Fundamental theorem of calculus example

We did an example today in class which I wanted to go through again here. The question was to calculate

 

\frac{d}{dx}\int_a^{x^4}\sec t dt

 

We spot the pattern immediately that it’s an FTC part 1 type question, but it’s not quite there yet. In the FTC part 1, the upper limit of the integral is just x, and not x^4. A question that we would be able to answer is:

 

\frac{d}{dx}\int_a^{x}\sec t dt

 

This would just be \sec x. Or, of course, we can show that in exactly the same way:

 

\frac{d}{du}\int_a^{u}\sec t dt=\sec u

 

That’s just changing the names of the variables, which is fine, right? But that’s not quite the question. So, how can we convert from x^4 to u? Well, how about a substitution? How about letting x^4=u and seeing what happens. This is actually just a chain rule. It’s like if I asked you to calculate:

 

\frac{d}{dx} g(x^4).

 

You would just say: Let x^4=u and then we have:

 

\frac{d}{dx} g(x^4)=\frac{du}{dx}\frac{d}{du}g(u)=4x^3 g'(u).…

Is MAM1000W Making You Anxious?

Hello, my name is Jeremy :-)

I am new to the MAM1000W team of tutors – if you want to read more about my background you can take a look at my bio in the MAM1000W document on Vula. In short, I returned to UCT last year to do my second undergraduate degree, a BSc in Applied Maths and Computer Science, at 25 years old, after not doing any maths for seven years. In the beginning, I found MAM1000W really hard; the pace of the content and the tutorials made me anxious and when test one came around I scored 50%. More anxiety. Luckily I have a great support system (inside and outside the Math department) and with some good advice and determination, I was able to figure out a new, way of studying and managing my time that worked better for me. When it came time for test 2, despite being super stressed out, I scored 81%.…

By | July 22nd, 2018|Uncategorized|1 Comment

Calculus for the ambitious, by Tom Korner, a review, by Henri Laurie

Amazon link

This is a lovely book: strong emphasis on ideas; a lively sense of humour; a sure logical touch; historical detail that is accurate, relevant, yet quirky (takes some doing!). What’s not to like?

Well, there’s this: it is not easy to decide whether to recommend the book to anybody who doesn’t already know calculus. I’ll return to that. Let me start by describing why this is such a good book.

Firstly, the light touch and the clarity, which together make it wonderfully accessible. Fans of Tom Korner, including yours truly, will be happy to hear that it it as good as his “The Pleasures of Counting” and “Fourier Analysis”, two of the best books on maths ever. Like them, it discusses applications, social context and history but always in a way that supports the maths, which remains the main focus.

Secondly, the balance between rigour and intuition is superbly judged and maintained.…

By | June 20th, 2018|Book reviews, Reviews|0 Comments

The Recamán sequence

In case you have watched the following video about the Recamán sequence.

and want to play around with it in Mathematica. Here is my code for doing so:

nums = {0};

For[i = 1, i < 66, i++,
If[nums[[-1]] – i > 0 && Position[nums, nums[[-1]] – i] === {}, nums = Append[nums, nums[[-1]] – i],
nums = Append[nums, nums[[-1]] + i]]
]

{{#[[1]], 0}, #[[2]]} & /@ Partition[Riffle[Mean[#] & /@ Partition[Riffle[nums, nums[[2 ;;]]], 2],
Abs[Differences[nums]]/2], 2];

Show[Show[
Table[Graphics[Circle[%[[i, 1]], %[[i, 2]], {(i) \[Pi], (i + 1) \[Pi]}]], {i, Length[%]}], ImageSize -> 1000], Plot[0, {x, 0, 91}],
Axes -> True]

(You may have to copy this by hand rather than copy/paste.)

