Compute the convolution L ★ II of the functions L and II, which are defined below: 1 if |æ|

Q: Q2: Iff (z) is analytic at zo. (b) Give an example to show that the converse of (a) is not…

A: If f(z)  is analytic at Z0  Then it must be continuous at Z0  We already prove.  Now we consider…

Q: Z→ Z be such that f(x) = 4x + 2. Is f invertible, and if so, find its inverse function.

A: Given query is to find the inverse function.

Q: Show that : f(x) = x2 + x + 2 e Zg[x} is irreducible. Further, find the order of f (x).

A: Definition: 1. A polynomial f(x) ∈ ℤp[x] is said to be irreducible if and only if it has no root in…

Q: Show that if f:R → R is odd and the restriction of fto the interval [0, co) is strictly decreasing,…

A: The odd functions are functions that return their negative inverse when x is replaced with –x. This…

Q: Form the interpolating polynomial of degree k that satisfies pG) (xo) = f(G) (xo) for 0<j< k.

A: To form: The interpolating polynomial of degree k that satisfies pjx0=fjx0 for 0≤j≤k

Q: Let f, g: R → R defined by if x 1 f(x) = %3D and g(x) = 3x + 2. %3D fog and go f.

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Q: (i) Construct the minimax polynomial p2 E P2 on the interval (-1,1] for the function g defined by…

A: As per the question we have to find the bset fit polynomial in p2 ∈ P , to to the curve g(x)=sin(x)…

Q: Prove that the function f:Q Q given byf(x) = 2x – 3 for all x E Q is bijection

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Q: The domain of the function f(x,y)= V1-x²-y² +In(x²+y² – 1) is empty Select one: O True O False

A: we have to find the domain for f(x)

Q: Find all a E Z such that f(x) =

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Q: Let f(x) = x² Inx. Use the Three-Point central Formula with h= 0.2 to approximate f"(1.5) %3D %3D…

A: Given that, f(x)=x2lnx and h=0.2. The three point formula for second derivative is given by,…

Q: Find the residue of the following functions: z3 i. f(z) : at z = 1,2,3. (z-1)(z-2)(z-3) 1 2 ii. f(z)…

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Q: {(u)= (u² -3)*

A: Given the function is  f(u) = 12u2-32. We have to find f'(u).

Q: B- Obtain PDE by eliminating the arbitrary functions from the following; 2- Ix + my + nz = Ø (x2 +…

A: We have to obtain PDE by eliminating the arbitrary function from  lx+my+nz = ϕx2+y2+z2.

Q: Inlegrate the function f(x,9) -x,3zy over the rectungle R: e6x41 and sy61

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Q: 2.Consider the following set of polynomials k kti %3D Pa(x) = x +x+* Show that this set is linearly…

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Q: Let ro = -a; xi = 0; and x2 = a. Compute the interpolation polynomial for f(x) = -r* using Lagrange…

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Q: Consider the following function: f (x) = x* + 2x + 81 + 5x Employ Golden-section to find the maximum…

A: Determine two intermediate points  x1 and x2 such that  x1=xl+dx2=xu-dd = 5-12(xu-xl) here  xl=-2…

Q: Derive a finite difference approximation formula for f"(x) using three points x;-1, x;, and x+1,…

A:

Q: Show that if m and b are fixed (but unspecified) real numbers then the function f(x) = mx + b is…

A: A function f(x) is said to be continuous at a point x=a, in its domain if the following three…

Q: Let f(x)=x*-2x. Use cubic Lagrange Interpolation based on the nodes Xo =-1,X1=0,x2= 3,X3=4 to…

A: Given, fx=x4-2x3 x0=-1, x1=0, x2=3, x3=4 To approximate the value of f3.5.

Q: 3. Determine whether the functions e2a and xe2 by calculating Wronskian.

A: Let f(x) = e2x and g(x) = xe2x Then,  f'(x) = 2e2x ,and g'(x) = e2x+2xe2x (by using chain-rule of…

Q: Prove, if f(x)= V1+x² –In +1+ 1 is increasing and concave on x X E (0;+∞) .

