Jonathan Shock – Page 25

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.

Circular base, semi-circular top, triangular cross-section

For those who were particularly confused with the shape we discussed today in class, I’ve created a little animation which may help…or not…

Remember a single cross-sectional slice looked like:

How clear is this post?

…

By Jonathan Shock| August 4th, 2015|Uncategorized|1 Comment

Average Value of a function

If we have a function in some range $[a,b]$ we can ask what the average value of the function is. We are going to do this in a very intuitive way. We split up the curve into $n$ points and take the average of those points, and then ask what happens to that number as we take the number of points to infinity. Let’s consider the curve $y=x^2-x^3$ between $-3$ and $3$ . If we take seven points along the curve (ie. at $x=-3,-2,-1,0,1,2,3$ ) We will get the function values: $f(x)=36, 12, 2, 0, 0, -4, -18$ . as can be seen here:

The average value of the function when we take just seven sample points is thus: $\frac{36+ 12+ 2+ 0+ 0 -4 -18}{7}=4$ . The figure below shows what happens when we take more and more points. We see that the mean value converges to a fixed number,in this case 3.

But there is a better way to do this and again, it involves integration.…

By Jonathan Shock| August 1st, 2015|Courses, First year, MAM1000, Uncategorized, Undergraduate|1 Comment

Arc lengths

We have now learnt a great deal about various properties of curves. We can study their gradients, the areas under them and between them, we can even study properties of functions which have discontinuities and which extend all the way to $\infty$ . We can look at the revolution of a curve about an axis and study the volume enclosed using cylinders of various forms and we have a good understanding of the link between finite sums of pieces which make up an area or volume and the limit that these pieces become infinitesimally small and thus how we end up with an integral from a Riemann sum.

There is one piece of the puzzle left, which is to know the length of a curve between two points. Let’s say we have some function $f(x)$ and we want to know the length of the curve (ie. how long a piece of string would be that went along the curve and stretched from some point $a$ to another point $b$ on the curve).…

By Jonathan Shock| August 1st, 2015|Courses, First year, MAM1000, Uncategorized, Undergraduate|1 Comment

Volumes by cylindrical shells

Sometimes it is easier to calculate the volume of a solid (formed by the revolution of a surface about an axis of revolution) by dividing it up in a different way than into thin cross-sections. We will still use cylinders, but this time they will be cylindrical shells, where the height of the cylindrical shell mimics the function. This figure illustrates how this works for a particular function (in this case $f(x)=\sqrt{1-x^2}$ between $x=0$ and $x=1$ .

Top Left image: the function $\sqrt{1-x^2}$ .
Top right image: The volume formed by revolving this about the $y$ axis.
Bottom left image: Breaking this shape up into cylindrical shells (think of taking a wire loop of varying diameters and slicing vertically through the function.
Bottom right image: The cylindrical shells when pulled apart (this is just for illustration purposes). In this case the central shell is taller than the outer shells because in the centre of the shape the function is highest and towards the edges it is lowest.

…

By Jonathan Shock| August 1st, 2015|Courses, First year, MAM1000, Uncategorized, Undergraduate|3 Comments

UCT MAM1000 lecture notes part 11 – Tuesday 4th August

We can choose to rotate our shapes around different axes and of course the volumes will be very different. If we choose the shape defined by the area in between the functions $y=x^2$ and $y=x$ and we choose to rotate it around the vertical line $x=-1$ , we get the shape on the left hand side of this figure:

The cross-section is shown here:

Again, we have to ask what the cylinder is going to look like at height $y$ (ask yourself what a thin slice of the shape would look like at height $y$ : It’s going to be an annulus). This time both the inner and the outer radii of the annulus are going to change as we change $y$ . Now the inner radius is going to be $1+y$ and the outer radius is going to be $1+\sqrt{y}$ .
Thus we can see that the annulus at height $y$ is going to have area $\pi ((1+\sqrt{y})^2-(1+y)^2)$ (the 1 is because we are going from the point $x=-1$ to the point $y$ and $\sqrt{y}$ ).…

By Jonathan Shock| August 1st, 2015|Courses, First year, MAM1000, Uncategorized, Undergraduate|5 Comments

The following is almost certainly going to seem a bit confusing until you’ve seen a fair few examples. Don’t stress now, just try and picture what’s going on and we will build up our understanding as we go along.

