(3) The exponential of a number x can be computed through the Taylor series e=1+2+2/2++"/n!+... e² = 1 + x + x² / 2 + ··· + x^ /n! + ... The exponential of a square matrix A is defined by the Taylor series: A² A³ 2 3! An e₁ = I + A+ + + + +... where I is the identity matrix of the same size as A. n! (a) Find e for 0 2 3 B == 004 000 As you will learn, the solution of the system of first order differential equations d x= Ax dt with initial condition (0) Fois = x(t) = e¹¹ão, where et=I+ At+ (At)² (At)³ (At)" + + + + 2! 3! n! A square matrix A is said to be diagonal, if all of its entries outside of the main diagonal, (a), are equal to zero. If A is a diagonal matrix, then it is very easy to calculate e, For example, let Then eA - (61) + ( 0 -(32) (1+3+ (3)² + .. -(63) 0 A 1-(2) + ( 3 0 02 2! 0 (2)2 + 3! +. 2! 0 3! 0 1+2+(2)²+ -1 (b) What is e, where D = 0 40 ? 0 00

Q: Find the Taylor series for f(x) centered at 1 if f(n)(1) = (-1)"(n+1)! In(7)7"

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Q: 1) Suppose f(0) = 3, f'(0) = 1, f"(0) = –2, f'(0) = 0, and f'(0) = 4. Write out the first five terms…

A: Here is the solution

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A: As per guide line we have to solve only one question at a time.

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Q: a) Find the first three terms of the Taylor series for f(x) = In x centered about x = 2. b)…

A: f(x) = ln x  a =2  f(2) = ln2  f'(X) = 1/X          f'(2) = 1/2  f"(X) = -1/x2      f"(2) = -1/4

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Q: Find the Taylor series for f(x) =- centered at a = 2.

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Q: centered at a=Z Find the taylor series for fcx) =

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Q: The Taylor series approximation of f(x) = 1/x at x = 1 is %3D

A: The taylor series approximation of f(x)=1x at x=1

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A: SOLUTION-

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Q: Taylor series

A: The Taylor sereis of f centered at x=a is given by…