2. Consider the function f : [0, 1] → R, 1 { f(x):= if 1 < x≤ for some n E N n+1 if x = 0 Prove that f is Riemann integrable on [0, 1]. Remark: It may be helpful to sketch f. "

Q: Prove that if ƒ and g are Riemann integrable on [a, b] then fg is also Riemann integrable on [a, b].…

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Q: -1+x+ 4x² cos (4) når r + 0 når x = 0. f(x) = %3D %3D

A: We have given function;

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Q: Let f be an integrable function of [a,b]and define F(x); =| f for x E[a,b] 1. Evaluate G(x): =] f in…

A: Solution of part (3): Given f is an integrable function on a,b, define Fx=∫axf  for x∈a,b

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Q: Exercise 3.2.4: Let f: RR be defined by Is f continuous? Prove your assertion. f(x):= sin(1/x) if x…

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Q: f(y) - f(z) = [²√ 1 (² + 1 dt = So f(z+t(y-2)) dt = ₁ Df (z+t(y-2)) dt (y-2).

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Q: 6. Let f(x)= : x = 0, 1, 2 : 0<x< 1 Prove f is Riemann integrable on [0, 2], and find 2- - 1<x< 2. +…

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Q: 3. Let fn(x) = ¹+cos(nx), x € R. Find f(x) so that fn → ƒ pointwise on R, then check whether fn → f…

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Q: Given n > 1, define f(z) = 1/z" for z +0. Show that f(z) = U(r, 0) + iV(r, 0) for cos(no) sin(no)…

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Q: Let ƒ : [0, 1] → R be the function given by { Is f Riemann integrable? Justify your answer. f(x) = X…

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Q: NJ Let f; R² R be of class C². We set Flr, 0) = f(rcose, rsone) where r and I are the pular…

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Q: Let f : R" → R be defined by f(x1,x2,..., xn) = x1x2··..xn on the cube ...x [0, 1] (i.e. for 0 < x1…

A: Evaluate ∫01∫01⋯∫01fx1, x2, ..., xndx1dx2⋯dxn ∫01∫01⋯∫01fx1, x2, ...,…

Q: -2 x< -1 Q1)a) For what values of a and b is g(x) = ах — b -1<x<1, x21 Continuous at x= -1

A: The suitable formula for continious       LEFT HAND LIMIT = RIGHT HAND LIMIT

Q: If f is bounded and integrable on [a, b] such that | f(x) | 0).

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Q: By, Where F(2, y, z) = sin yi+(2 cos y+cos z)?+y sinză und (sint, £,2),0 < t < 2

A: Value of the line integral

Q: Let f [0, 6] → R be the function defined as : f(x) = = -2, [4/x], if 0 < x < 1, if 1 ≤ x ≤ 6. Show…

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Q: Let f [0, 6] → R be the function defined as : f(x) = = -2, [4/x], if 0 < x < 1, if 1 ≤ x ≤ 6. Show…

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Q: nu² – u?v² + nv². u² +v² h(u, v) = In

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A: Let's prove this.

Q: 2. Let function f [0, 1] → R defined by f(x) = integrable on [0, 1]. Evaluate f f(x) dr. 659 0, if a…

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Q: Exercise 2. Consider the function f: [0, 1] → R defined by O ifx e Q f(x) =- Vx ifx eR\Q Show in…

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