Appendix
1. Recall from course MATH10242 that we used the definition of convergence of a sequence to test a given sequence converges by assuming that
an > 0 is given and then trying to find an appropriate N . Similarly,
we will check a given function has
Theorem 1.2.7 Sandwich Rule:
Suppose that f, g and h are three functions such that
h (x) f (x) g (x)
for all x in some deleted neighbourhood of a.
If limxa h (x) = L and limxa g (x) = L then limxa f (x) = L.
Proof By the assumption in the Theorem there ex
Part 4. Integration
4.1 Definition
Definition 4.1 A partition P of [a, b] is a finite set of points cfw_x0 , x1 , ., xn
with a = x0 < x1 < . < xn = b.
1 3
Example 4.2 0, 14 , 13 , 78 , 1 and 0, 100
, 100 , 1 are partitions of [0, 1].
Definition 4.3
Appendix 4.5
The previous appendix contained material that sometimes
is covered in lectures, if there is enough time. This appendix covers material
that is either omitted from lectures due to lack of time (often left to students
to do) or is background to
Appendix
Contents
Extrema and local extrema
Example tan x = x1 ,
Examples sin x = x3 ,
Example p (x) = x3 + x +
Example 66 8,
x x2 /2 < ln (1 + x) for x > 0,
Increasing-Decreasing Theorem,
n
limn 1 + n1 = e,
What not to do with LHopitals Rule.
1
Part 3.2 Differentiation
3.2.1 Derivative Results
We now come to the three main results of this section on Differentiation.
The first, Rolles Theorem is a special case of the second, the Mean Value
Theorem. In the section following this we will give Taylo
Appendix
1. If S is a set of real numbers then is called an upper bound for S if
x for all x S.
Similarly, is a lower bound for S if
x for all x S.
We say that S is bounded if it has both an upper bound and a lower
bound.
The least upper bound U for S sa
Appendix
1. From a geometric point of view, a function is continuous on an interval
if you can draw its graph without lifting your pencil from the paper.
For, if you have to lift your pencil from the graph there has to be a
jump in the graph. At that poin
Part 3.1 Differentiation
3.1.1 Definition
y=f(x)
Q
f(x)
nt
tan ge
f(a)
P
x
a
Consider a nice, smooth function f , such as the one above, with a fixed
point P = (a, f (a) . The slope, or gradient, of the chord from P to another
point Q = (x, f (x) on the c
Appendix
Contents.
The Sum Rule for derivatives.
dxn
for n N and x 6= 0.
dx
dex
on R.
dx
How not to prove the Product or Quotient Rules.
How not to prove the Composite Rule for differentiation.
Another proof of the Composite Rule for differentiation.
Appendix
1. Recall from course MATH10242 that we used the definition of convergence of a sequence to test a given sequence converges by assuming that
an > 0 is given and then trying to find an appropriate N . Similarly,
we will check a given function has
1.1.5 Divergence or when the limit is infinite.
In the theory of sequences we say that a sequence cfw_an n1 diverges if it
fails to converge. There are a number of ways it could fail to converge, e.g.
it could oscillate, i.e. an = (1)n , n 1, or it could
Part 2.3 Monotonic functions and their properties
2.2.2 Monotonic functions
Definition 2.2.1 A function is
increasing if x1 < x2 implies f (x1 ) f (x2 ),
strictly increasing if x1 < x2 implies f (x1 ) < f (x2 ),
decreasing if x1 < x2 implies f (x1 ) f
Example 4.12 Let
1
.
x2
Find U (Qn , f ) and L (Qn , f ) for the geometric partitions Qn .
f : [2, 4] R : x 7
Solution In the notation above a = 2, b = 4, so
1/n
4
= 21/n ,
=
2
in which case n = 2. Then
Qn = 2 i : 0 i n .
So
[xi1 , xi ] = 2 i1 , 2 i
an
Appendix 4.3 Integration as the inverse of differentiation.
