If {S.) is a sequence of continuous functions on [0, 1] such that 0sf, s1 and such that f(x)0 as n- co, for every x e [0, 1], then lim f,(x) dx Try to prove this without using any measure theory or any theorems about Lebesgue integration. (This is to impress you with the power of the Lebesgue integral. A nice proof was given by W. F. Eberlein in Communications on Pure and Applied Mathematics, vol. X, pp. 357-360, 1957.)

Q: 6 (b). Let 0 < f(x) < cos x and 0 < g(x) < sin x for x E I. Show that there is ro E I such that…

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Q: (0,1). Prove the following functions is uniformly continuous on sin x f(x) =

A: Uniformly Continuous:-  Here we prove the result using a theorem of uniformly continuous.

Q: Suppose F and G are continuously differentiable functions defined on [a, b] such that F'(x) = G'(x)…

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Q: Which of the following is an NOT Riemann integrable over the interval [-1,1]? example of a function…

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Q: Letf be a differentiable function over [0, 2]. The values of the constants a and b such that the…

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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: Find all the values of a, b e R such that the function sin (3x) ar if x > 0 if x = 0 (arctan() if x…

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Q: On the interval [0, 8] the continuous func- tion f has graph 6- 21 2 4 I 6 Determine the value of…

A: The graph of the function f(x) over the given interval 0, 8 is:

Q: 2.6. Let ƒ be continuous on [0, a]. If x e [0, a], define fo(x) = f(x) and let %3D 1 fa+1(x) = -…

A: We will use the basic knowledge of integral calculus to answer this question. Note that we can only…

Q: f is continuous, then show the integral from 0 to π of xf(sin(x)) dx = π/2 times the integral from…

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Q: Let constant c > 0. Suppose that f is continuous on [ 1,c]. Use ONLY- the Riemann sum definition of…

A: Riemann sum: Any definite integral value can be approximated using Riemann sum. This is done by…

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: Let f : (0, ∞) → R be twice differentiable and |f| < Mo, |f"| < M2 for some positive constants Mo…

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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: 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: Show the following functions are continuous P• 18²R" such that fix,y) = (x₂ - y)

A: AS PER GUIDELINES WILL SOLVE ONLY FIRST QUESTION , PLEASE MENTION WHICH QUESTION TO BE SOLVED OR…

Q: Suppose f and g are continuous on [a, b] and f f(x) dx = S% 8(x) dr. Prove that there is cE [a, b]…

A: Suppose f and g are continuous functions on [a, b] such that ∫abf(x) dx=∫abg(x) dx We need to prove…

Q: Prove that there exists a unique function f : R → R such that f (0) = 0 and f ′(x) = sin(x)

A: We have to prove that there exists a unique function satisfying given conditions.

Q: Find a linear model, g(t)=mx+b, and an exponential model, g(t)=ab^t, if g(0)=5 and g(-2)=10. (Round…

A: Here the exponential model is given. So from that, we will find the value of a and b. then solving…

Q: Consider the following functions. f₁₁f₂ R² OR + S₁ (x₁, x₂) = x² sin(x₂) + X₂ exp(x₁) 12 (X₁1×₂) =…

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Q: If f is continuous on [0, π], use the substitution u = π-x to show that π [" xf(sin(x)) dx =…

A: To prove

Q: Find the value of the constant c that makes the following function continuous on (-0, 00). са + 4…

A: We will solve the problem

Q: If f is continuous on [0, π], use the substitution u = -x to show that -76 2 ["xf(s xf(sin(x)) dx =…

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Q: Theorem 6.1.2 (Differentiation of the Integral) Let f be a continuous function on (a, b], and let…

A: We have to prove the Theorem 6.1.3 from Theorem 6.1.2

Q: |Consider the function f with domain (-2, 0): 3|1² – 1| 21² + 2r – 4 if – 2<1<1 f(z) = if x = 1 3 if…

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Q: 2. The following funchion O, x = 0 flad = f(x) |-X, 0<x<l is differentiable on (0,1) and satisfies…

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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: ( g(x) + a, xsO 13-b cos x, x>0 Let f(x) be given by the function f(x) = where a and b are constants…

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