300 ) = 12 and ∂ f ∂ y ( 200 , 300 ) = 10.…

A: Given, f(200,300)=1000 ∂f∂x(200,300)=12∂f∂y(200,300)=10

Q: Is g = {(x, y) ∈ [−5, 5] × R : x^2 + y^2 = 25} a function from [−5, 5] to R.

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Q: Assume that f is an odd function defined on R and t e Rt. Compute

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Q: Is the function f : Z→Z defined as f(n)=n^3+2 one-to-one ?

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Q: Show that f1x2 Vin x is right-continuous at x = 1.

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Q: Let f(x, y) = x* /I Evaluate f(3, 0)

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Q: KB is ∀xy F(x, y). Prove using resolution-refutation that ∀xy F(y, x).

A: Solution: First, we convert clauses to CNF : (1) ∀xy Fx,y from KB (2) ∀x Fx,Ax (3) Fx,Ax CNF

Q: Let f(x, y) /150- 4а2 — 2у? 4x? – 2y? Evaluate f(1, 3)

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Q: Q2] Find the analytic function F(z) if u = coshy.cosx, (B.C. v-0 at x=0)

A: at

Q: Define a function f : R² → R² by f 2.у 1. Show that f is differentiable on A = E R² |x > 0, y > 0}…

A: To show that the function f defined as below is differentiable on A and compute its derivative

Q: Consider the function f with domain (-o, o0) whose second derivative is given 20x(r +3) (22 – 1)3 a)…

A: Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and…

Q: Finel the ralves of a anel b such that the function aV5-x tb-1 it Hax cos(TT Ix) 9 +1+ b(x²-7)…

A: Note: There is an error in question, for the given function a has 2 values but it cannot be possible…

Q: Let f: A-XX Y be defined as f (a) = (f, (a). f2 (a)) each a A, Then f is continuous if and only if…

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Q: find f c^) : 2x+3 5X+4

A: Topic :- Functions

Q: If f (s Continuous then on -fex) True or felse?

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Q: Verify by differentiation that

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Q: Let /: X →Y be a function, A, BC X and C, DCY. Show that: 1. f(AUB) = f(A) uf(B) 2. If AC B, then…

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Q: (-) = { 2*. x 2 Find the values of a and b so that f will be differentiable at x = 2.

A: Given function is fx=7x2        ,x≤2ax+b    ,x>2. By continuity, 722=a2+b74=2a+b2a+b=28     ...1…

Q: For the function Hlo>x'tle, fnd a)flxth) 5bf(xth)- flx) c) f(x th)-f(x)

A: Given: f(x)=x2+6 To find: a) f(x+h) b)f(x+h)-f(x) c)f(x+h)-f(x)h

Q: Given F = (xyz², x²yz³, y²), find the following. (a). div (b). grad (divF) (c). curl (grad (divF))

A: We need to find the divergence of the field. F=xyz2, x2yz3, y2

Q: Which of the following functions maps the red function (y=x)onto the image function? a) y = -x+2|…

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Q: Let esinx SoN f(t) dt and f is a continuous function, determine the value of f ()

A: If we have F(x)=∫0xf(t)dx then we have F'(x)=f(x). This is called the fundamental theorem of…

Q: Find the orginal function of this A Fx) =-1 (B) Frz)= X +1 ニー re Fez) s -x+3 tan x - sec3x

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Q: Let f. ROR be a Continuous function with f(x) ≤ 10 for = all XER. Explain why f is not homeomorphism

A: A homeomorphism is a continuous bijection with continuous inverse.

Q: Let x, y € R. Then x^2+y^2 = 16 defines y as a function of x. True False

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Q: Differentiate the given function y = cosh(In(6x*)). 1 а) О Зг? + 4 x4 1 b) O 92? 4 x4 c) O 3z2 1 3x²…

A: Given function is y=cosh(ln6x3)

Q: Construct an analytic function of the form f (z) = u + iv, where v is tan- (y/x), x # 0, y # 0

A: f(z)=u+iv where v=tan-1(yx)  We need to find the analytic function f(z)=u+iv  we apply Milne Thomson…