테일러 급수의 활용

Use the identity $2\sin x\cos x=\sin (2x)$ to find the power series expansion of ${\sin }^{2}x$ at $x=0.$ (Hint: Integrate the Maclaurin series of $\sin (2x)$ term by term.)

If $y=\sum_{n=0}^{\infty }{a}_{n}{x}^{n},$ find the power series expansions of $x{y}^{'}$ and ${x}^{2}{y}^{''}.$

[T] Suppose that $y=\sum_{k=0}^{\infty }{a}_{k}{x}^{k}$ satisfies ${y}^{'}=-2xy$ and $y(0)=0.$ Show that ${a}_{2k+1}=0$ for all $k$ and that ${a}_{2k+2}=\frac{-{a}_{2k}}{k+1}.$ Plot the partial sum ${S}_{20}$ of $y$ on the interval $[-4,4].$

[T] Suppose that a set of standardized test scores is normally distributed with mean $\mu =100$ and standard deviation $\sigma =10.$ Set up an integral that represents the probability that a test score will be between $90$ and $110$ and use the integral of the degree $10$ Maclaurin polynomial of $\frac{1}{\sqrt{2\pi }}{e}^{-{x}^{2}/2}$ to estimate this probability.

[T] Suppose that a set of standardized test scores is normally distributed with mean $\mu =100$ and standard deviation $\sigma =10.$ Set up an integral that represents the probability that a test score will be between $70$ and $130$ and use the integral of the degree $50$ Maclaurin polynomial of $\frac{1}{\sqrt{2\pi }}{e}^{-{x}^{2}/2}$ to estimate this probability.

[T] Suppose that $\sum_{n=0}^{\infty }{a}_{n}{x}^{n}$ converges to a function $f(x)$ such that $f(0)=1,{f}^{'}(0)=0,$ and ${f}^{''}(x)=-f(x).$ Find a formula for ${a}_{n}$ and plot the partial sum ${S}_{N}$ for $N=20$ on $[-5,5].$

[T] Suppose that $\sum_{n=0}^{\infty }{a}_{n}{x}^{n}$ converges to a function $f(x)$ such that $f(0)=0,\,{f}^{'}(0)=1,$ and ${f}^{''}(x)=-f(x).$ Find a formula for ${a}_{n}$ and plot the partial sum ${S}_{N}$ for $N=10$ on $[-5,5].$

Suppose that $\sum_{n=0}^{\infty }{a}_{n}{x}^{n}$ converges to a function $y$ such that ${y}^{''}-{y}^{'}+y=0$ where $y(0)=1$ and $y'(0)=0.$ Find a formula that relates ${a}_{n+2},{a}_{n+1},$ and ${a}_{n}$ and compute ${a}_{0},...,{a}_{5}.$

Suppose that $\sum_{n=0}^{\infty }{a}_{n}{x}^{n}$ converges to a function $y$ such that ${y}^{''}-{y}^{'}+y=0$ where $y(0)=0$ and ${y}^{'}(0)=1.$ Find a formula that relates ${a}_{n+2},{a}_{n+1},$ and ${a}_{n}$ and compute ${a}_{1},...,{a}_{5}.$

[T] ${\int }_{0}^{\pi }\frac{\sin t}{t}dt;{P}_{s}=1-\frac{{x}^{2}}{3!}+\frac{{x}^{4}}{5!}-\frac{{x}^{6}}{7!}+\frac{{x}^{8}}{9!}$ (You may assume that the absolute value of the ninth derivative of $\frac{\sin t}{t}$ is bounded by $0.1.)$

[T] ${\int }_{0}^{2}{e}^{-{x}^{2}}dx;{p}_{11}=1-{x}^{2}+\frac{{x}^{4}}{2}-\frac{{x}^{6}}{3!}+\cdots -\frac{{x}^{22}}{11!}$ (You may assume that the absolute value of the $23\text{rd}$ derivative of ${e}^{-{x}^{2}}$ is less than $2\,\times \,{10}^{14}.)$

The Fresnel integrals are defined by $C(x)={\int }_{0}^{x}\cos ({t}^{2})dt$ and $S(x)={\int }_{0}^{x}\sin ({t}^{2})dt.$ Compute the power series of $C(x)$ and $S(x)$ and plot the sums ${C}_{N}(x)$ and ${S}_{N}(x)$ of the first $N=50$ nonzero terms on $[0,2\pi ].$

[T] The Fresnel integrals are used in design applications for roadways and railways and other applications because of the curvature properties of the curve with coordinates $(C(t),S(t)).$ Plot the curve $({C}_{50},{S}_{50})$ for $0\le t\le 2\pi ,$ the coordinates of which were computed in the previous exercise.

