Let $f$ be the function given by $f(x) = e^{-x^2}$. (a) Write the first four nonzero terms and the general term of the Taylor series for $f$ about $x = 0$. (b) Use your answer to part (a) to find $\lim_{x \to 0} \frac{1 - x^2 - f(x)}{x^4}$. (c) Write the first four nonzero terms of the Taylor series for $\int_{0}^{x} e^{-t^2}\, dt$ about $x = 0$. Use the first two terms of your answer to estimate $\int_{0}^{1/2} e^{-t^2}\, dt$. (d) Explain why the estimate found in part (c) differs from the actual value of $\int_{0}^{1/2} e^{-t^2}\, dt$ by less than $\frac{1}{200}$.
Let $f$ be the function given by $f(x) = e^{-x^2}$.
(a) Write the first four nonzero terms and the general term of the Taylor series for $f$ about $x = 0$.
(b) Use your answer to part (a) to find $\lim_{x \to 0} \frac{1 - x^2 - f(x)}{x^4}$.
(c) Write the first four nonzero terms of the Taylor series for $\int_{0}^{x} e^{-t^2}\, dt$ about $x = 0$. Use the first two terms of your answer to estimate $\int_{0}^{1/2} e^{-t^2}\, dt$.
(d) Explain why the estimate found in part (c) differs from the actual value of $\int_{0}^{1/2} e^{-t^2}\, dt$ by less than $\frac{1}{200}$.