Q5 13 Points Let f(x) = (1+20) Q5.1 7 Points Show that the Taylor series of...
7. (-/5 Points) DETAILS MY NOTES Find the Taylor series for f(x) centered at the given value of a, assuming that f(x) has a power series expansion about a. f(x) = x - x3 = --3 Submit Answer Find the Taylor series for f(x) centered at the given value of a, assuming that f(x) has a power series expansion about a. 1 f(x) a = 2 х 20 8( (-1)". „n+1(x - 2) n=0 Find the Maclaurin series for f(x),...
5. Let f(x)- arctan(x) (a) (3 marks) Find the Taylor series about a 0 for f(x). Hint: - arctan(x) - dx You may assume that the Taylor series for f(x) converges to f (x) for values of x in the interval of convergence (b) (3 marks) What is the radius of convergence of the Taylor series for f(x)? Show that the Taylor series converges at x-1. (c) (3 marks) Hence, write T as a series (d) (3 marks) Go to...
QUESTION 6 Compute the Taylor series of f(x)= sin 2x at Then show for the series above that linck; f(x) = 0 for each r QUESTION 7 Let f (x) =-x + 3, x E [0, 1] and let P be a partition of [0,1] given by 1 2 n-1 Calculate L(P) and U(P) and prove using these summations that f is Riemann integrable on [0, 1]. Also evaluate o f(x)dx.
4. Taylor series. (15 pi:) The Taylor series of a real function f(r) that is indefinitely differen- tiable at a real number o is the power series n-0 n! where ỡnf To Write down the Taylor series of the following functions around x = 0: ear, In(1 r), and (x+a)m, where a and m are constants.
19. . 20 . 21 Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] f(x) = e-3x f(x) = Σ n = 0 Find the associated radius of convergence R. R = Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) = 0.] f(x)...
5. Let f(z) = arctan(z) (a) (3 marks) Find the Taylor series about r)Hint: darctan( You may assume that the Taylor series for f(x) converges to f(x) for values of r in the interval of convergence (b) (3 marks) What is the radius of convergence of the Taylor series for f(z)? Show that the Taylor series converges at z = 1 (c) (3 marks) Hence, write as a series. (d) (3 marks) Go to https://teaching.smp.uq.edu.au/scims Calculus/Series.html. Use the interactive animation...
Let f(x) = (1 + x2),1. Find the radius of convergence of the Taylor series of f about x, = 0. fix 2 Let f(x) = (1 + x2),1. Find the radius of convergence of the Taylor series of f about x, = 0. fix 2
15. Let f(x) = 2 . 2+x a) Find the power series representation of f(x) centered at o b) Use the power series representation to find ½ dx. 16. Find the Taylor Series for f(x) = sin (2x) centered at T. 15. Let f(x) = 2 . 2+x a) Find the power series representation of f(x) centered at o b) Use the power series representation to find ½ dx. 16. Find the Taylor Series for f(x) = sin (2x) centered...
Q5 8 Points Let A = -4 0 -3 2 3 0 -2 -1 3 Q5.1 6 Points Compute PTA (x) (the characteristic polynomial of TA), Write monomials in descending order of their exponent, x^n for 3" and a/b for Example: 5/2 (polynomial of degree 0) -2x+1/3 (polynomial of degree 1) 2x^2-3x+12 (polynomial of degree 2) Enter your answer here give the eigenvalue(s) of TA (in ascending order if multiple exists, separated by a comma and with a blank after...
Differential Equations (3) Computing Taylor Series quickly from Other Power Series: Use your result for the Taylor series for f(x) = V r to find the first 3 (non-zero) terms of the Taylor-Maclaurin series of f(r) = v1-r2, by replacing with 1-2 in your series and expanding and combining the coefficients of powers of x. (The Taylor-Maclaurin series is the Taylor series centered around o 0. Note that when a is near 0, 1-2 is near 1.) (3) Computing Taylor...