Jonathan Shock – Page 3

About Jonathan Shock

I'm a lecturer at the University of Cape Town in the department of Mathematics and Applied Mathematics. I teach mathematics both at undergraduate and at honours levels and my research interests lie in the intersection of applied mathematics and many other areas of science, from biology and neuroscience to fundamental particle physics and psychology.

The Fundamental Theory of Calculus part 2 (part i)

OK, now we come onto the part of the FTC that you are going to use most. We are finally going to show the direct link between the definite integral and the antiderivative. I know that you’ve been holding your breaths until this moment. Get ready to breath a sign of relief:

The Fundamental Theorem of Calculus, Part 2 (also known as the Evaluation Theorem)

If $f$ is continuous on $[a,b]$ then

$\int_a^b f(x) dx=F(b)-F(a)$

where $F$ is any antiderivative of $f$ . Ie any function such that $F'=f$ .

————-

This means that, very excitingly, now to calculate the area under the curve of a continuous function we no longer have to do any ghastly Riemann sums. We just have to find an antiderivative!

OK, let’s prove this one straight away.

We’ll define:

$g(x)=\int_a^x f(t)dt$

and we know from the FTC part 1 how to take derivatives of this. It’s just $g'(x)=f(x)$ . This says that $g$ is an antiderivative of $f$ .…

By Jonathan Shock| July 11th, 2019|Courses, First year, MAM1000, Uncategorized, Undergraduate|0 Comments

The Fundamental Theorem of Calculus part 1 (part iii)

So, we are now ready to prove the FTC part 1. We’re going to follow the proof in Stewart and add in some discussion as we go along to motivate what we are doing. What we are going to prove is that:

$\frac{d}{dx} \int_a^x f(t) dt=f(x)$

for $x\in [a,b]$ when $f$ is continuous on $[a,b]$ .

Proof:

we define $g(x)=\int_a^x f(t)dt$ and we want to find the derivative of $g$ . We will do this by using the fundamental definition of the derivative, so let’s look at calculating this function at $x$ and $x+h$ – ie. how much does it change when we change $x$ by a little bit?

$g(x+h)-g(x)=\int_a^{x+h}f(t) dt-\int_a^x f(t) dt$

But remember that the definite integral is just the area, so this difference is the area between a and x+h minus the area between a and x. Which is just the area between x and x+h. Using the properties of integrals, we can write this formally as:

$g(x+h)-g(x)=\int_a^{x+h}f(t) dt-\int_a^x f(t) dt=\left(\int_a^{x}f(t)+\int_x^{x+h}f(t)\right)-\int_a^{x}f(t)=\int_x^{x+h}f(t)dt$

and we can write, for $h\ne 0$ :

$\frac{g(x+h)-g(x)}{h}=\frac{1}{h}\int_x^{x+h}f(t)dt$

Restated, we can think of this as the area between x and x+h divided by h.…

By Jonathan Shock| July 11th, 2019|Courses, First year, MAM1000, Uncategorized, Undergraduate|0 Comments

The Fundamental Theorem of Calculus part 1 (part ii)

OK, so up to now we can’t actually use the FTC (Fundamental Theorem of Calculus) to calculate any areas. That will come from the FTC part 2.

For now, let’s take some examples and see what the FTC is saying. I’ll restate it here:

The Fundamental Theorem of Calculus, part 1

If $f$ is continuous on $[a,b]$ then the function $g$ defined by:

$g(x)=\int_a^x f(t) dt$ , for $a\le x\le b$

is continuous on $[a,b]$ and differentiable on $(a,b)$ , and $g'(x)=f(x)$ .

——

Let’s look at some examples. We’re going to take an example that we can calculate using a Riemann sum. Let’s choose $f(x)=x^2$ .

If we integrate this from $0$ to some point $x$ – ie. calculate the area under the curve, we get:

$\int_0^x t^2 dt=\frac{x^3}{3}$ .

Make sure that you can indeed get this by calculating the Riemann sum.

So, what does the FTC part 1 tell us? It says that if we take the derivative of this area, with respect to the upper limit, $x$ , then we get back $f(x)$ .…

By Jonathan Shock| July 11th, 2019|Courses, First year, MAM1000, Uncategorized, Undergraduate|0 Comments

Prime Suspects – The anatomy of integers and permutations, by Andrew Granville and Jennifer Granville, illustrated by Robert Lewis – a review

NB I was sent this book as a review copy.

From Princeton University Press.

What a spectacular book! I am rather blown away by it. This is a graphic novel written about two bodies discovered by cops in an American city some time around the present day, and the forensic investigation which goes into solving the case, and somehow the authors have managed to make the whole book about number theory and combinatorics.

I have to admit that when I started reading the book I was worried that it was going to have the all-too-common flaw of starting off very simple and then suddenly getting way too complicated for the average reader, but they have managed to somehow avoid that remarkably well.

It is however a book that should be read with pen and paper, or preferably computer by one’s side. As I read through and mathematical claims were made, about prime factors of the integers and about cycle groups of permutations, I coded up each one to see if I was following along, and I would recommend this to be a good way to really follow the book.…

By Jonathan Shock| July 9th, 2019|Book reviews, Reviews|1 Comment

The Fundamental Theorem of Calculus, part 1 (part i)

We’ve seen some intriguing things in this course so far, and we’ve developed some clever tricks, from how to find the gradient of just about any function we can throw at you, to proving statements to be true for an infinite number of cases.

