Question

Use zero- through third order Taylor series expansions to predict f(3) for: f(x) = x^3-10x^2 + 6 Using a base point at "x_i = 1" Compare the true percent relative error (epsilon_t) for each approximation

Answer 1

Taylor series expansions from zero to third order are used to predict the value of the function f(x) at x = 3, with comparisons made of each approximation's accuracy against the true value using the true percent relative error.

Step-by-step explanation:

The student is asked to predict the value of the function f(x) = x^3 - 10x^2 + 6 at x = 3 using Taylor series expansions of zero through third order with a base point at xi = 1. To do so, the nth derivative of the given function at the base point is evaluated and used in the Taylor series formula.

Then, the true percent relative error (εt) for each approximation is calculated to compare the accuracy of these approximations with the true value of the function at x = 3.

• Zero-order approximation uses only the function value at the base point.
• First-order approximation includes the first derivative.
• Second-order approximation takes into account up to the second derivative.
• Third-order approximation involves up to third derivative.

For each approximation, the corresponding Taylor polynomial is evaluated at x = 3, and the absolute difference between this predicted value and the true value (f(3)) is divided by the true value and multiplied by 100 to give εt.