Review problems for Math 261 nal exam
(1) Dene the derivative f (a). Calculate (from the denition) the derivative of
f (x) = 1/x.
f (a + h) f (a)
.
h0
h
f (a) = lim
If f (x) = 1/x, then
f (a) = lim
= lim
h0
h0
h
a(a+h)
h
1
a+h
h
1
a
= lim
h0
a(a+h)
a(a+h)
Review problems for Math 261 nal exam
(1) Dene the derivative f (a). Calculate (from the denition) the derivative of
f (x) = 1/x.
(2) Let
f (x) =
x2 sin(1/x) x = 0
0
x = 0.
Find f (0).
(3) Calculate the derivatives of the functions below. You may use that
REVIEW PROBLEMS FOR MATH 261 FIRST EXAM
Your exam will have around 4-6 questions, which will look much like some of the questions below.
From the axioms for the real numbers given on page nine of the text, prove the following four statements.
1. (a + b) c
REVIEW PROBLEM SOLUTIONS FOR MATH 261, SECOND
MIDTERM.
(1) Say precisely what it means for the limit of f (x) as x approaches a to
be l (that is, limxa f (x) = l).
Solution: For every > 0, there is a number > 0 such that if x
satises 0 < |x a| < then |f (
REVIEW PROBLEMS FOR MATH 261, SECOND MIDTERM.
(1) Say precisely what it means for the limit of f (x) as x approaches a to
be l (that is, limxa f (x) = l).
(2) Dene what it means to say the function f (x) is continuous at a.
(3) Prove, using the denition o