#1: By Taylor's theorem, we can find a Taylor polynomial P3(1) of degree 3 for the function g(t) = cos(2t) sin(31) near/= 0 such that g(1) = P3(1) + R3(0, 1) in some interval where R3(0, 1) is the remainder term. Writing P3(1) as P3(1)= a+at+az² + azt³, calculate the coefficient a3.

Problem 1. Determine the Taylor polynomial of order two of the function  $f(x, y)=\operatorname{arctan}\left(\frac{x-y}{x+y}\right)$ at point (1,1).

Find T5(x): Taylor polynomial of degree 5 of the function f(x)=cos(x) at a=0. T5(x)=  Using the Taylor Remainder Theorem, find all values of x for which this approximation is within 0.00053 of the right answer. Assume for simplicity that we limit ourselves to |x|≤1. |x|≤

1-2. Find the third Taylor polynomial P3(x) for the function f(x) = x(e^x )+ 1 expanded about x0 = 0. 1-3. Use P3(0.5) to approximate f(0.5) and nd an upper bound for the error |f(0.5) - P3(0.5)| using the error formula and compare it to the actual error.  1-4. Find n, the degree of the polynomial Pn(x) such that |f(0.5) - Pn(0.5)| < 10^(-6)

Calculate the Taylor polynomial T3 centered at x = a for the given function and values of a andEstimate the accuracy of the 3th degree Taylor approximation, f(x) ≈T3(x), centered at x = a onthe given interval. 2) f(x) = ln(1 + 2x), a = 1, and [0.5,1.5]

find the Taylor polynomial of degree n for x near the given point a. ln(x^2) a=1 n=4

Find T4(x) Taylor polynomial of degree 5 of the function f(x)=cos(x^3) at a=0

Problem 1. Determine the Taylor polynomial of order two of the function  $f(x, y)=\frac{x y}{x^{2}+y^{2}}$ at point (1,1).

Determine the first three nonzero terms in the Taylor polynomial approximation for the given initial value problem. 4x'' + 3tx=0; x(0)=1, x'(0)=0 The Taylor approximation to three nonzero terms is x(t)=?

Find the degree 2 Taylor polynomial of f(x)=ln(x+1) at a=1