Second Taylor Polynomials:

For the following functions, find the second Taylor polynomial $T_2(x)$ for the given $f(x)$ and $b$, bound the error $|f(x) - T_2(x)|$ on the given interval $I$, and find an interval on which $|f(x) - T_1(x)|\lt 1/10$ near $b$.

• $\frac{1}{1-x}$ at $b = 0$
• $\tan x$ at $b = 0$
• $\cos (x - \pi/6)$ at $b = \pi/6$
• $\frac{1}{3-2x}$ at $b = 2$

Integral Estimate:

Use the second Taylor polynomial $T_2(x)$ of $f(x) = \cos (x^2)$ at $b = 0$ to approximate $\int_0^{\pi/6} \cos(x^2) dx$.