Chapter 10

Find a power series that has (2,6) as an interval of convergence.

Consider the following common approximations when \(x\) is near zero. a. Estimate \(f(0.1)\) and give the maximum error in the approximation. b. Estimate \(f(0.2)\) and give the maximum error in the approximation. $$f(x)=\tan ^{-1} x \approx x$$

An essential function in statistics and the study of the normal distribution is the error function $$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}} d t$$ a. Compute the derivative of erf \((x)\) b. Expand \(e^{-t^{2}}\) in a Maclaurin series, then integrate to find the first four nonzero terms of the Maclaurin series for erf. c. Use the polynomial in part (b) to approximate erf (0.15) and erf ( -0.09 ). d. Estimate the error in the approximations of part (c).

Find the next two terms of the following Taylor series. $$\sqrt{1+x}: 1+\frac{1}{2} x-\frac{1}{2 \cdot 4} x^{2}+\frac{1 \cdot 3}{2 \cdot 4 \cdot 6} x^{3}-\cdots$$

Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is $$J_{0}(x)=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{2^{2 k}(k !)^{2}} x^{2 k}.$$ a. Write out the first four terms of \(J_{0}\) b. Find the radius and interval of convergence of the power series for \(J_{0}\) c. Differentiate \(J_{0}\) twice and show (by keeping terms through \(x^{6}\) ) that \(J_{0}\) satisfies the equation \(x^{2} y^{\prime \prime}(x)+x y^{\prime}(x)+x^{2} y(x)=0\)

Find the next two terms of the following Taylor series. $$\frac{1}{\sqrt{1+x}} 1-\frac{1}{2} x+\frac{1 \cdot 3}{2 \cdot 4} x^{2}-\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} x^{3}+\cdots$$

Use the identity \(\sec x=\frac{1}{\cos x}\) and long division to find the first three terms of the Maclaurin series for \(\sec x\)

Use composition of series to find the first three terms of the Maclaurin series for the following functions. a. \(e^{\sin x}\) b. \(e^{\tan x}\) c. \(\sqrt{1+\sin ^{2} x}\)

Errors in approximations Suppose you approximate \(\sin x\) at the points \(x=-0.2,-0.1,0.0,0.1,\) and 0.2 using the Taylor polynomials \(p_{3}=x-x^{3} / 6\) and \(p_{5}=x-x^{3} / 6+x^{5} / 120 .\) Assume that the exact value of \(\sin x\) is given by a calculator. a. Complete the table showing the absolute errors in the approximations at each point. Show two significant digits. $$\begin{array}{|c|l|l|} \hline x & \text { Error }=\left|\sin x-p_{3}(x)\right| & \text { Error }=\left|\sin x-p_{5}(x)\right| \\ \hline-0.2 & & \\ \hline-0.1 & & \\ \hline 0.0 & & \\ \hline 0.1 & & \\ \hline 0.2 & & \\ \hline \end{array}$$ b. In each error column, how do the errors vary with \(x\) ? For what values of \(x\) are the errors the largest and smallest in magnitude?

Suppose \(f\) and \(g\) have Taylor series about the point \(a.\) a. If \(f(a)=g(a)=0\) and \(g^{\prime}(a) \neq 0,\) evaluate \(\lim _{x \rightarrow a} f(x) / g(x).\) by expanding \(f\) and \(g\) in their Taylor series. Show that the result is consistent with l'Hôpital's Rule. b. If \(f(a)=g(a)=f^{\prime}(a)=g^{\prime}(a)=0\) and \(g^{\prime \prime}(a) \neq 0\) evaluate \(\lim _{x \rightarrow a} \frac{f(x)}{g(x)}\) by expanding \(f\) and \(g\) in their Taylor series. Show that the result is consistent with two applications of I'Hôpital's Rule.

Choose a Taylor series and a center point a to approximate the following quantities with an error of \(10^{-4}\) or less. $$\sqrt[3]{83}$$

Explain why the Mean Value Theorem is a special case of Taylor's Theorem.

Assume that \(f\) has at least two continuous derivatives on an interval containing \(a\) with \(f^{\prime}(a)=0 .\) Use Taylor's Theorem to prove the following version of the Second Derivative Test: a. If \(f^{\prime \prime}(x) > 0\) on some interval containing \(a,\) then \(f\) has a local minimum at \(a\) b. If \(f^{\prime \prime}(x) < 0\) on some interval containing \(a,\) then \(f\) has a local maximum at \(a\)

Best expansion point Suppose you wish to approximate \(e^{0.35}\) using Taylor polynomials. Is the approximation more accurate if you use Taylor polynomials centered at 0 or \(\ln 2 ?\) Use a calculator for numerical experiments and check for consistency with Theorem 2. Does the answer depend on the order of the polynomial?

Tangent line is \(p_{1}\) Let \(f\) be differentiable at \(x=a\) a. Find the equation of the line tangent to the curve \(y=f(x)\) at \((a, f(a))\) b. Find the Taylor polynomial \(p_{1}\) centered at \(a\) and confirm that it describes the tangent line found in part (a).

Local extreme points and inflection points Suppose that \(f\) has two continuous derivatives at \(a\) a. Show that if \(f\) has a local maximum at \(a,\) then the Taylor polynomial \(p_{2}\) centered at \(a\) also has a local maximum at \(a\) b. Show that if \(f\) has a local minimum at \(a,\) then the Taylor polynomial \(p_{2}\) centered at \(a\) also has a local minimum at \(a\) c. Is it true that if \(f\) has an inflection point at \(a,\) then the Taylor polynomial \(p_{2}\) centered at \(a\) also has an inflection point at \(a ?\) d. Are the converses to parts (a) and (b) true? If \(p_{2}\) has a local extreme point at \(a,\) does \(f\) have the same type of point at \(a ?\)