Prove the given theorem and provide (1) examples. Theorem 1.7 (Maclaurin's expansion) It states S(x) = f(0) + xf'(0) +f"(0)- + ... + (m)(0) +. .. n!

Q: 2. Find all possible expansions of f (z) = 7(2²-9) 1 about z=1.

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Q: Question 4: Consider the function (kt+1) = kt+1, where St = S(rt+1) = S(r(kt+1)). St Differentiate…

A: Given- Consider the function Ωkt+1 ≡ kt+1st, where st = srt+1 = srkt+1.  To find- Differentiate the…

Q: solve this x(dv/dx)= (1-4v2)/(3v)

A: Solution

Q: In applications it is sometimes useful to use an approximation for factorials. The factorial of a…

A: consider the provided question, fx=lnxFx=xlnx-xFx is an antiderivative of fxtherefore, fx=F'x Now by…

Q: Obtain the 1st three non-zero terms of the Fourier expansion for the following function, f(x)…

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Q: Let T(N) = 1. T(floor (N/3)) + 1 and 7(1) = T(2) Prove by induction that T(N) ≤ log N + 1 for all N…

A: Given: TN=TfloorN3+1 T1=T2=1   To Prove: TN≤log3N+1 for all N≥1, using induction.

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: Assume that the coefficients in the expansion f(x) = anx" n=0 satisfy Enanx" – E Anx"+1 = 0. n=1 n=0…

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Q: Find the Fourier expansion of f (x)=-x; – x<xST;

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Q: 1 h(s) = (s – 10)2 Find the largest value of A such that the function h(s) is increasing for all s…

A: Given: h(s) = 1s-102  Increasing interval :(-∞ , A ). The function f(x) is increasing on an interval…

Q: Prove if equal to MPn (W) dx= 2" (n!)" (2n+)! that m is an integer

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Q: 2k +1 All zeros of T+1(r) are in the interval [-1,1] and given by r = cos where k E {0,1, 2, . , n}.…

A: Given, the Chebyshev polynomials of the first kind Tnx=cosncos-1x.   (d) Consider…

Q: ) Compute the Fourier transform of the function. (a) f(x) = h(x + 1) – 2h(x) + h(x - 1), where 1, if…

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Q: Determine the first 3 terms in the Maclaurin expansion of y = esin² T + X + x²

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Q: A function f(x) = 1 - e-²x is defined over the range 0 < x < 1.

A: The functionTo find the even and odd extension of the function along with the half range Fourier…

Q: If 00 f(a)Σα, π", an a", n=0 then f"(x) =(n+1)(n + 2)an+2x". n=2 Select one: O True O False

A: Given: f(x) = ∑n=0∞anxn Now expanding f(x) will give  fx=a0x0+a1x1+a2x2+a3x3+…+anxn+...…

Q: 6. Find the sum-of-products expansion of F(x, y) = x + y.

A: The given expression is F(x,y)=x¯+y. The sum of the product of expansion can be obtained by the…

Q: A discrete time function x[n+2] = x[n+1]+x[n] defined as. x[0]=18 x[1]=33 x[n] using the…

A: Given: xn+2 =xn+1 +xnwhere x0 = 18 , x1 = 33  To find : function xn  using z transform expression .…

Q: Derive the taylor expansion around x = 0 for the following functions a). cos(x)

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Q: In Exercise 15, you found the second solution to the Legendre equation for 0 and 1 Qo(z) - log (1).…

A: What is Legendre Function: A second order differential equation 1-x2y"-2xy'+nn+1=0 is referred to as…

Q: Let P(x) be a polynomial that interpolates f(x) at x; for i = 0, 1,...,l−1,1+1, ---, n (i.e.,…

A: Given Information:The polynomial  is a polynomial which interpolates  at ,  exclude .The polynomial…

Q: Let f (n) = n²+ 3 and g(n) = 5n - 11 for n in N. Thus %3D %3D f: N N and g: N→ Z. Calculate f(10) +…

A: To find f(10) + g(10) when fn=n2+3 and gn=5n-11 for all n in N.

Q: Compute the coefficient Fourier f the f(x) = {" - T<x< 0 0<x<T

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Q: Consider the funetion f(r) = The value of f(0) i (z+1) (a) 1 (b) -2 (e) 3 (d) None of the above

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Q: 1 x+1 64

A: Here we have to find the value of x.

Q: Let Dn = {ar , a; b | 0<k< n}, where a " = e, b² = e , and ba =alb (a) Show that (a* b)? = e for…

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Q: (a) Let n, k € Z with 0 ≤ k ≤n. Write down the formula for the binomial coefficient (7) and use it…

A: a) Given: n,k∈ℤ with 0≤k≤n To prove that  nk+nk+1=n+1k+1

Q: (1- 2xt + t?) = (x-t)(x,t) (1) at ag (1- 2xt + t) t g(x,t) (2) ax ag t= (x - t) ag at (3) ere g(x,t)…

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Q: Given the periodic function over the period [-π/2, π/2] f(x) = x|x|

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Q: The McLaurin expansion of the second order of the function f (x, y)= e2x4y 3 is (A) – 2 + y + 2x2 +…

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Q: - For a positive integer n, le Pn(r) = (x + 1)(r+2) (x + n). Show that the set of polynomials Pn (x)…

A: Here, we want to show that the set {pn(x)x+1,pn(x)x+1,...pn(x)x+1} is linearly independent. A set…