24. Applications of Taylor Series

Homework

1. Use a power series to compute each limit.

   1. \(\displaystyle \lim_{x\to0}\dfrac{ {\large e}^{\small x^2}-1-2x-2x^2}{x^3}\)

   2. \(\displaystyle \lim_{x\to 0} \dfrac{6x-36x^3-\sin(6x)}{x^5}\)

   3. \(\displaystyle \lim_{x\to 0} \dfrac{2x-\ln(1+2x)}{x^2}\)

2. If \(f(x)=x^2\sin(x^2)\), find \(f^{(16)}(0)\) and \(f^{(17)}(0)\).

3. Consider the function \(\displaystyle f(x)=\int_0^x \dfrac{e^t-1}{t} \,dt\).

   1. Find the Maclaurin series for \(f(x)\).
      Notice the series is alternating for \(x\) negative but positive for \(x\) positive.

   2. Using the Maclaurin polynomial of degree \(3\) for \(f(x)\) approximate \(f(-0.1)\). Keep \(20\) digits.

   3. If you approximate \(f(0.1)\) by the Maclaurin polynomial of degree \(3\) for \(f(x)\), find a bound on the error in the approximation.

   4. Approximate \(f(0.1)\) by the Maclaurin polynomial of degree \(3\) for \(f(x)\). Keep \(20\) digits.

   5. If you approximate \(f(0.1)\) by the Maclaurin polynomial of degree \(3\) for \(f(x)\), find a bound on the error in the approximation.

      The Taylor Remainder theorem says \(|R_{k}f(x)| \le \dfrac{M}{(k+1)!}|x-a|^{k+1}\) where \(M \ge f^{(k+1)}(c)\) for \(c\) between \(a\) and \(x\).

      You will need to find the \(4^\text{th}\) derivative of \(\displaystyle f(x)=\int_0^x \dfrac{e^t-1}{t}\,dt\). Here is its plot.

      

4. Use a Maclaurin polynomial for \(e^x\) to estimate \(\dfrac{1}{\sqrt{e}}\) to within \(\pm 10^{-6}\). What degree polynomial did you need? Why?

5. (Honors only) Consider the initial value problem: \[ \dfrac{dy}{dx}=2x+2xy \qquad \text{with} \qquad y(0)=2 \]

   1. Find the Maclaurin polynomial of degree \(4\) for the solution to this initial value problem.

   2. This differential equation is separable. Find the exact solution to the initial value problem, \(y=f(x)\).

   3. This differential equation is also linear. Find the exact solution to the initial value problem, \(y=f(x)\), again.

   4. Find the Maclaurin series for the solution to this initial value problem as found in parts (b) and (c). Check the Maclaurin polynomial of degree \(4\) agrees with that found in part (a).