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

Q: Let wi and wa be k-forms defined on all of IR". Show that wi and wy have the same exterior…

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Q: Define f(x) = x if x is rational and f(x) = 0 if x is irrational. Compute So f dx and ſ f dx. Is f…

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Q: If f is lebesque integrable  over [a,b] the f is Riemann integrable over [a,b] . The converse is…

A: This is a problem of measure theory.

Q: Let ƒ : ℝ → ℝ be continuously differentiable function and k > 0 be a number such that ƒ '(x) ≥ k for…

A: Let me show f is one one first,then onto then continuously differentiable....

Q: (b) Show that 1/2 1/2 9. 1/2 [f(x)+ g(x)]² dx, g (x) dx a This is the Minkowski inequality for…

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Q: Assume that f is continuous and | f) dt 3 sin х — х сos x. 0, (a) Determine f(;n). (b) Find f'(x).

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Q: Prove that the function f(x):= { Lo sin r x 1. x > 0, x = 0. is Riemann integrable on [0, ∞),i.e., f…

A: The function f(x) has only one point of discontinuity at x=0 in [0,∞)Therefore if we consider the…

Q: Prove that if f is continuous, then f(t)(x-t) dt = f(u) du ) dt

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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: On the interval [-3, 3] (x², x) = / dx = 1 ||x²|| = dx II

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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: Use the Second Fundamental Theorem of Calculus to find F"(x). F(: •¤ /1+ sin t dt t2 + 5t – 6 x /1 –…

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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: 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: Show that at (0,0). f(x, y) = x² sin 0, (²7)(y² +1), (x, y) ‡ (0,0), (x, y) = (0,0). of (0,0)…

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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: Show that if f is Riemann integrable on [a,b] and f(x) ≥ 0 for all x ∈ [a,b], then

A: A bounded function on a compact interval [a, b] is Riemann integrable if and only if it is…

Q: Show that, if f and g are Reimann integrable on [a, b] then f - g is Reimann integrable on [a, b],…

A: We have to prove that following identity.