
MA1S11 (Calculus portion)
Dr. David R. Wilkins
Lectures — Michaelmas Term 2016

Lecture Material
 Lecture 1 (September 26, 2016)
 This was an introductory lecture,
delivered extempore
 Lecture 2 (September 27, 2016)
 This lecture was essentially a revision of topics
in the Irish Leaving Certificate Syllabus, including
unions, intersections and differences of sets,
sums of arithmetic sequences and the Principle of
Mathematical Induction. Some formal proofs of such
results were presented.
 Lecture 3 (September 29, 2016)
 This lecture covered results concerning laws of
indices, with integer exponents, and also
concerning intervals in the real line,
and included an example of finding subsets of
the real line constructed out of intervals using
the operations of union, intersection and
set difference. The Binomial Theorem was not covered
in the lecture as delivered, but is included in the
notes to preserve the formal ordering of the material.
in the Irish Leaving Certificate Syllabus, including
unions, intersections and differences of sets,
sums of arithmetic sequences and the Principle of
Mathematical Induction. Some formal proofs of such
results were presented.
 Lecture 4 (October 3, 2016)
 The first part of this lecture covered the statement
and proof of the Binomial Theorem, formally included
in the online lecture notes for Lecture 3 (September 29,
2016). The lecture continued with a
“chalk and talk”
solution of a problem proved using the Method of
Mathematical Induction.
 Lecture 5 (October 4, 2016)
 This lecture began with some remarks on differences between
the study of mathematics in Senior Cycle and at university.
The lecture continued by covering basic definitions of
least upper bounds and greatest lower bounds, up to and
including Subsection 1.12 in the notes covering
this lecture.
 Lecture 6 (October 6, 2016)
 This lecture began by completing the discussion of
least upper bounds and greatest lower bounds contained
in the lecture slides for Lecture 5, and then reviewed
some results in Lecture 3 concerning laws of indices
with nonnegative integer exponents. Most of the lecture
was then devoted to a discussion of the definition and
basic properties of fractional powers of positive real
numbers. Laws of indices were proved in the case
where the base is a positive real number and the exponents
are rational numbers.
 Lecture 7 (October 10, 2016)
 This lecture began by reviewing an interpolation
into the notes for Lecture 2 explaining notation
for summations. An additional slide appended to
Lecture 6 summarizing laws of indices was visited.
The remainder of the lecture concerned standard
techniques for the solution of quadratic equations.
The lecture concluded with a selection of problems
whose solution used standard results on roots of
quadratic polynomials.
 Lecture 8 (October 11, 2016)
 This lecture covered the algorithm for dividing
one polynomial by another, and the use of this
algorithm in order to factor cubic polynomials
with at least one integer root.
 Lecture 9 (October 13, 2016)
 This lecture covered the basic definitions and
results concerning functions between sets, the
domain, codomain and range of a function,
injective functions, surjective functions,
bijective functions and inverses of functions,
with examples.
 Lecture 10 (October 18, 2016)
 This lecture consisted of a discussion of
examples of realvalued functions, identifying
intervals where such functions increase or
decrease, and identifying local maxima and
minima. The functions studied included the
function sending real numbers x to
x^{3}  x and the function
sending nonzero real numbers x
to ax + c/x.

 Lecture 11 (October 20, 2016)
 This lecture discussed Hamilton's construction
of the complex number system, identifying complex
numbers with algebraic couples (i.e.,
ordered pairs of real numbers with
appropriatelydefined operations of addition,
subtraction, multiplication and division).
 Lecture 12 (October 24, 2016)
 This lecture discussed the local minima and
maxima of the polynomial
x^{3}  9 x^{2}
+ 24 x  16.
The lecture continued with a discussion of the secant
lines of the function sending nonzero real
numbers x to ax + c/x.
The concepts of limit and derivative
were introduced in the context of this example. 
 Lecture 13 (October 25, 2016)
 This lecture reviewed some of the material presented
in the previous lecture, particularly the formal definition
of limits, and the preceding example in which it was
introduced. The lecture continued by discussing limits,
as x tends to zero, of the functions whose values
for nonzero real numbers are
sin( π/(2x)),
3 x sin( π/(2x)) and
2 x^{½} sin( π/(2x)).
 Lecture 14 (October 27, 2016)
 This lecture covered limits of polynomial functions,
and derived the formula for the derivative of a polynomial
function. It was shown that, for a polynomial function,
the derivative of the polynomial must be zero at a
local maximum or local minimum.
 Lecture 15 (November 1, 2016)
 This lecture began with a discussion of absolute values
of real numbers, noting in particular that
x + y ≤ x + y
and
xy = x y
for all real numbers x and y.
The lecture continued by discussing various
ways in which the formal definition of limit
could be stated using ε and δ,
with or without the use of absolute values.
A concrete example discussed the application of
the ε—δ limit criterion to
show formally that the limit of
√ (1 + 3x^{2})
as x tends to 1 is equal to 2.
The concept of limit points was
introduced: a real number x is
a limit point of a set D of real
numbers if and only if s can be
approximated to within any prescribed
degree of precision by elements of D
not equal to s itself. Given a
realvalued function defined over a
subset D of the set of real numbers,
it only makes sense to consider whether the
limit of the function exists as x tends to
s in cases where s is a
limit point of the domain D
of the function.
