Question $1$: Taylor Series ExpansionProblem: Expand the function f$($x$)=$ e$\text{'}$ using the Taylor seriesabout x $=0$ up to the fourth$-$order term. Write the general formof the expansion as well.Question $2$: Approximation Using Taylor SeriesProblem: Use the Taylor series expansion of sin$($x$)$ around $\backslash$times $=0$ approximate sin $($x$)$ for x $=0\mathrm{.}1$ using the first three terms.Compare the result with the exact value of sin $(0\mathrm{.}1)($use a calculator$).$Question $3$: Truncation ErrorProblem: For the function f$($x$)=$ In $(1+\backslash$times $),$ find the Taylor seriesexpansion about x $=0$ up to the third$-$order term. Calculate thetruncation error if the series is truncated after the second$-$orderterm when x $=0\mathrm{.}5.$Question $4$: Higher$-$Order ApproximationProblem: Determine the fourth$-$order Taylor polynomial for thefunction f$($x$)=$ cos$($x$)$ centered at x $=$ pi$/4.$ Use this polynomial toestimate cos$($pi$/3).$Question $5$: Error EstimationProblem: Given the function f$($x$)=$ sqrt $(1+\backslash$times $),$ find the Taylorseries expansion up to the second$-$order term around x $=0.$Estimate the error when using this expansion to approximate sqrt $(1\mathrm{.}1).$

Question $1$: Taylor Series Expansion

Problem: Expand the function $f(x)=e^x$ using the Taylor series about $x=0$ up to the fourth$-$order term. Write the general form of the expansion as well.

Question $2$: Approximation Using Taylor Series

Problem: Use the Taylor series expansion of $sin(x)$ around $x=0$ approximate $sin(x)$ for $x=0\mathrm{.}1$ using the first three terms. Compare the result with the exact value of $sin(0\mathrm{.}1)($use a calculator$).$

Question $3$: Truncation Error

Problem: For the function $f(x)=ln(1+x),$ find the Taylor series expansion about $x=0$ up to the third$-$order term. Calculate the truncation error if the series is truncated after the second$-$order term when $x=0\mathrm{.}5.$

Question $4$: Higher$-$Order Approximation

Problem: Determine the fourth$-$order Taylor polynomial for the function $f(x)=cos(x)$ centered at $x=\frac{}{4}.$ Use this polynomial to estimate $cos(\frac{}{3}).$

Question $5$: Error Estimation

Problem: Given the function $f(x)=\sqrt[\phantom{2}]{1+x},$ find the Taylor series expansion up to the second$-$order term around $x=0.$ Estimate the error when using this expansion to approximate sqrt $(1\mathrm{.}1).$