In lecture 10, timestamped in this link:
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?
OCW Scholar - Single Variable Calculus
Stacey Warren - Expert brainly.com
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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.
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.