To some extent, this is what we have looked at so far (at least in terms of calculus, and building up to calculus):

However, we’re about to see some magic. We’re about to see the most important thing yet on this course, and indeed one of the most important moments in all of mathematical history.

We are going to see…actually, we are going to prove, that there is a relationship between rates of change and the area under a graph. This doesn’t sound that amazing, but its consequences have essentially allowed for the development of much of modern mathematics over the last 350 years.

The link that we are going to prove will allow us to find the area under graphs of functions for which taking the Riemann sum would be really hard.…

By Jonathan Shock| July 9th, 2019|Courses, First year, MAM1000, Uncategorized, Undergraduate|1 Comment

Deborah Kent (Drake University) Omar Khayyam’s Geometrical Solution of the Cubic: An Example of Using History in the Teaching of Mathematics

Second talk at the Diversifying the curriculum conference in Oxford.

The following was taken down live, and as such there may be mistakes and misquotes. It is mostly a way for me to keep notes and to share useful resources and thoughts with others. As such, nothing should be used to quote the speaker from this article

From https://www.drake.edu/math/faculty/deborahkent/

How to generate interesting conversations with students surrounding mathematical diversity.

Historical figures (From wikipedia):

Omar Khayyam 18 May 1048 – 4 December 1131) was a Persian mathematician, astronomer, and poet. He was born in Nishapur, in northeastern Iran, and spent most of his life near the court of the Karakhanid and Seljuq rulers in the period which witnessed the First Crusade.

As a mathematician, he is most notable for his work on the classification and solution of cubic equations, where he provided geometric solutions by the intersection of conics.

…

By Jonathan Shock| June 19th, 2019|Conference, Diversifying the curriculum, Uncategorized|0 Comments

June Barrow-Green (The Open University) Towards a diversity resource for undergraduate mathematics

First talk at the Diversifying the curriculum conference in Oxford.

from https://en.wikipedia.org/wiki/June_Barrow-Green

When reading through the Open University’s textbook “Pure Mathematics M208”, in the historical margin notes, the only female mathematician was Emmy Noether. The vast majority of characters were white European men.

The question is how do we create a resource which can tackle issues of diversity in mathematics? This is a recent project begun by June Barrow Green.

It is important to avoid tokenism when thinking about diversity.

Athena Swan – ECU Gender charter – a very useful resource for statistics about women in STEM

What do we mean by diversity: Ethnicity, gender, culture/ Images of mathematicians, who are the students?…

By Jonathan Shock| June 18th, 2019|Conference, Diversifying the curriculum|0 Comments

Relativity, The Special and General Theory, 100th anniversary edition – by Albert Einstein

NB. I was sent this book as a review copy.

From Princeton University Press.

In 1917, two years after publishing his work on The General Theory of Relativity, Einstein published a popular science account of both The Special, and General Theories of relativity. It is with some embarrassment that I have to admit that I’d never read this before, despite taking a number of undergraduate and postgraduate courses in relativity. Einstein understood the importance that his results had on our understanding of the universe, but also that the profundity of them could not truly be grasped by the general public, despite the headlines which covered many newspapers around the world on his results, without a popular exposition. 1917 was the publication of the first edition of this explication, but he continued to update them up until 1954. This allowed him to extend the theoretical discussion with the experimental verifications and discoveries which occurred over the next decades, including that of the expanding cosmology, spearheaded by Hubble’s observations.…

By Jonathan Shock| April 19th, 2019|Book reviews, Reviews|1 Comment

Data Visualization, a practical introduction – by Kieran Healy, a review

NB. I was sent this book as a review copy.

From Princeton University Press.

I’m not an expert on the R programming language, but I have dabbled, which meant that while this book is perhaps aimed at slightly more advanced users (I’ve used it a half a dozen times for Coursera courses), I had enough to appreciate the value of this really lovely resource.

The book can be seen, I think, in two ways. One of the ways, which is the one which most interests me, is in explaining what it is that makes good data visualization captivating, clear and unambiguous. Interleaved in these ideas of aesthetics are the precisel methods to go about making such visualizations using the ggplot package in R.

The other way to look at the book is as a way to really get to grips with the advanced features of the ggplot package, which is taught via interesting examples of data visualization.…

By Jonathan Shock| March 27th, 2019|Book reviews, Reviews|1 Comment

All you’ve ever wanted to know about absolute values (and weren’t afraid to ask)

I’ve been getting a lot of questions about absolute values, and so I thought I would try and clarify things here as much as possible. I’ll give some basic definitions and intuition, and then go through some examples, from easier to harder.

The absolute value function is just….a function. You give it a number, and it returns a number. In the same way that $f(x)=x^2$ is a function. You give it a number and it returns that number multiplied by itself. So the absolute value function, which we write as $f(x)=|x|$ takes a number and returns the same number if the number was positive, and the negative of the number if it was negative, thus returning always a positive number.

We can think of this as the function “how far away from the point 0 (the origin) on the real number line is x?”. It doesn’t care about what direction it is, only how far away it is.…

By Jonathan Shock| February 22nd, 2019|Courses, First year, MAM1000, Uncategorized, Undergraduate|3 Comments

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