1. In this problem, you will estimate fol/2 e'm2 dzr: using Taylor series, and also review some earlier parts of the course. Let All Taylor notation refers to the Taylor series for f centered at 0. In this problem all conceptual parts should be justied briey (e.g. cite a theorem, write a sentence). For computational parts, show your work. At a certain point you may be tempted to justify a bound using just a picture of a graph; you need more justication than that. (a) Why does f have a Taylor series? (Eg. why is f differentiable at 0? twice dierentiable at 0? etc.) $2 (b) Assume that the Taylor series for e- at 0 converges on R. Does the Taylor series T for f at 0 converge on R? What is the largest domain on which T is continuous? (c) Calculate f(")(m) for n S 4 and Tn(m) for n S 3. (d) Use T3 to estimate fol/2 e'f2 dzr. (e) Use Taylor's remainder formula to nd a bound for the error associated to the approximation in the previous part. Hint: if |g(:r)| 3 M1 and |h(a)| 3 M2 for all a: E S, then lg(m)h(a:)| g MlMg. (f) Explain why your calculations above show that fol/2 632 (in: