the following theorem. Theorem If f is integrable on [a, b], then b [ºr(x) dx f(x) dx = lim f(x₁)AX n→∞ where Ax = i = 1 (x² b-a n and x; = a + iAx. iven theorem to evaluate the definite integral. 9 [₁²³ 4x + 9) dx

Q: Prove that the function f(x) = sin r x¹/3 is absolutely Riemann integrable over [0, 1].

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Q: (i) Using scaling and translation invariance only, find a, b,c such that for some Rie- mann…

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Q: 3. Consider a function f(x) = sin() : R → R. x²+1 (a) Show that f is continuous on R.

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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: Which of the following functions f: R → R is continuous at the point 0 € R? f(x) = { √x 0 f(x) = {…

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Q: Let f be continuous on the interval [0, b], where f(x) + f(b − x) ≠ 0 on [0, b].

A: Let I=∫0bfxfx+fb-xdx   I=∫0bfxfx+fb-xdx                     ....... (1) use ∫0afxdx=∫0afa-xdx…

Q: Evaluate function given assuming ƒ is integrable on [a, b].

A: We have to calculate,∫abf(x)dx+∫baf(x)dx = ?            ...................(1)As function f(x) is…

Q: Is the set {sin nx , cos mx} an orthogonal set with respect to the inner product on the set of all…

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Q: 7. If f is continuous on [0, 1] and if [s f(x)x" dx = 0 (n = 0, 1, 2, ...), prove that f(x) = 0 on…

A: We use Weirstrass Theorem

Q: Evaluate the definite integral using the Fundamental Theorem of Calculus. 64 (1 Vx)2 dx 16 2x

A: we need to evaluate ∫16641 - x22x   using fundamental theorem of calculus if F(x) be any…

Q: 3. Let f be a bounded continuous function on C and a E C. Show that f(z) dz lim 0. iz]=r (z – a)²…

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Q: 1. Let f: R² →→ R be defined by f(x, y) = sin(+²) and f(0,0) = 0. Is f continuous at (0, 0). Is it…

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