Taylor Polynomials and Series

Exercises

Exercise 1

Find the Taylor series for the function f(x) = cos(2x) at x = 0.

Hint

Solution

Exercise 2

Find the Taylor expansions for ln(x) about x = 1 and ln(x + 1) about x = 0 and compare your results.

Solution

Exercise 3

Take the derivative of the series for sin(x) and show that it equals the Taylor expansion (series) for cos(x), therefore verifying our formula for the derivative:

Solution

Exercise 4

Taylor was a student of John Machin, who had derived an exact formula for calculating π given by

Unfortunately, arctan (the inverse of the tangent function) is a transcendental function, so it was difficult to evaluate. Of course, Taylor came to the rescue by providing a general expansion formula to represent a function by polynomial approximations.

Find the 10th-order Taylor polynomial for arctan and evaluate the formula. What is the error with the actual value of π?

Hint

Solution