The following exercises deal with Fresnel integrals.

Estimate ${\int }_{0}^{1/4}\sqrt{x-{x}^{2}}dx$ by approximating $\sqrt{1-x}$ using the binomial approximation $1-\frac{x}{2}-\frac{{x}^{2}}{8}-\frac{{x}^{3}}{16}-\frac{5{x}^{4}}{2128}-\frac{7{x}^{5}}{256}.$

[T] Use Newton’s approximation of the binomial $\sqrt{1-{x}^{2}}$ to approximate $\pi$ as follows. The circle centered at $\left(\frac{1}{2},0\right)$ with radius $\frac{1}{2}$ has upper semicircle $y=\sqrt{x}\sqrt{1-x}.$ The sector of this circle bounded by the $x$-axis between $x=0$ and $x=\frac{1}{2}$ and by the line joining $\left(\frac{1}{4},\frac{\sqrt{3}}{4}\right)$ corresponds to $\frac{1}{6}$ of the circle and has area $\frac{\pi }{24}.$ This sector is the union of a right triangle with height $\frac{\sqrt{3}}{4}$ and base $\frac{1}{4}$ and the region below the graph between $x=0$ and $x=\frac{1}{4}.$ To find the area of this region you can write $y=\sqrt{x}\sqrt{1-x}=\sqrt{x}\,\times \,\left(\text{binomial expansion of}\sqrt{1-x}\right)$ and integrate term by term. Use this approach with the binomial approximation from the previous exercise to estimate $\pi .$

Use the approximation $T\approx 2\pi \sqrt{\frac{L}{g}}\left(1+\frac{{k}^{2}}{4}\right)$ to approximate the period of a pendulum having length $10$ meters and maximum angle ${\theta }_{\max }=\frac{\pi }{6}$ where $k=\sin \left(\frac{{\theta }_{\max }}{2}\right).$ Compare this with the small angle estimate $T\approx 2\pi \sqrt{\frac{L}{g}}.$

Suppose that a pendulum is to have a period of $2$ seconds and a maximum angle of ${\theta }_{\max }=\frac{\pi }{6}.$ Use $T\approx 2\pi \sqrt{\frac{L}{g}}\left(1+\frac{{k}^{2}}{4}\right)$ to approximate the desired length of the pendulum. What length is predicted by the small angle estimate $T\approx 2\pi \sqrt{\frac{L}{g}}?$

Evaluate ${\int }_{0}^{\pi /2}{\sin }^{4}\theta d\theta$ in the approximation $T=4\sqrt{\frac{L}{g}}{\int }_{0}^{\pi /2}\left(1+\frac{1}{2}{k}^{2}{\sin }^{2}\theta +\frac{3}{8}{k}^{4}{\sin }^{4}\theta +\cdots \right)d\theta$ to obtain an improved estimate for $T.$

[T] An equivalent formula for the period of a pendulum with amplitude ${\theta }_{\max }$ is $T({\theta }_{\max })=2\sqrt{2}\sqrt{\frac{L}{g}}{\int }_{0}^{{\theta }_{\max }}\frac{d\theta }{\sqrt{\cos \theta }-\cos ({\theta }_{\max })}$ where $L$ is the pendulum length and $g$ is the gravitational acceleration constant. When ${\theta }_{\max }=\frac{\pi }{3}$ we get $\frac{1}{\sqrt{\cos t-1/2}}\approx \sqrt{2}\left(1+\frac{{t}^{2}}{2}+\frac{{t}^{4}}{3}+\frac{181{t}^{6}}{720}\right).$ Integrate this approximation to estimate $T\left(\frac{\pi }{3}\right)$ in terms of $L$ and $g.$ Assuming $g=9.806$ meters per second squared, find an approximate length $L$ such that $T\left(\frac{\pi }{3}\right)=2$ seconds.