6.4 Working with taylor series  (Page 8/11)

[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 $\frac{1}{\sqrt{2\pi }}{e}^{\text{−}{x}^2\text{/}2}$ 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 $\frac{1}{\sqrt{2\pi }}{e}^{\text{−}{x}^2\text{/}2}$ to estimate this probability.

[T] Suppose that ${\displaystyle \sum _{n=0}^{\infty }{a}_n{x}^n}$ converges to a function $f\left(x\right)$ such that $f\left(0\right)=1,f^{\prime }\left(0\right)=0,$ and $f\text{″}\left(x\right)=\text{−}f\left(x\right).$ Find a formula for $a_n$ and plot the partial sum $S_N$ for $N=20$ on $[\mathrm{-5},5].$

[T] Suppose that ${\displaystyle \sum _{n=0}^{\infty }{a}_n{x}^n}$ converges to a function $f\left(x\right)$ such that $f\left(0\right)=0,f^{\prime }\left(0\right)=1,$ and $f\text{″}\left(x\right)=\text{−}f\left(x\right).$ Find a formula for $a_n$ and plot the partial sum $S_N$ for $N=10$ on $[\mathrm{-5},5].$

Suppose that ${\displaystyle \sum _{n=0}^{\infty }{a}_n{x}^n}$ converges to a function $y$ such that $y\text{″}-y^{\prime }+y=0$ where $y\left(0\right)=1$ and $y\prime \left(0\right)=0.$ Find a formula that relates $a_{n+2},a_{n+1},$ and $a_n$ and compute $a_0,\mathrm{...},a_5.$

Suppose that ${\displaystyle \sum _{n=0}^{\infty }{a}_n{x}^n}$ converges to a function $y$ such that $y\text{″}-y^{\prime }+y=0$ where $y\left(0\right)=0$ and $y^{\prime }\left(0\right)=1.$ Find a formula that relates $a_{n+2},a_{n+1},$ and $a_n$ and compute $a_1,\mathrm{...},a_5.$

[T] ${\displaystyle {\int }_0^{\pi }\frac{\text{sin}t}{t}dt};P_s=1-\frac{x^2}{3\text{!}}+\frac{x^4}{5\text{!}}-\frac{x^6}{7\text{!}}+\frac{x^8}{9\text{!}}$ (You may assume that the absolute value of the ninth derivative of $\frac{\text{sin}t}{t}$ is bounded by $0.1.)$

[T] ${\displaystyle {\int }_0^2{e}^{\text{−}{x}^2}}dx;p_{11}=1-x^2+\frac{x^4}{2}-\frac{x^6}{3\text{!}}+\text{⋯}-\frac{x^{22}}{11\text{!}}$ (You may assume that the absolute value of the $23\text{rd}$ derivative of $e^{\text{−}{x}^2}$ is less than $2\times {10}^{14}.)$

The Fresnel integrals are defined by $C\left(x\right)={\displaystyle {\int }_0^x\text{cos}\left(t^2\right)dt}$ and $S\left(x\right)={\displaystyle {\int }_0^x\text{sin}\left(t^2\right)dt}.$ Compute the power series of $C\left(x\right)$ and $S\left(x\right)$ and plot the sums $C_N\left(x\right)$ and $S_N\left(x\right)$ of the first $N=50$ nonzero terms on $[0,2\pi ].$