5 Is the function f(#) = E (-1)" X[n,n+1) n21 integrable on R in the Lebesgue sense. Here XA is the characteristic function on A. Prove the assertions you make.

Q: 2. If J = [c, d), or J = (c, d], or J = (c, d) and / C [a, b], show that g =x, is Reimann b.…

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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: a. Show that W[5, sin2 t, cos(2t)] = 0 for all t by directly evaluating the Wrosnkian. b. Establish…

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Q: The set of all values of p which makes the integral x* dx is divergent is a) (-1, 0) b) (-n,-1) c)…

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Q: 59. Prove the general formula iwollo oi ons dx 1 x - In x – b + C %3D (x – a)(x – b) a – b - - where…

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Q: 3 Evaluate the integral de using the two-point Gaussian quadrature rule O -|4 1035 N/N N/W 끔

A: We have to evaluate the integral ∫−3311+x4dx by using two-point Gaussian quadrature rule. Here,…

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Q: Let f (x) = e* cos x. Use the second-degree (n = 2) %3D Taylor polynomial P2(0.5) about xo = 0 to…

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Q: 3.10 Show that an arbitrary function f(x) can be expanded on the interval 0<xsa in a modified…

A: Let f(x)=∑AnJYyrnxa Now multiply both sides by x JYyrmxa , x f(x) Jvyrmxa=∑x AnJvyrmxaJvyrnxa Now…

Q: 5. Let D = [0, 1] × [0, 2] be a rectangle in R? and define the function f : D → R by f (x, y) = 4 –…

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Q: Let f (x) = cos(6x) + 7. Compute the following Taylor polynomials of f. For any approximations, you…

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Q: 4. Let f(x) = x, Vx E [0, b]. Show that f is integrable and that 62 f(x) dx : Answer:

A: Given function is  fx=x, ∀ x∈0,b

Q: 2. Consider the space L ([1,2)) of complex square integrable functions on the interval [1, 2).…

A: Consider the  equation

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

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Q: dx 2 -o x+4x+5

A: The general form of a complex number is given by , where a, b are two real numbers and . Any complex…

Q: Compute the residue of the following functions in two different ways and what is the type of…

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Q: -пя for all x € 0, 00). t fal(x) — же па for all z € [0, co). Find || fn||sup for all n. Prove that…

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Q: 1 Use the Cauchy integral formulas to evaluate the e cos z *zp- 22(2: - 37) where C is the region…

A: Given: Let  I =∮C  eiz cos zz2 2z-3π dz  .....1 And,  C is the region covered by circle z-2=3.

Q: 6. Let r>0. Evaluate the following improper integral:

A: the given integral is ∫-∞∞rr2+x2dx First solve the integral ∫rr2+x2dx r∫1r2+x2dx substitute x = ru…

Q: FIAD THE OTER FIXF FUH aiON OF ĦF xauTF XHGUE X O SIn X =3

A: The given function is:sinX=35To find other five function

Q: 3) Suppose that g is a Riemann integrable function on [0.3,31.5] and f(x) = g(x) except for the…

A: This is a question of the Theory of Reimann Integration.

Q: Prove the following result: Lf(at) = 1 S F a a Evaluate L(cosh(5t)).

A: To prove that, Lfat=1aFsa. By definition, Lft=∫0∞ft e-stdt=Fs ⋯⋯⋯1. Therefore,  Lfat=∫0∞fat e-stdt…

Q: (4 b Prove that 5 foxx dx = K (6-a) using definition of integral, where fø= k [a,b], on

A: Given function is fx=k on a,b. We have to prove that ∫abfxdx=kb−a. Take Left hand side  Use Property…

Q: Compute the suitable integral representation of the following function 0, 0.5, —2 2

A: The given, G1x=0,x&lt;-20.5,-2&lt;x&lt;-11+x,-1&lt;x&lt;01-x,0&lt;x&lt;10.5,1&lt;x&lt;20,x&gt;2

Q: ii. Compute f (x) = In(1 + 2x) at xị+1 = 1.1 by first, second, third, and fourth degree Taylor…

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Q: Let f (x) = cos(7x) + 8. Compute the following Taylor polynomials of f. For any approximations, you…

A: From the given problem:   fx=cos7x+8   Let, the centered at a=0,   As we know that the Taylor…

Q: Prove the irrationality of n² by expressing the integral In - = 6.² P(x) sin x dx as a linear…

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Q: 11. The function f(x) is defined on [1, o0) and satisfies VTO dt. 10 Find f(x).

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Q: Consider the function f(x) = lIn(x). Show that x(In(x) - 1) is an antiderivative of the function f.…

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Q: Express the limit lim E (4 cos (272;) + 6)A¤; [2, 4] over as an integral. n- 00 i=1

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Q: Let f : [0, 1] → R defined by if æ = 0, f (x) - L otherwise is Riemann integrable on [0, 1].

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Q: Evaluate the following integral using Cauchy's Residue Theorem. 2+3 $ √√2-51=3 (z − 3)(z - 6)(z- 9)…

A: Given ∮z-5=3z+3z-3z-6z-9dz

Q: Let f be an integrable function on [-1,3], with the property that for each E [-1,3], we have | F(t)…

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Q: 4. Use the Fundamental Theorem of Calculus and the definition of the integral to compute the…

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Q: Let f (x) = cos(6x) + 7. Compute the following Taylor polynomials of f. For any approximations, you…

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Q: Evaluate the improper integral t* e-2t 3+e-4t at

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Q: . Let Show that, Hence show that •2T F₁(x) = 2/1/² 0 Fn(2) = eix cos ∞ N=-∞ eix cose eine de in Jn…

A: As per the question we are given the sequence of functions Fn(x) in form of a Fourier integral. And…

Q: 2. Evaluate the following integral using the Fundamental Theorem of Calculus. (t-3 – 8) dt

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Q: Evaluate the integrals

A: Given information: ∫141ti+15-tj+12tkdt

Q: show that f(z)=(z-1) cos(1/z+2) has an essential singularity at z=-2

A: Introduction: Laurent series has an important role in determining the singularities. Suppose, a…

Q: Let f (x) = cos(6x) + 7. Compute the following Taylor polynomials of f. For any approximations, you…

A: Given  functions fx=cos6x+7

Q: 11. Show that the definite integral of f(x) = c over [a, b] is c(b-a) by applying the definition of…

A: Here given function as fx=c where the function fx is defined on the interval a, b.

Q: Use the Integral Test to determinOwheher the scrie sure to show that the hypothescs of the Intogr al…

A: Given

Q: 7. Evaluate the following integrals : (») f (- 4 - z) dz, (a) z – 1 |z|=3

A: Since you have asked multiple questions in a single request we would be answering only the first…

Q: 1. Determine the discrete least squares trigonometric polynomial S2 (x) on the interval [-T, T] for…

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Q: Expand the function f(x) = tan x up to the first two non-zero terms in the Taylor's polyno- xn+1…

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Q: Suppose that f is continuous on [0, 1], ƒ(0) = 0, and f'(0) exists. Prove that the Lebesgue integral…

A: This is a question of measure theory.

Q: 2.2 Let y = f(x) be a C2 function. Use the first order Taylor Theorem to show that f is concave in…

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Q: 2. Let f(t) be a continuous function on the interval [0, 27]. Suppose the 1st order Fourier…

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