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 non-negative 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 real-valued 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 x³ - x and the function sending non-zero 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 appropriately-defined operations of addition, subtraction, multiplication and division).

Lecture 12 (October 24, 2016)

This lecture discussed the local minima and maxima of the polynomial

x³ - 9 x² + 24 x - 16.

The lecture continued with a discussion of the secant lines of the function sending non-zero 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 non-zero 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 + 3x²) 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 real-valued 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 non-zero around s and tends to a non-zero 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 non-negative function is always non-negative, and the lecture concluded with a proof of the Squeeze Theorem, which asserts that if f, g and h are real-valued 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 one-sided 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 twice-differentiable function and the “second derivative test” for local minima and maxima. It was shown that the graph of a twice-differentiable 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 Newton-Raphson 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 Riemann-Darboux 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 non-decreasing and non-increasing 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

• Draft Lecture Slides for Forthcoming Lectures

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Dr. David R. Wilkins, School of Mathematics, Trinity College Dublin.