Question

In lecture 10, timestamped in this link:

https://www.youtube.com/watch?v=eRCN3daFCmU&t=21m13s

The prof is finding the quadratic approximation of a product of two functions. To find that, he simply takes the product of the quadratic approximations of the two functions.

My question is: how can we know that the product of two approximations is the approximation of a product? Is this guaranteed by some theorem somewhere?

Answer 1

if you have f(x) and g(x) both represented as a power series (i.e. increasing powers of x)
then you can find the series representation of f(x)*g(x) by multiplying the two series.
If you are only interested in a quadratic approximation, you would drop the "higher order" terms.

We could use infinite series as an example, but let's just
say both f(x) and g(x) are represented by a series up to x^8.
You do the f(x)*g(x) multiplication, and collect the terms of the same power, and then
at the end, drop all terms higher than x^2. You will see that you only needed to start with the terms up to x^2 in f(x) and g(x), and most of your work was wasted.

In other words, if you only want a quadratic series for the product, you only need the terms up to x^2 in the multiplicands.

Answer 2

Thanks, that was quite helpful, although it doesn't address the case where f(x) and g(x) are not power series. It seems to me that there could be some crazy functions f and g that would not behave if you put them through the product rule prior to finding the quadratic approximation.