A: f is increasing if first derivative is positive in the interval  and f has concave if second…

Q: Let F(x)= ^4sqrt(x) A. Find the linearization of F(x) at x=16 B. Use your answer in part (a) to…

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Q: Prove that the function f(x) = x3 – 6x2 + 15x - 18 is strictly increasing on R. %3D

A: Given function f(x) = x3 - 6x2 +15x -18 We need to differentiate the given equation  f'(x)= 3x2 -12x…

Q: Suppose that the function f(x)= qx<+4x-6 satisfies the necessary conditions of Rolle's theorem in…

A: The given function is f(x)=qx2+4x-6 ....................................... (1)  Satisfied the…

Q: 3. Let f(x) = { ,x<2 ax + b, x22 Find a and b so that f (2) exists.

A: To find the value of a and b

Q: . Suppose that f is a polynomial of degree 1 and s1, is the linear spline that interpolates f at the…

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Q: 5. Find all pairs (m, n) of positive integers such that y(y(nm)) = n, where p is Euler's totient…

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Q: Determine the Wronskian of the following functions: f1(x) = x, f2(x) = e*, f3(x) = xe*

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Q: Let f(r) = x – 8rš. (a) Prove that f has a maximum and minimum value on [0,4]. (b) Find the maximum…

A: Here we have to find the maximum and minimum value of the given function.

Q: Let f(x) = cos(E) and P(x) be the interpolation polynomial of degree at most 3 which agrees with…

A: Neville’s method generates Lagrange’s approximations recursively. In Neville’s method, we denote…

Q: Let P be a monic polynomial function of degree d, i.e. P(x) = xd + cd–1æd-1 + cd-2æd-2 + ·..+ c2x² +…

A: Let P be a monic polynomial function of degree d. Let Px=xd+cd-1xd-1+cd-2xd-2+⋯+c2x2+c1x+c0. Prove…

Q: Q3: a) Let f : Z → Z be such that f(x) = Vx² – x + 1. If invertible, and if it is, what is its -…

A: the function is invertible if its one -one and onto. The function is one - one if f(x1)=f(x2)…

Q: O Show that the function f: N2 N defined by f(x, y) = 2X3Y is injective but not surjective.

A:

Q: Find the Wronskian of the given pair of functions: x, xe* O xe* O x O et

A: The solution is given as

Q: Q3. Prove that if g interpolates the function f at ro, T1, . .. , In-1 and if h interpolates f at…

A: We have to prove that if g interpolates the function f at x0,x1,…,xn-1 and if h interpolates f at…

Q: .a) Prove by induction that P(n):2+6+12+ 20+..+ 1+2)%3Dn(n+1)(n+2) Vn 21 ,xeR. Find the x-1 b) Let…

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Q: 1- Prove the following relations: (1– 2xt + t?). dg = (x-t) g(x, t) at (1) og (1 – 2xt + t?) = t…

A: Since you have asked multiple questions  so as per guidelines we will solve the first question for…

Q: Derive a finite difference approximation formula for f"(x) using three points x;_1, x¡, and x;41,…

A: Using Taylor series for approximating f(xi+1) about the point xi, we get…

Q: Find the Wronskian for the set of functions.{e−x, xe−x, (x + 3)e−x}

A: To Find theWronskianfor the set of functions. {e-x,xe-x,(x+3)e-x}

Q: 5.11. Prove that the function f(z) = Arg(z) is discontinuous at each %3D point on the nonpositive…

A: Solution

Q: inf f. is not measurable Vn. i

A: False ; is sequence of measurable functions fn is measurable then inf fn is measurable .

Q: Find the upper bound for the absolute error on the interval [xo, x4] for the Hermit polynomial…

A: Given:- f(x)=cos x x0=1 , x1=1.1

Q: Find the residue of f(z)=- at z=1 (z-1)* (z-2)(z-3)

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Q: Find the linearization L(x) of f(x) at x = a. f(x) = 3square root of x, a = 27 show all work

A: fx=3x, a=27 Differentiating with respect to x, we get f/x=3·12x    ∵dxdx=12x=32x Now,…

Q: Consider the function f defined by the formula ( ax2 + bx – 3b if x 1. Determine the values of a…

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