Calculus II delves into advanced integration techniques, series, and applications of calculus to real-world problems.
Sometimes, complex functions are hard to work with directly. Power series let you write these functions as infinite sums of powers of \( x \), just like polynomials.
The Taylor series is a special power series centered at a point \( a \), allowing you to approximate functions near that point.
\[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots \]
Calculators and computers use Taylor series to quickly estimate function values!
\[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n\]
Write the Taylor series for \( \sin(x) \) centered at \( x = 0 \).
Use a Taylor polynomial to estimate \( e^{0.1} \).
Power and Taylor series are powerful tools for approximating functions and solving complex problems.