# [Calc II] quadratic Chebyshev approximation

In summary, to find the quadratic Chebyshev approximation of e^x on [-1, 1], we can plug in the given points (-sqrt(3)/2), 0, (sqrt(3)/2) into e^x = ax^2 + bx + c to get 3 equations with 3 unknowns. Solving these equations gives us a ≈ 0.53204, b ≈ 1.12977, and c = 1. As for part (b), we can use the Maclaurin series for the found quadratic to approximate the truncated Taylor series for e^x centered at 0.

## 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?

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).

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