Though the primitive of a continuous function need not be unique, because of a possible constant, the value of definite integral is the difference
between the primitive evaluated at two points, w
Part 2.1 Continuous functions and their properties
v1 2016
2.1.1 Definition
Definition 2.1.1 A function f is continuous at a R if, and only if,
lim f (x) = f (a) ,
xa
that is
> 0, > 0, x, |x a| < = |f (x) f (a)| < .
(1)
Notes
In the definition of limxa
Part 1.2 Limits of functions
v1 2016
1.2.1 Limit Rules
An important result says that if the limit of f (x) as x a exists and is
non-zero then, for x sufficiently close to a, the values of the function f (x)
cannot be too large nor too small.
Lemma 1.2.1 I
Now we come to the first Theorem of the course.
Theorem 1.1.12 If limxa f (x) = L exists then the limit is unique.
Proof in lectures.
Note you can now see why you need to learn the definitions. In the Theorem
you are told that limxa f (x) = L exists. You
1.1.5 Divergence or when the limit is infinite.
In the theory of sequences we say that a sequence cfw_an n1 diverges if it
fails to converge. There are a number of ways it could fail to converge, e.g.
it could oscillate, i.e. an = (1)n , n 1, or it could
Appendix
1. If cfw_an n1 is a sequence that converges to as n and the an are
bounded for all n 1, i.e. |an | B for some B > 0, then | B.
Proof by contradiction. Assume | > B. Choose = (| B) /2 > 0
in the definition of limn an = to find N 1 such that if n
Appendix
1. Drawing a continuous function. From a geometric point of view,
a function is continuous on an interval if you can draw its graph
without lifting your pencil from the paper. For, if you have to lift
your pencil from the graph there has to be a
Note We cannot say
limx+ (4x2 + 2)
4x2 + 2
=
,
x+ 2x2 + 4x
limx+ (2x2 + 4x)
lim
because neither of the limits on the right hand side exist.
Theorem 1.2.7 Sandwich Rule:
Suppose that f, g and h are three functions such that
h (x) f (x) g (x)
for all x in s
Appendix 4.2
The observant student would have seen that we assumed
the following result in the proof of the Fundamental Theorem of Calculus.
Theorem 4.31 Assume that the bounded functions f and g are Riemann
integrable on [a, b]. Then
(i) Linearity f + g
Appendix 4.4 Improper Integrals
We have only defined the integral for closed intervals and functions bounded
on such intervals. We finish with a list of definitions that try to extend the
situations in which our definitions are meaningful.
Definition 4.41
Appendix
Inverse Function Theorem It may look impossible to remember the
choice of
= min (k f (g (k) ) , f (g (k) + ) k)
(2)
in the proof of this Theorem. But a little Rough Work might help,
starting by looking at what we want, namely |g (y) g (k)| < fo
Appendix
1. If cfw_an n1 is a sequence that converges to as n and the an are
bounded for all n 1, i.e. |an | B for some B > 0, then | B.
Proof by contradiction. Assume | > B. Choose = (| B) /2 > 0
in the definition of limn an = to find N 1 such that if n
Appendix 4.1
This contains material sometimes seen in lectures if there
is enough time.
Definition 4.26 If f is continuous on (a, b) and F is continuous on [a, b]
and differentiable on (a, b) with F (x) = f (x) for all x (a, b) then F is a
primitive for f
Part 2.1 Continuous functions and their properties
2.1.1 Definition
Definition 2.1.1 A function f is continuous at a R if, and only if,
lim f (x) = f (a) ,
xa
that is
> 0, > 0, x, |x a| < |f (x) f (a)| < .
(1)
Notes
In the definition of limxa f (x) we h
Part 1.2 Limits of functions
1.2.1 Limit Rules
An important result says that if the limit of f (x) as x a exists and is
non-zero then, for x sufficiently close to a, the values of the function f (x)
cannot be too large nor too small.
Lemma 1.2.1 If limxa
1.2.2 Special limits
In this course we assume that ex is defined by the series
x
e =
X
xr
r=0
r!
.
From your first year course you know this converges (absolutely) for all x R.
We will further assume that ex ey = ex+y for all x, y R. To prove this
you nee