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.