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: 6.(a) Show that (f.g) = f f(x)g(x)dx satisfies the axioms of an inner product on C[a, b], the space…

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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: 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: Question 1 Determine the interval of convergence of the following power series. (x + 4)" Vn 8n n=0

A: We have to find the interval of radius of convergence.

Q: Prove that m-l dx =0 m +n (し+x)" where m,n are both Positive

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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: Consider the following integral equation. f(t) + f(T) dt = -10 Take the Laplace transform of the…

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Q: 2) Let f be a (1:1) Pinction such that differenhable f(3) = 7 P(7) = 8, f'(3)=2, P(1) -3 If glad = f…

A: topic - derivative of inverse function if f(a)=bg(x)=f-1(x)then g'(b)=1f'(a)

Q: # ∙1.) (Cauchy form f defined f(n) exists (i) For each (ii) for of some. on and Taylor Thm) (a, b)…

A: 1) Suppose that  is  defined  on     Also  given that    exists  and  continuous  on  c)  We  have…

Q: 1 =dx <1 Prove 4 4

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Q: Stan"( 1 -tan m+2 -tan m+1 m+1 1 m+3 Evaluate A B Ⓒ D tan" (x) sec¹x dx. (m + -2)(x) + ¹) (x) - 1)…

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Q: az + b If T(z) cz + d a, b, c, d e R. Show that T(RU{∞}) = RU {∞} if and only if exist

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Q: Prove that + J2 (x) + J, (x) = 0

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Q: Suppose that f: R (0, ∞) and that f'(x) = f(x) ‡ 0. Prove that (f-¹)(x) = 1/x for x > 0.

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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: K=Zp(t) ,then f(x)=x^p-t is irreducible over K[x]

A: GIVEN: K = Zp(t)

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: Find f(2021) if function f satisfies following conditions: f(1)=1, f(x+y)=f(x)+f(y), for all x,y…

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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: Let g(x) = 2 f(u) + 3 where . u= u = with respect to x? ○ 2. f' (u) +3 O (2. f' (u) + 3) (2x + 2)…

A:

Q: Evaluate tan" (x) sec4x dx. 1 1 A -tan 2) (x). + (m+2 -tan" (x) + C m+2 m (m+1) ¹) (x) (m- m+1 1…

A: Evaluate: I=∫tanmxsec4xdx

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: The linear functional F: R^2[x] -&gt; R is defined by the formula F: p -&gt; ∫(from 1 to 0) p(t)dt.…

A: Step 1: Step 2: Step 3: Step 4:

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: ( b) For n=1,2,K , show that (x² –1)y"*) +2(nx + p)y") +(n² –n)y"-1) =0 where %3D d" y p") = (Hint:…

A: * Given that y = x+1x-1pa dydx = px+1x-1p-1ddxx+1x-1    dydx = px+1x-1p-1x-11-x+1.1x-12   dydx =…

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: Assume that f: R→ R and g: R → R are infinitely differentiable functions. Prove the following…

A: We have given that ,  are both infinitely differentiable.We need to prove that , We will use…

Q: F(x+y=z₁ x² + y²) where 2= z(x,y) Prove that x2² - y2/²/² = x-y dy if it was

A: We have to show that x∂z∂y-y∂z∂x=x-y for the given integral surface of the partial differential…