In class November 14, 2007 - Turn in November 28, 2007
A typed paper is preferred, but a neat hand written paper is OK.
Group Work Rules:
•
You are encouraged to work together!
•
Away from the group, do your own neat write up of the
problems.
•
Acknowledge the group members and any other
person/resource you use.
1.
Let f be continuous on [0,∞) and
differentiable on (0,∞); suppose that
lim
h → 0+
f(0 + h) − f(0)
h
exists.
Show that there is a function g, continuous and differentiable
on (−∞,∞), such that, for all x ≥ 0, g(x) = f(x).
2.
Spivak Chapter 9, Problem 14.
Let f(x) = x2 if x is rational, and f(x) = 0 if x is
irrational. Prove that f is differentiable at 0. (Don't be
confused by this function. Just write out the definition of
f′(0).)
3.
Spivak Chapter 9, Problem 22 (Part (b) modified).
(a)
Suppose that f is differentiable at x. Prove that
f′(x)
=
lim
h → 0
f(x +h) −f(x−h)
2 h
(B)
Give an example of a function of a function f
which is not differentiable at 0, but
lim
h → 0
f(0 +h) −f(0 − h)
2 h
exists.
4.
Spivak Chapter 9, Problem 28.
(a)
Find f′(x) if f(x) = |x|3. Find
f′′(x). Does f′′′(x) exist for
all x?
(b)
(Statement slightly modified) Let f(x) = x4 for x ≥ 0 and f(x) = − x4 for x ≤ 0. Find f′(x),
f′′(x), and f(3)(x). Does f(4)(x) exist
for all x?
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