f defined (a, b) € (a, b) n f" exists and cont on (a, b) X (i) For each x € (a, b), R₁ (x) = f (x-t)__ fin (t) at Jc (n-1)! Induction (ii) c ‡ x € (a, b) Integral By fart R₁ (x) = (x-c) = (x-y)n-1 fn (y) for some y (n-1)! C, x between

Q: Let fn, f be real functions on S2. Show ∞ ∞ ∞ {w: fn(w) + f(w)} = UNU {w: | fn(w) - f (w)| ≥ } }.…

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Q: let f:(a,b)→R be de fined by X€QN(a,b) x€Q' n(a,b) -2 f(x)=' 3

A: This is a problem of Differential Calculus.

Q: F(x, y, z) = yzeX²i + e×zj + xyexZk, C: r(t) = (t2 + 1)i + (t² – 1)j + (t² – 4t)k, %3D 0 <t< 4 (a)…

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Q: Find the divergence of = ea + sin(xy) a + cos² (xz) a X O yey+x cos(xy) - 2x cos(xz) sin(xz) ọ xe…

A: U→=exyax+sin(xy)ay+cos2(xz)az

Q: ,10 If f is a continuous function and F'(x)=f(x) for all x € R, then A 2[F(5) – F(1)] B 2[F(10) –…

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Q: Theorem: The Chain Rule provided If m(x) = EI(x) is a composite function, then that E'[I(x)) and…

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Q: 2. Trapezoid Rule f(x)dx= [(zo) + 25 (2₁) + 2ƒ (2₂) + ... + 2ƒ (2n-1) + ƒ(2₂)]. and z, = a +iAx, i =…

A: ∫abfxdx≈∆x2fx0+2fx1+2fx2+..+2fxn-1+fxn Where ∆x=b-an and xi=a+i∆x, i=0,1,2,..,n

Q: sume that f(t) is continuous on [a, b| and that u = g(t) is cont ferentiable on [a, b], then f(u)…

A: Note- Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: let d(x) be adunction on (0, c) satistying the dollowfng: SCx) So on the interval (0,0) S(x) So…

A: We have to go one by one option wise here.

Q: Suppose f(x, y, z) = xyz + 2xy + 2² · Compute the following values -(2,8₁16) - (-2,-8, 16) (1,-1,0)…

A: We have to find functional value.

Q: Please

A: Given function is

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Q: dt Use parseval's identity to prove that J a? +6²+?) 2ab (a + b) TC

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Q: 5. Let f(z) = 3z* + iz3 + z + 7. Prove that all roots of f (z) are in the annulus 1 < |z| < 2.

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Q: Let f(x) = 1+ VI-1, then f 1'(x) = %3D Select one: O a. (r +1)2 +1 of O b. (z - 1) -1 O c (x – 1)² +…

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Q: find all pairs (m, n) of positive integers such that ϕ(ϕ(n^m)) = n, where ϕ(n) is Euler's totient…

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Q: Let f(x) = (Vx + b² - b)/x, b > 0. What is the domain of f? (Select all that apply.) O (-∞, ∞)…

A: f(x)=x+b2-bx The given query is to find the domain of f(x) Also, to define the function such that it…

Q: 1. Let f(z) = (z+1) (7-3z++2)z and I = P.V. Aiirear . dx. (x²+1)*(x²-3x+2)x If p Res f(z)|z=-i»r =…

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Q: prove that ਰ Apply the calculus of residues to dx (1 + x²)² - π 4 .

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Q: Suppose that f: [1,9] →→ R² such that f(1) =(2,-2) and f' (t) Find f(9) (v₁, 71) √t, √t

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Q: Let f1, f2, .. fn be n real-valued, differentiable functions.- Prove that e :[fif2 ... fn] = (fif2…

A: We have to prove that the given formula.

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Q: Integral Representation r(c) Г(В)Г(c-b) Jo -2 F₁ (a, b; c; x) = for F, : ¹ (1 - ₁)¤-b-¹ (1-x₁)¯ª di…

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Q: Q1] (A) If F(x+y-z₁x² + y²)=0 where z = z(x, y) дz дz x -y =x-y ax (B) If f(x, t) = f²e-y²dy, then…

A: (.)  Leibniz rule :      Given ,(A).        where   and   .(B).

Q: 10 If f is a continuous function and F(x)=f(x) for all x € R, then x)dx=? A © FS) - F)] B ©2[F(5) –…

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Q: Suppose f(z) is defined on all of C, is differentiable at z = a, and satisfies f(a) = a. Show that…

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Q: show the function is continous dis continou or 2=(2x+3 3メ→1」Xハz F(x)ニ

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Handwritten fast Please.. 2) Use Cauchy Residue Theorem to calculate the inverse transforms a) F(s)=…

A: Please find attachment below.  I use formula for residue.

Q: Let F(x, y, =) = (6x° In(5y² + 9) + 4=5) i + ( 10yx + 7:)j + (20xz4 + 7y – 37 sin 72) k 5y2 + 9 and…

A: To evaluate

Q: Consider the function F whose criterion is = / dt. 1+ t2 F(x) sena When determining dF/dx the result…

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Q: Let f(x) = x - cos() +1 on [1,1] a) Use Bisection Method to find P3 (use 5-chopping digit) b)…

A: As per the question we are given a function f(x) = x - cos(πx/4) + 1 on the interval [-1, 1] And we…