Taylor polynomials and Taylor series aren't just for getting numerical values of known functions. They also help us represent integrals (that are otherwise too hard for us) and solutions to differential equations as power series.

Examples:

There is no difficulty computing $e^{-2x^2}$ – any scientific calculator will do it. But how do you compute $\int_{0.1}^{0.2} e^{-2x^2} \,dx$? Answer: write a power series for $e^{-2x^2}$ and integrate it term-by-term.

How do you solve differential equations like $\frac{dy}{dx} = y + x$ that aren't separable? Answer: write $y$ as a power series and compare the two sides of the differential equation. This allows us to recursively figure out all of the coefficients.

Both of these techniques are explored in the video below.