Problem $1$: Find a general formula $($if possible$)$ for the Maclaurin series of the following functions:

a$.f(x)=cos(x)$

b$.f(x)=sin(x)$

c$.f(x)=e^x$

d$.f(x)=ta{n}^{-1}(x)$

Problem $2$: Find a general formula $($if possible$)$ for the Taylor series centered at $a,$ of the following functions:

a$.f(x)=\frac{1}{x},a=1$

b$.f(x)=\sqrt[\phantom{2}]{x},a=4$

c$.f(x)=x^4+3x-1,a=2$

d$.f(x)=e^{3x},a=-1$

Problem $3$: Approximate $f(x)$ by a Taylor polynomial with degree $n$ about $x=a$ on the interval provided. Then, estimate the accuracy of the approximation.

$($a$)f(x)=sinx,a=\frac{}{6},n=4,0x\frac{}{3}$

$($b$)f(x)=xlnx,a=1,n=3,\mathrm{.}5x1\mathrm{.}5$

$($c$)f(x)=ln(1+2x),a=1,n=3,t0\mathrm{.}5x1\mathrm{.}5$

Problem $4$: a$.$ Express the antiderivative

$F(x)={\displaystyle \underset{0}{\overset{x}{}}}{e}^{-t^2}dt$

as a power series.

b$.$ Use your result from a$.$ to express the definite integral

${\displaystyle \underset{0}{\overset{1}{}}}{e}^{-x^2}dx$

as an infinite series.

c$.$ Use your infinite series from part b$.$ to approximate the value of

${\displaystyle \underset{0}{\overset{1}{}}}{e}^{-x^2}dx$

to within an error of at most ${10}^{-4}.$

d$.$ Repeat the exercise in a$.$ to approximate

${\displaystyle \underset{0}{\overset{1}{}}}ta{n}^{-1}(x^2)dx$

to within an error of at most ${10}^{-4}.$

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Math $141-$ Calculus with Analytic Geometry II

Homework $13$

Problem $5$: Use power series representations to evaluate the following limits$.$

a$.\underset{x0}{lim}\frac{cosx-1+\frac{1}{2}{x}^2}{x^4}$

b$.\underset{x0}{lim}\frac{ta{n}^{-1}x-xcosx-\frac{1}{6}{x}^3}{x^5}$

Problem $6$: Solve the following initial value problem

$\frac{dy}{dx}=2xy,y(0)=1$

a$.$ using separation of variables and integration,

b$.$ using the power series method.