Other Uses of Taylor Polynomials

Taylor polynomials and Taylor series aren't just for approximating 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.

Example:  There is no difficulty computing $e^{-2x^2}$ – any scientific calculator will do it.  But how do you compute $\displaystyle \int_{0.1}^{0.2} e^{-2x^2} \,dx$?  A good solution is to find a Taylor polynomial from the Taylor series for $e^{-2x^2}$ and integrate it term-by-term.

Example:  How do you solve differential equations like $\frac{dy}{dx} = y + x$ that aren't separable?   A good solution is to 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 examples are discussed in the video.