Problem Set - Trapezoidal and Simpson approximations

Problem Set - Trapezoidal and Simpson approximations

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 $f\left(x\right)={x}^{1/3}$ 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 $x>0$, $\frac{{x}^{n+1}}{\left(n+1\right)\text{!}}<{R}_{N}<{e}^{x}\frac{{x}^{n+1}}{\left(n+1\right)\text{!}}$ 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 $x>0$.