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- P11) Consider the function f(x) =?? This problem explores two methods for finding the third degree Taylor polynomial centered at x = 0 for f(x). Method 1: Find a Taylor Series, then integrate. .... Method 2: Integrate, then find a Taylor Series. ... PLEASE see detailed question in image attached and show process for all of them8.) F(x)= sin x, a =5pi/3 A. Find the first three nonzero terms of the Taylor series for the given function centered at a.(1) The function f(x) = ln(x) has a Taylor Series representation at x = 1 given byX∞n=0cn(x − 1)n where cn =f(n)(1)n!(a) Find the first five coefficients (c0 through c4) of the Taylor series for f(x) = ln(x).(b) Use the coefficients you found in part (a) to write the formula for the fourth-degreeTaylor polynomial of f(x) = ln(x) centered at x = 1:(c) For this question, use a graphing tool such as Desmos, Geogebra, or a graphing calculator to graph ln(x) and the Taylor polynomial you created in (b).What do you notice about the graphs? (You do not need to include the graph here,just talk about what you observe.) *please write a detail as much as possible
- f(x) = 1/√x = x-1/2 a. Determine the Taylor series about x=1 up to n=3 expressed as a simplified polynomia b. Evaluate f(1.5) c. Determine an approximate value of f(1.5) using the taylor series in a.Found the derivatives, cant seem to find the values to plug into the Taylor series summation notation I have written in the screenshota. Determine the 6th order polynomial approximation of f(x) = cos(2x) by getting the 6th order polynomial approximation of cos (x) from a table of power series and then replacing x with 2x. b. Determine an approximate value of cos (1.5) using the polynomial approximation of cos(2x) in a.
- Let f(x)=e−xf(x)=e−x. We're going to calculate the Taylor series for this function near a=0a=0 . Calculate f(0),f′(0),f′′(0),f(0),f′(0),f″(0), and f(3)(0)f(3)(0) . What patterns do you notice here? Write down the degree 3 Taylor polynomial approximation for f(x)f(x) near a=0a=0 . Make a graph that shows the function and the degree 3 Taylor polynomial. For what interval of x-values is this a relatively good approximation? For which x-values is this not a very good approximation? Based on the patterns you observed in (a), find a general formula for the k-th derivative of this function when evaluated at 0.f(k)(0)=?f(k)(0)=? Write the Taylor series fo f(x)f(x) near a=0a=0, using sigma notation. What is the interval of convergence for this series? What would be different about your result if we instead calculated the Taylor series for f(x)f(x) near a=4a=4 ?a) Find the Maclaurin series for the function f(x) = 1/1 + x b) Use differentiation of power series and the result of part a) to find theMaclaurin series for the function g(x) = 1/(x + 1)^2 c) Use differentiation of power series and the result of part b) to find theMaclaurin series for the function h(x) = 1/(x + 1)^3 d) Find the sum of the series ∞ SUM n(n − 1)/(2n)n=3 This is a Taylor series problem, I understand parts a - c but I do not understand how to do part d where the answer is 7/2(a) Use any method to find the power series representation of f(x) =ln(x2)centered at 1. (b) Approximate the definite integral from 1 to 2, ln(x2)dx using a 2nd-order Taylor Polynomial for f(x) = ln(x2).
- Conduct Fourier series expansion for the figure. Along with the solution and how to choose which form to use. (f(x)=a0/2+Σ(an*cosnx+bn*sinnx) or using e^(-jωt)a) Express f(x) = x2ex as a power series.b) Determine the 3rd degree Maclaurin polynomial of f.c) Use (2) to approximate the value of (e^1/10) / 100Let P2(x) be the second-order Taylor polynomial for cos x centered at x=0 . Suppose that P2(x) is used to approximate cos x for |x| < 0.2. The error in this approximation is the absolute value of the difference between the actual value and the approximation. That is, Error = |P2(x)-cos x|. Use the Taylor series remainder estimate to bound the error in the approximation. Your answer should be a number; that is, you should give a bound for the error which works for all x in the given interval. Hint: Notice that the second- and third-order Taylor polynomials are the same. So you could think of your approximation of cos x as a second-order approximation OR a third-order approximation. Which one gives you a better bound? Error≤ Use the alternating series remainder estimate to bound the error in the approximation. Your answer should be a number; that is, give a bound for the error which works for all x in the given interval. Error≤…