[Calc II] quadratic Chebyshev approximation

[adillhoff]

Homework Statement

(a) The quadratic Chebyshev approximation of a function on [-1, 1] can be obtained by finding the coefficients of an arbitrary quadratic y = ax^2 + bx + c which fit the function exactly at the points (-sqrt(3)/2), 0, (sqrt(3)/2). Find the quadratic Chebyshev approximation of e^x on [-1, 1]. Approximate a, b, c to 5 decimal places.

(b) Find the quadratic truncated Taylor series centered at 0 for e^x.

Homework Equations

e^x = ax^2 + bx + c
Maclaurin for e^x

The Attempt at a Solution

For part (a), I did not know how to find the quadratic Chebyshev approximation of e^x, but I plugged in the different values of x that were given into e^x = ax^2 + bx + c in order to get 3 equations with 3 unknowns. This gives me a ≈ 0.53204, b ≈ 1.12977, and c = 1.

As for part (b), since it asks for a truncated Taylor series, I assume I need a value to stop at. I really don't feel confident doing this. Do I need to simply write the Maclaurin series for e^x?

 

[rude man]

Just a guess, but maybe what's appropriate here is the Maclaurin series for your found quadratic. You could then compare with the series for exp(x).

 

[lanedance]

maclaurin series is just a taylor series about zero, and it asks for quadratic so just find terms up to x^2