Question

Use zero- through third-order Taylor series expansions to predict f (2.5) for f(x) = 25x²-6x² + 7x - 88 using a base point at x = 1. Compute the true percent relative error & for each approximation?

Answer 1

To predict f(2.5) using Taylor series expansions, we calculate the derivatives of f(x), find the Taylor series expansion centered at x = 1, and plug in the values to approximate f(2.5). We can then compute the true percent relative error by comparing the actual value to the approximation.

Step-by-step explanation:

To predict f(2.5) using Taylor series expansions, we start by finding the derivatives of f(x) up to the third order. The derivatives are: f'(x) = 50x - 6, f''(x) = 50, and f'''(x) = 0. Next, we can calculate the Taylor series expansion of f(x) centered at x = 1 as follows:

f(x) = f(1) + f'(1)(x - 1) + (f''(1)/2!)(x - 1)^2 + (f'''(1)/3!)(x - 1)^3

Substituting x = 2.5 into the expansion gives us:

f(2.5) ≈ f(1) + f'(1)(2.5 - 1) + (f''(1)/2!)(2.5 - 1)^2 + (f'''(1)/3!)(2.5 - 1)^3

Now, plug in the values of f(1), f'(1), f''(1), and f'''(1) to calculate the approximation for f(2.5).

To compute the true percent relative error for each approximation, you can compare the actual value of f(2.5) to the approximated value and calculate the percent difference. True percent relative error = (|Actual - Approximation| / |Actual|) * 100%.