AP Calculus BC / Unit 10: Infinite Sequences and Series
10.11 Finding Taylor Polynomial Approximations of Functions
Notes
$$P_n(x)=\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k$$
- The coefficient of $(x-a)^k$ is $\displaystyle \frac{f^{(k)}(a)}{k!}$.
- Centered at 0, it's a Maclaurin polynomial.
Practice
The second-degree Maclaurin polynomial for $e^x$:
All derivatives equal 1 at 0.
$f(2)=1$, $f'(2)=3$, $f''(2)=4$. The coefficient of $(x-2)^2$ in $P_2$:
$\displaystyle \frac{4}{2!}$.
The third-degree Maclaurin polynomial for $\sin x$:
$\displaystyle \sin x=x-\frac{x^3}{3!}+\dots$
Ready for more? Take the Unit 10 test.