Question 1.

Find Taylor series for ; 𝑓(𝑥) = ln(𝑥) at x=1.
Use MATLAB to plot 𝑓(𝑥) = ln(𝑥) for 𝑥 ∈ (0, 5], along with Taylor approximations around 𝑥 = 1 with 𝑛 = 0, 2, 4, 6.

Question 2.

Find Taylor series for 𝑓(𝑥) = sin(2𝑥) at 𝑥 = 𝜋

Use an appropriate Taylor series to write a MATLAB function M–file that approximates 𝑓(𝑥) = sin(2𝑥) for all x with a maximum error of 0.01. That is, your M–file should take values of x as input, and return values of sin(2𝑥) as output, but should not use MATLAB’s built–in sine function i.e., sin.m. Hint. You might find MATLAB’s built–in function mod useful in dealing with the periodicity.