1- Prove the following relations: ag (1 – 2xt + t?)= (x - t) g(x, t) (1) at ag (1- 2xt + t2)= t g(x, t) (2) ax ag t at ag (x - t) ax (3) Where g(x,t) is the generating function of Legendre's polynomials

Q: Interpolate the function f(x) = ev3 on [-1, 1] with a second degree polynomial by using the zeros of…

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Q: Q7; Functions (a) Prove that the function f : Z4 → Z12 given by f(T) = [3x] is well %3D defined.

A: we need to prove homomorphism

Q: If q(x) is an arbitrary polynomial in Pn it follows that                                      q(x) =…

A: Given, if q(x) is an arbitrary polynomial in Pn it follows that                   q(x) = c0p0(x) + .…

Q: Exercise 2.7: Let f(x) = x² - 2x - 3=0 (a) Find the two roots for f(x) analytically. (b) Show that…

A: “Since you have posted a question with multiple sub parts, we will provide the solution only to…

Q: If we multiply the Legendre polynomial of degree n by an appropriate scalar, we can obtain a…

A: We use the generating function of legendre polynomial.

Q: The domain of the function g(t) = vt – 2 is O R-{2} O [2, 00) O R о (-о, — 2) n (? m)

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Q: = + (-x² + xy 7) =

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Q: Given the two points [a, f(a)} [b. f(b)]. the linear Lagrange polynomial fi(x) that passes through…

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Q: (a)Verify Rolle's Theorem for f (x)=x - 16x+10 over the interval [0,4].

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Q: Consider functions f(t) = (t − 1)/(t+2) and g(t) = (2t + 4)/(t−3) Show that the expression for (f ·…

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Q: f (x) + 5 f(x) – 3 g(x) (4) Let f(x) = mx+b, where m and b are constants and m 6= 0. Prove that the…

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Q: Let a be a repeated root of a quadratic equation f(x) = 0 and A(x), B(x), C(x) be polynomials of…

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Q: Transform f(x) = 0 to g(x) = X. Z1 Z2 f(x) = 0 k2+x k1+x +

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Q: 2. Consider the Newton polynomial Pn(z)= ao+a1 (1-10) + a₂(1-10) (1-11)+... +an(1-10)(1-1₁).…

A: Given that   that interpolate then we have Now  ,       putting

Q: Use the table to determine Df(g(x)) at x = 3. |1 2 f(x) 1 3 4 3 2 f(x) 3 -5 1 4 g(x) 2 4. 4 1 3…

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Q: * Q1 In(x²-1) is: Vx²-2x The domain D, of the function f (x) = Df = (-00, –1) U (2, ∞) (A) D, =…

A: Solution:Givenf(x)=ln(x2-1)x2-2xFormula:ln ab= ln a+ln b(x-y)(x+y)=x2-y2

Q: 3. (a) Find the first-order Maclaurin polynomial for the function f(u) = with a bound on the…

A: Introduction: A Maclaurin series is a special case of a Taylor series, which is a mathematical…

Q: The domain of the function g(t) = vt – 8 is O (8, 00) O [8, 0) O (-∞, –8) O R O R-{8}

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Q: 6. Let 1 f (x) - cos x + sin(2x), x E [0, 1] 2 Find the least squares polynomial approximation of…

A: Let fx=12cos x-sin2x, x∈0,1. To find quadratic fitting  fx≈P2x=a0+a1x+a2x2 Consider, s0=∫011dx=1

Q: The Domain of f (x) = V1 – x is 1. [-1, 0) 2. (-∞, 1] 3. (-0, 1) 4. [1, c0)

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Q: Find f(t). &{ 4s - 1 3 \s² (s + 1)³√ f(t) = 7H(t)-t-7e --(61-5-12) 6t- - e X

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Q: Function f that is defined and differentable on all of R.  a. suppose h(x) = (x3 − 4x-1 + 5) ·…

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Q: Find the linearization L(x) of f(x) at x =a. f(x) = 2x² + 3x-2, a = 4

A: Find the linearization L(x) of f(x) at x=a.f(x)=2x2+3x−2,a=4

Q: (x + 3)³ 12 16 11 f(x) Consider the function 4(х — 1)2 4 1 (x – 1)2 4. Note that - (x – 9)(x + 3)²…

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Q: 5. Find all polynomials f(x), g(x) satisfying (2x³ + 1) ƒ(x) + (x² + 3x + 1)g(x) = 1 in Z5 [x]. Are…

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Q: f(x,y) = –x² + 2xy + 3y² – 6x – 2y – 4 | - Serialize the function around the point (-2,1). a) f(x,y)…

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Q: Q4) Consider the Lagrange coefficient polynomial L2x(x) that are used for quadratic interpolation at…

A: The Lagrange coefficient polynomials are given as,…

Q: 0st <4 f (t) = | 2 ,t24 O A.) f (t) = t -tu (t - 4) – 2u (t – 4) B.) f (t) = t +tu (t – 4) – 2u (t –…

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Q: Let g be the function given by g(x) = C (2 – 5t – 14) dt. What is the x-coordinate of the point of…

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Q: Express f(2)= Inz in the form u + iv and test the analyticity of f(z).

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Q: Prove using the generating function of the Legendre Polynomials: t2 [xP(x) – nP,(x)]t": (1- 2xt +…

A: We have to prove ∑n=0∞xPn'(x)-nPn(x)tn=t21-2xt+t232 We know the generating function of the Legendre…

Q: 1-t2 -i+ t+2 j+ In(t + 2)k. Find the domain of r(t) t2+1

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Q: z? –1} (z +2

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Q: Let f(x) 0 = c-8 Oc==3 Oc=2/20 Oc-4/ 11 2xc + 1 C + 3x 3 ▸ Determine the value of c so that x = 9 is…

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Q: 1. (a) Draw the graphs of the functions x and x in the plane. Then find all integers n > 2 such that…

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Q: Let f(x) be a function on (- 0, 0) and f(x + 2) = f(x - 2). If f(x) = 0 has only three real roots in…

A: Introduction :