This produces the following rather beautiful graphic (and answers the question posed in the video):

RecamanEvidence away my dear Watson…evidence away.…

By | June 14th, 2018|Uncategorized|0 Comments

Music by the Numbers, From Pythagoras to Schoenberg – By Eli Maor, a review

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

Music by the numbers leads us on a journey, as stated in the title, from Pythagoras to Schoenberg. In many ways the endpoint is stated early on, giving us clues that a revolution in mathematical thinking about musical scales will be encountered in the early twentieth century. Indeed the journey through musical practice, mathematics, physics and the biology of hearing is woven rather beautifully together, giving the account of our step by step explorations of tonal systems and their links to the physics of vibration. The development of calculus and the triumph of Fourier take as from the somewhat numerological and empiric realms of musical experimentation to the age of a true understanding of timbre – the way different instruments express harmonics and their overtones in different admixtures. A lot of emphasis is placed on the development of scales based on subtly different frequency ratios, which were developed over the years (particularly within European music, non-European music being given only very brief comment) to balance the physical, mathematical and aesthetic qualities of the various possible tunings of instruments.…

By | June 2nd, 2018|Book reviews, Reviews|1 Comment

Kovalevskaia research grants for female mathematicians in Southern African region

KovaleskaiaAward2018

By | May 28th, 2018|Advertising|0 Comments

What can be computed? A practical guide to the theory of computation – by John MacCormick, a review

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

It’s not often that a textbook comes along that is compelling enough that you want to read it from cover to cover. It’s also not often that the seed of inspiration of a textbook is quoted as being Douglas Hofstadter’s Pulitzer prize-winning book Godel, Escher Bach. However, in the case of “What can be computed”, both of these things are true.

I am not a computer scientist, but I have spent some time thinking about computability, Turing machines, automata, regular expressions and the like, but to read this book you don’t even need to have dipped your toes into such waters. This is a textbook of truly outstanding clarity, which feels much more like a popular science book in terms of the journey that it takes you on. If it weren’t for the fact that it is a rigorous guide to the theory of computability and computational complexity, complete with a lot of well thought through exercises, formal definitions and huge numbers of examples, you might be fooled by the easy-reading nature of it into thinking that this book couldn’t take you that far.…

By | May 27th, 2018|Book reviews, Reviews|1 Comment

On Gravity, a brief tour of a weighty subject – By Tony Zee, a review

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

In the era when our eyes are being opened to the Universe in the gravitational spectrum via the recent gravitational wave observations, this book is exactly what is needed to communicate to the general public the beauty and depth of Einstein’s theory of gravity, as well as the interplay between gravity and quantum mechanics which takes place at the event horizon of a black hole.

Starting with the observations of merging black holes black in 2016, Tony Zee takes the reader on a clear and swift journey through the ideas of the incredible weakness of gravity, the basics of field theory, relativity, curved space-times, quantum weirdness, black holes and Hawking radiation, back, full circle to the consequences of General Relativity including the existence of gravitational waves and the detectors now observing them and those which will give us a far clearer picture of the gravitational universe in the near future.…

By | May 27th, 2018|Book reviews, Reviews|1 Comment

PDE: Physics, Math and Common Sense. Part I: Conservation Law

wing-flow
Source: CFDIinside blog

INTRODUCTION

The course of Partial differential equations (PDEs) usually is a tough one. There is a number of factors contributing to this toughness:

  • PDE course combines the knowledge from calculus, algebra, ordinary differential equations (ODEs), complex analysis and functional analysis. Simply put, there is a lot that you need to know about!
  • PDE methods often (or should I say, mostly?) come from physics, but this aspect is not always emphasized and, as a result, the intuition is lost.
  • There is lots of abstraction in the PDE course material: characteristics, generalized functions (distributions), eigenfunctions, convolutions and etc. Many of these concepts actually have simple interpretations, but again, this is not emphasized.
  • PDEs themselves are tough. In contrast to ODEs, there are no general methods for all kinds of PDEs. The field is young and a bit messy.

This series of posts aims to demystify PDEs and show some general way of handling PDE problems by combining physical intuition and mathematical methods.…

By | May 21st, 2018|Level: intermediate, Uncategorized|0 Comments

Advice for MAM1000W students from former MAM1000W students – part 5

While I resisted Mam1000W every single day, I even complained about how it isn’t useful to myself. Little did I know when it all finally clicked towards the end that even though I wasn’t going to be using math in my life directly, the methodology of thinking and applying helps me to this day.

Surviving Mam1000W isn’t really a miraculous thing. While everyone tends to make it seem like it’s impossible, it is challenging (Not hard) and I said that because I have seen first-hand that practice makes it better each time. Getting to know the principles by actually doing the tuts which is the most important element of the course in my opinion will make sure that even though you feel like you aren’t learning anything when the time comes (usually 2nd semester) it will all click on how you actually are linking the information together.

Another important aspect is playing the numbers game.

By | May 14th, 2018|Courses, First year, MAM1000, Undergraduate|1 Comment