Problem Set - Trapezoidal and Simpson approximations

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 7 December 2009

Taylor approximations

	Define the following and give an example of each: (a)   converges pointwise (b)   converges uniformly
	Write a quadratic approximation for $f\left(x\right)=x^{\mathrm{1/3}}$ near 8 and approximate 91/3. Estimate the error and find the smallest interval that you can be sure contains the value.
	Write a quadratic approximation for $f\left(x\right)=x^{-1}$ near 1 and approximate 1/1.02. Estimate the error and find the smallest interval that you can be sure contains the value.
	Write a quadratic approximation for $f\left(x\right)=e^x$ near 0 and approximate $e^{-0.5}$. Estimate the error and find the smallest interval that you can be sure contains the value.
	(a)   From Taylor's theorem write down an expansion for the remainder when the Taylor polynomial of degree $N$ for $e^x$ (about $x=0$) is subtracted from $e^x$. In what interval does the unknown constant $c$ lie, if $x>0$? (b)   Show that the remainder has the bounds, if $x>0$, $\frac{x^{n+1}}{(n+1)\text{!}}<R_N<e^x\frac{x^{n+1}}{(n+1)\text{!}}$ and use the sandwich rule to show that $R_N\to 0$ as $N\to \infty$. This proves that the Taylor series for $e^x$ does converge to $e^x$, for any $x>0$.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)