What is the quadratic polynomial g(x)=ax^2+bx+c which best fits the function f(x)=sin(x) at x=0?

Answer 1

The sine function has a graph which looks like a wave.
To get a parabola fitting the wave function, then choose 3 points (include (0,0)) close to the origin. (The distances between the origin and the other points should not exceed pi/2.)

Those two other point will be points to your liking, just note the restriction. It might be ideal to choose the points at x = +- pi/4 as these are the x values where the sine function changes concavity.

But clearly c = 0. Since (0,0) is included.
d:

Answer 2

The slope of sinx at x = 0 is cos 0 = 1
2ax+b = 1 = slope of parabola so when x = 0, b =1.
We want the parabola to go through (0,0) just as sin x does so c = 0. The axis of symmetry of the parabola should be at - pi/2 so -b/2a= pi/2 --> a = b/Pi = 1/pi
The requiresd equation is x^2/pi +x = 0

Answer 3

To find the equation of the parabola with the best fit you need 3 points on the curve. If you have three points on the curve you can solve for the 3 unknowns by using simultaneous equations.
The problem with this question is that the sinx curve can be approximated with a parabola just after (0,0) or just before (0,0) but NOT through the point (0,0).
The curve y=x^2 will do because both it and the sine curve have (0,0) on them.

Answer 4

The well-known power series for sin(x) is

sin(x) = x - x^3/3! + x^5/5! - ...
= x - x^3/6 + x^5/120 - ...

So, if you're limited to quadratic, take

sin(x) ~ x