Taylor Polynomials I, Part 3

Taylor Polynomials I

Part 3: Polynomial Approximations to sin x

We now apply the concept of Taylor approximation to another familiar function, sin x. As with the exponential function, it is easy to calculate values of its derivatives at x = 0.

How do we know that the sine function is not a polynomial? State at least one property of this function that could not be a property of any polynomial.

For g(x) = sin x, list the numbers g(0), g'(0), g''(0), and so on. You may quit when the pattern in the list is clear.

List the first six Taylor coefficients for the sine function, i.e., those for k = 0 through 5. Enter these numbers in your worksheet as a₀, a₁, ..., a₅.

Your worksheet contains definitions for the Taylor polynomials for g(x). For each of the polynomials p₁(x), p₃(x), and p₅(x),

plot p_n(x) and g(x) together;
plot the error function, g(x) - p_n(x);
describe the extent to which p_n(x) does and does not approximate g(x).

Assume you are locked in a room with only pencil and paper. You will not be released until you calculate a value of sin 1 to within 0.01. Extend your calculations in step 4 (if necessary) to find a way to do the calculation, and write out the calculation in your report. You may use your computer algebra system (or a calculator) to do the arithmetic, but it should be arithmetic that you could do by hand if you had to. Use your CAS to check your answer.