Author Archives: lukewarren

Parametric Curves-Luke

The lab I have been looking at is on Parametric Curves. These are where you have two variables and a function that graphs them. But instead of using one function in terms of x and/or y:

$y=f(x)$ or $z=f(x,y)$,

parametric curves give the values for x and y as a function of t. So:

$x=x(t)$ and $y=y(t)$

This way of representing curves can be very useful with circles and polar coordinates. The questions that they ask deal mostly with parameters of $sin(at)$ and $cos(bt)$. The first few questions start exploring functions and seeing what they do with different values, how they intersect, etc. And then forming conjectures based on the data.

Numerical Integration: Summary and Conclusions

In this lab we looked at some various methods to compute integrals.
Those methods were, Riemann Sums: left rectangle, midpoint, and right rectangle, Trapezoidal, and Simpsons Rule.

We tested various methods and got results that showed that Simpsons Rule was very accurate, in many cases exact for cubic functions. We explored this more saw that Simpsons rule and the actual integration yielded the same result. This was also a proof of our conjectures about this method Finally we showed where the Simpsons Rule came from.

Now with all of that there are many other things that can be explored. Exploring more functions than cubics, (trig functions, logarithmic, exponential, etc.) and seeing which methods work best for those. Other routes one might take would be to explore higher dimensions. Some questions that could be explored would be how to test the accuracy for functions that don’t have an antiderivative? What methods would seem to work best? How small of an interval would be needed?

Equation for Simpsons Rule: Proof and Derivation

We get in the book that an equation for a parabola for any three points is

$ q(x)=A\frac {(x-b)(x-c)}{(a-b)(a-c)}+ B \frac{(x-a)(x-c)}{(b-a)(b-c)} +C\frac{(x-a)(x-b)}{(c-a)(c-b)}$

This is the formula for some parabola $ q= mx^2+nx+p$ through the points, (a, A), (b,B), (c,C), we will take this as a given and go from there.

First we need to say we are working on a interval [-h, h]. Then our three points will be (-h, f(-h)), (0,f(0)), (h,f(h)).

Photo on 2013-11-18 at 09.09

First we write a generic equation for this quadratic. We can do this using the formula given and replacing the values:
$ q(x)=f(-h)\frac {(x)(x-h)}{(-h)(-2h)}+ f(0) \frac{(x+h)(x-h)}{(h)(-h)} +f(h)\frac{(x+h)(x)}{(2h)(h)}$

Reducing this we get:

$ q(x)=f(-h)\frac {x^2-xh}{2h^2}+ f(0) \frac{(x^2-h^2)}{-h^2} +f(h)\frac{x^2+xh}{2h^2}$

Now we integrate from [-h, h].

Photo on 2013-11-18 at 09.20

$\int^h_{-h} f(-h)\frac {x^2-xh}{2h^2}+ \int^h_{-h} f(0) \frac{(x^2-h^2)}{-h^2} +\int^h_{-h} f(h)\frac{x^2+xh}{2h^2}$

$=f(-h) (\frac{x^3}{6h^2} – \frac{x^2}{4h}) |_{-h}^h + f(0) (\frac{x^3}{-h^2} + x) |_{-h}^h + f(h) (\frac{x^3}{6h^2} + \frac{x^2}{4h}) |_{-h}^h$

Just taking the first integral:

$f(-h)((\frac{h^3}{6h^2} – \frac{h^2}{4h})-(\frac{-h^3}{6h^2} – \frac{h^2}{4h}))$

$=f(-h)\frac{h}{3}$

Evaluating the other integrals and reducing we get

$ f(0)\frac{4}{3}h$ and $f(h)\frac{h}{3}$

So we have: $f(-h)\frac{h}{3} + f(0)\frac{4}{3}h +f(h)\frac{h}{3}$
$=\frac{h}{3}[f(-h)+4f(0)+f(h)]$

Now the points if you change the points we were working with to

$(x_{i-1},f(x_{i-1})),(x_i,f(x_{i})), (x_{i+1},f(x_{i+1}))$

we get
$=\frac{h}{3}[f(x_{i-1})+4f(x_{i})+f(x_{i+1})]$

This is the formula for Simpson’s Rule. This has major implications. It shoes how accurate it can be. This is accurate because when you have three points and a function that goes through them Simpsons rule finds a best fit quadratic for that region. Then we see that the formula does not depend on the actual equations themselves, but the points that one is evaluating them at and the interval they are going across.

Numerical Integration Methods-Motivation

I’m sure most of us have at least some experience with integrals being tricky and hard to compute. Whether it be trig substitution or some other method integrals can be very difficult. There are also functions, real functions that do not have an antiderivative, functions like:

$e^{-x^2}$ or $\frac{sin x}{x}$

However, there are ways to evaluate these integrals, and these are contained in the numerical methods. It is worth studying these methods one, to find the area under the cover for these certain functions, and two, to test the accuracy of these methods. There are certain methods in this chapter which my teammates will tell you more about.