2-3. If f is integrable on [a, b], then ("f(x) dx = lim E f(x) Ax i=1 • In the theorem above, define the ff: •, Ax Xi,

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A: To prove:

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Q: 1 14. If f is integrable on [a, b), then f is also integrable en [a, b].

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Q: d x"J,(x)=-x"Jns(x) d, n+1 X.

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Q: f(x)=0 for x EQ and f(x)=x otherwise. We can conclude f is not integrable on [-1,1] sim

A: In this question we will use basic concepts of calculus i.e concepts of integration and continuity.…

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Q: 1. Consider the function f defined on [0, 0), f (x) = {x* sin- x + 0 x = 0 where r > 0. Determine…

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Q: 4. ·| cotl t• dt 0. lim limitinin değeri kaçtır? n In(x + 1) a) 3 b) 1 c) O 2 d) 4 Boş bırak

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Q: Suppose that f and g are integrable and that -1 f(x) dx = 5 h(x) dx = 4 7, dx = %3D Use the rules to…

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Q: 3) Examine the limit of 1 (x² +y²).sin- x*+ (x, y) (0,0) f(x, y) = (0, (x, y)=(0,0)

A: Solution:Givenf(x,y)=x2+y2 sin 1x2+y2To Eamine the limit of a functionFormula:limx→0 sin xx=1

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Q: 7. Prove that if the functions f and o are bounded and integrable in [a, b], then 6. 9. f(x) $(x) dx…

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Q: 68. Show that the asymptotic representation of the Bessel function Jn (kr) for large kr is Jn (kr) =…

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Q: 3. Suppose that f and g are integrable and that f(x) dx = -1; , f(x) dx = 5; and f, h(x) dx ='4,…

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Q: Evaluate the integral f 2x ¡dx using the two-point Gaussian quadrature rule. 16 \\frac{2}{3}\)

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Q: (5) Prove that if f is continuously differentiable on [a, b], then |f(a) – f(b)| a – b max f'(x)|…

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Q: 1) If f(x)=x2, find f(x+2)-f(2) .what is bijective h mapping? 2) Does the integral J, dx exist?

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Q: 11. Verify Rolle's theorem for: c) f(x) = In () on [1,3] %3D

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Q: 2. Find a function f(x) such that n n 1 n+ i i=1 n n i=1 3. By evaluating an integral, compute the…

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