Question

Compute the third order Taylor polynomial for the cube root function expanded about an arbitrary point. b). Choose the point x₀ so as to be able to get the best rational approximation (ie, written as a fraction), to 10¹/³, and find the approximation. Compute a bound on the error from Taylor's theorem and show that the estimate obeys it.

Answer 1

The third order Taylor polynomial of the cube root function is computed, and choosing x₀=8 gives us a good rational approximation for 10^(1/3). We also calculate the error bound to verify the accuracy of the estimate.

Step-by-step explanation:

To compute the third order Taylor polynomial for the cube root function, we expand it about an arbitrary point x0. The cube root function is f(x) = ∛x.

First, we will find the derivatives needed for the Taylor expansion:

• f'(x) = (1/3)x-2/3
• f''(x) = (-2/9)x-5/3
• f'''(x) = (10/27)x-8/3

The Taylor polynomial is given by:

P3(x) = f(x0) + f'(x0)(x - x0) + f''(x0)(x - x0)2/2! + f'''(x0)(x - x0)3/3!

To get the best rational approximation for 101/3, we can choose x0 = 8 since 8 is a perfect cube. Substituting x0 into the Taylor polynomial gives us a rational approximation. The error bound can be calculated using the remainder term R4(x) of Taylor's theorem.

To ensure the estimate obeys the error bound, we verify that the inequalities prescribed by Taylor's theorem hold for our approximation.