1.
Calculate the Taylor Polynomial and the Taylor residual for the function .
Prove that as , for all .
Find the Taylor series of f.
What is the radius of convergence for the Taylor series? Justify your answer.
2.
Let f:[0,1] be a bounded function. Show that, for every n, the upper Riemann sum and the lower Riemann sum satisfy the inequality .
Define for . Use upper and lower Riemann sums to prove that h is Riemann integrable.
3.
Use mathematical induction to prove that
Let with . Calculate the upper Riemann sum and the lower Riemann sum of f on [0,1].
Calculate the upper Riemann integral and the lower Riemann integral.
Show that f is Riemann integrable in [0,1] and find the Riemann integral.