Calculus 2

Look at the Taylor polynomial for $\cos(x)$ and we're cutting it off at degree 4, $T_4(x)$. We want to figure out what values of $x$ you can plug in there for it to be accurate to two decimal places.

Using a Taylor polynomial, approximate a function when x is in the range $[7, \, \text{upper bound}]$.

Find the Taylor series for the function $f(x) = e^x$ centered at $a = 2$.

Find the sum $\sum_{n=0}^{\infty} \frac{(-1)^n \pi^{2n+1}}{4^{2n+1} (2n+1)!}$ and evaluate it.

Find the sum $\sum_{n=0}^{\infty} \frac{(-1)^n \pi^{2n+1}}{3^{2n-2} (2n)!}$.

Use series to find the limit of $\frac{e^x - 1 - x}{x^2}$ as $x \to 0$.

Integrate $e^{-x^2}$ using power series.

Find the 100th derivative of $f(x) = x^3 e^{2x}$ at $x = 0$.

Find the 100th derivative of $f(x) = x^3 e^{2x}$ at $x = 0$.

Find the first three non-zero terms in the Maclaurin series for $e^x \sin x$.

Evaluate the limit of $\displaystyle \frac{e^x - \cos x}{x^2}$ as $x \to 0$.

Find the fourth degree Taylor polynomial for the function $f(x) = \ln x$ centered at $c = 1$ and use it to approximate $\ln(1.1)$.

Find the fourth degree Maclaurin polynomial for $f(x) = e^x$ and use it to approximate $e^2$.

Using the Taylor series, decompose the function $e^x$.

Compute the Taylor series for $e^x$ centered at $x = 0$.