 Lecture 16 (November 3, 2016)
 This lecture contained statements and proofs of many
basic properties of limits of functions of a real variable.
In particular it was proved that if f and g
are functions of a real variable x, and if
lim_{x → s} f(x)
and
lim_{x → s} g(x)
exist, then
lim_{x → s}
(f(x) + g(x))
= lim_{x → s} f(x)
+ lim_{x → s} g(x),
lim_{x → s}
(f(x)  g(x))
= lim_{x → s} f(x)
 lim_{x → s} g(x),
lim_{x → s}
(f(x) g(x))
= lim_{x → s} f(x)
× lim_{x → s} g(x),
and if moreover g(x) is nonzero around s
and tends to a nonzero limit as x approaches s then
lim_{x → s}
(f(x) / g(x))
= lim_{x → s} f(x)
/ lim_{x → s} g(x).
It was also shown that the limit of a nonnegative
function is always nonnegative, and the lecture
concluded with a proof of the Squeeze Theorem,
which asserts that if f, g and
h are realvalued functions that satisfy
f(x) ≤ g(x) ≤ h(x)
for values of the real variable x close to but
distinct from a fixed value s, and if
lim_{x → s} f(x)
= lim_{x → s} h(x)
= l,
for some real number l then
lim_{x → s} g(x)
= l.
 Lecture 17 (November 14, 2016)
 This lecture defined the concepts of continuity, and
continued by applying algebraic identities and limit theorems
in order to obtain the derivative of the function sending
positive real numbers x to x^{q},
where q is a fixed rational number. The lecture
continued by showing that any composition of continuous
functions is continuous, and that limits are preserved
under composition with continuous functions.
 Lecture 18 (November 15, 2016)
 This lecture discussed onesided limits, limits as the
variable increases or decreases without limit and situations
where functions increase without limit.
 Lecture 19 (November 17, 2016)
 This lecture discussed functions increasing and decreasing
without bound.
 Lecture 20 (November 21, 2016)
 This lecture began with proofs of the basic rules
of differential calculus, including the Product Rule,
the Quotient Rule and the Chain Rule. The lecture
continued by showing that the derivative f'(x)
of a differentiable function f(x) is zero at
local minima and maxima where x lies in the interior
of the domain of f.
 Lecture 21 (November 22, 2016)
 This lecture began with proofs of Rolle's Theorem and
the Mean Value Theorem, and continued with a discussion
of the second derivative of a twicedifferentiable function
and the “second derivative test” for local
minima and maxima. It was shown that the graph of a
twicedifferentiable function is concave upwards on
intervals in the domain where the second derivative
is positive, and concave downwards on intervals in the
domain where the second derivative is negative. Also
points of inflection were discussed, and it was shown
that the second derivative of the function must be
zero at points of inflection. The lecture concluded
with a discussion of the NewtonRaphson Method.
 Lecture 22 (November 24, 2016)
 This lecture covered the basic definitions and
properties of trigonometrical functions.
 Lecture 23 (November 28, 2016)
 This lecture covered the derivatives of
trigonometric functions. Also the inverse
trigonometric functions arctan,
arcsin and arccos were
introduced, and their derivatives were
obtained.
 Lecture 24 (November 29, 2016)
 This lecture covered the essential definitions
in the theory of the RiemannDarboux integral,
defining in particular the lower and upper
Darboux sums and the lower and upper Riemann
integrals for a bounded function on a
closed bounded interval.
 Lecture 25 (December 1, 2016)
 This lecture covered some basic properties
of Darboux lower and upper sums, showing in
particular that the passage from a partition
to a refinement of that partition does not
decrease lower sums or increase upper sums.
 Lecture 26 (December 5, 2016)
 This lecture began by proving that nondecreasing
and nonincreasing functions are always integrable,
and continued with the statement and proof
of the Fundamental Theorem of Calculus.
The basic formulae for Integration by Substitution
and Integration by Parts were obtained
as corollaries of the Fundamental Theorem
of Calculus. The lecture included a
preview of
basic properties of the natural logarithm and exponential functions
 Lecture 27 (December 6, 2016)
 This lecture discussed the rules for
Integration by Parts and Integration by Substitution,
and included several applications of the
latter rule.
 Lecture 28 (December 8, 2016)
 This lecture contained an extended investigation
into the natural logarithm and exponential functions
and their derivatives and integrals.
 Lecture 29 (December 12, 2016)
 This lecture concerned the use of calculus to
study the motion of particles under the action
of forces directed towards a fixed point.
 Lecture 30 (December 13, 2016)
 This lecture continued the study of
the motion of particles under the action
of forces directed towards a fixed point.
 Lecture 31 (December 15, 2016)
 This lecture included a discussion of the
Wave Equation
Draft Lecture Slides
Back to D.R. Wilkins: MA1S11 (Calculus)
Back to D.R. Wilkins: Lecture Notes
Dr. David R. Wilkins,
School of Mathematics,
Trinity College Dublin.