Math 521, Lecture 2 - Spring 2009
Advanced Calculus
Homework 10
Exercise 1 Let f:R→Rbe a function. Assume that f0(0) exists and that f(x+y) = f(x)f(y)for
every x, y ∈R. Prove that fis differentiable on R.
Exercise 2 Let f, g :I→Rbe two differentiable functions, satisfying f(a) = g(a) = 0 at a point a, and
g0(a)6= 0. Prove that limx→af(x)
g(x)=f0(a)
g0(a).
Exercise 3 Let f:R\ {0} → Rdefined by f(x) := sin x
x.
(1) Using Exercise 2, define f(0) so that fis continuous at 0.
(2) (different way to prove (1)) Using the Taylor expansion of sin, define f(0) so that fis continuous
at 0.
Exercise 4 Determine if wheter the following functions have a local maximum, local minimum, or neither
f1(x) = x2ex3, f2(x) = x3ex2, f3=1 + x2
1 + x3, f4=1 + x3
1 + x2.
Exercise 5 Determine the Taylor expansion of 1
1+xof up to order 1255.
1