Volumes

Having understood how to calculate the areas between two curves, or simply the area under a curve as a limit of the Riemann sum we can start to think about how to approximate not areas but volumes. If we can calculate an area by adding together small rectangles, perhaps we can calculate volumes by adding together small boxes, or other shapes. In fact we will use not boxes but cylinders, though perhaps not cylinders as we normally think of them.

You probably think of a cylinder as a tube with a circular cross-section, but in fact that is a particular type of cylinder called a circular cylinder. The general idea of a cylinder is a three dimensional object with a constant cross section.…

By Jonathan Shock| July 31st, 2015|Courses, First year, MAM1000, Uncategorized, Undergraduate|1 Comment

On shifting the integration region in an improper integral

I gave a rather unclear explanation of the following in a lecture a couple of days ago and promised to (at least attempt to) clarify a bit.

You know that:

$\int_0^{10}\frac{1}{x} dx$

is divergent and gives an answer of $\infty$ . If you have an integral of the form:

$\int_3^{10}\frac{1}{x-3}$

as we had in class, you don’t need to calculate this separately if you already know about the behaviour of $1/x$ about $x=0$ as they are exactly the same thing, just shifted (at least their convergence properties are the same).

The take-home message is that if you have:

$\int_a^b \frac{c}{(x-a)^p}dx$

and you already know the behaviour of:

$\int_0^d \frac{1}{x^p}dx$

then the integral you are interested in will have the same convergence or divergence behaviour.

How clear is this post?

…

By Jonathan Shock| July 31st, 2015|Uncategorized|0 Comments

Areas between curves

We know how to calculate the area between a curve and the horizontal ( $x$ ) axis of a graph. We have learnt a number of very sophisticated techniques which allow us to get an analytic answer.

It is a fairly simple extension to study not the area between a curve and the horizontal axis, but between two curves. You can see here the link between the two:

We can see in the figure below how we can approximate the area by a sum of rectangles. In this case we have used the left point approximation. Each rectangle here has $\Delta x=0.02$ and the heights are $f(x_i)-g(x_i)$ where the $x_i$ are the left points of each rectangle, in this case $=0.02(i-1)$ . So $x_1=0$ , $x_2=0.02$ , etc.

We started off studying integration by taking the limit of a Riemann sum. We can do exactly the same thing here, taking the simplest possible Riemann sum, with rectangular regions which stretch between the two curves (see the diagrams on page 448 in Stewart) and then taking the limit to find an expression in terms of integrals.…

By Jonathan Shock| July 30th, 2015|Courses, English, First year, Level: intermediate, MAM1000, Uncategorized, Undergraduate|4 Comments

Paradoxes of infinity and some more bonus material on prime numbers

The bonus material from yesterday:

Perfect numbers are those numbers which are equal to the sum of their divisors (other than themselves). For instance, 6 is divisible by 1, 2 and 3 (and 6, but we don’t include this in what follows). If you add these together, you get 6. 6 is a perfect number. 8 is divisible by 1, 2 and 4. Add these together and you get 7 – so 8 is not a perfect number (actually it is called deficient, the other alternative is an abundant number).

It was shown by Euclid that $2^{p-1}(2^p-1)$ is an even perfect number whenever $2^p-1$ is a prime (actually a prime of this form is called a Mersenne prime). In fact, it was shown by Euler two millennia later that every even perfect number could be written in this form! This is known as the Euclid-Euler theorem. So far there are 48 Mersenne primes known, and therefore 48 even perfect numbers known.…

By Jonathan Shock| July 29th, 2015|Uncategorized|0 Comments

UCT MAM1000 lecture notes part 8

Last time we looked at integrals which weren’t proper integrals because the limits of integration were infinite (either on one side, or both). This was one of the constraints we had on a well defined Riemann sum. The other constraint we had was that there were no infinite discontinuities in the integrand. Here we will show that sometimes we can indeed define an improper integral which does include such a discontinuity within the limits of integration.

Improper Integrals of the second kind: Infinite discontinuities in the integrand

We have seen what happens when we integrate $x$ from $\infty$ or to $-\infty$ . Sometimes it gives a convergent value and we can find the integral, sometimes the improper integral does not converge and it gives us an answer of $\pm\infty$ . Indeed sometimes it doesn’t blow up, but it just doesn’t converge – as in the integral of $\cos x$ over an infinite range.

Now we are going to look at what happens if the function itself has an infinite discontinuity in it and we want to integrate over this region, or include it at one of the limits of integration.…

By Jonathan Shock| July 28th, 2015|Courses, English, First year, Level: intermediate, MAM1000, Uncategorized, Undergraduate|3 Comments

Previous 23 24 25 26 27 Next