Review of prerequisite concepts

DISCUSSION

On Wed, Aug 18, 2010 at 17:10, Erica Leung <ericahk124@gmail.com> wrote:

Dear Sid,

Two questions I am asked by the students during the tutorial are:

- thanks for letting me know. encourage them to post their questions on the wiki (or if you're asked during the tutorial, post them yourself during the tutorial, and the results of your discussions)

1. What is Taylor series for? Why we want to approximate the function with a polynomial?

- good question.

(at least) two answers.

a. sometimes we don't know the function, but want to approximate it. for example, suppose we're measuring the weather. the weather is a complicated function. we can measure it as a function of time, but do not know how to write down the function. can you think of a way of using the taylor series to approximate the "weather function"? (here's a hint -- to approximate the function, we'll first need to approximate the derivatives of the function. but we defined the derivative f'(x_0) as the limit of [f(x_0+epsilon)-f(x)]/epsilon as epsilon goes to zero. Does this help you approximate f'(x_0)? how would you approximate f''(x_0)? f'''(x_0)? how about higher order derivatives?

b. sometimes, even if we can write down the formula for a function, it's a complicated formula, and it's hard to do some calculations with it. but it's easier to do calculations with polynomials. for example, suppose someone asked us to calculate the integral of sin(x)/x from -2 to 2. we don't know a formula for this.

try using maxima to integrate it -- it won't be able to

integrate(sin(x)/x, x);

however, we could write the taylor series expansion of sin(x)/x, and choose the first few terms. now this would be a polynomial. this is easy to integrate. try doing it -- use maxima to find the taylor series expansion of sin(x)/x

for example, the command is for the first four terms of the taylor series for x_0 = 0,

taylor(sin(x)/x,x,0,4)

try using maxima to find the taylor series expansion for different values of x_0, and for different number of terms.

now, plot these against sin(x)/x, to see how close the two functions are. for example, try these different values

wxplot2d([taylor(sin(x)/x,x,0,2),sin(x)/x], [x,-5,5], [y,-1.5,1.5]);

wxplot2d([taylor(sin(x)/x,x,0,4),sin(x)/x], [x,-5,5], [y,-1.5,1.5]);

wxplot2d([taylor(sin(x)/x,x,0,6),sin(x)/x], [x,-5,5], [y,-1.5,1.5]);

wxplot2d([taylor(sin(x)/x,x,0,8),sin(x)/x], [x,-5,5], [y,-1.5,1.5]);

finally, try using maxima to use the taylor series approximation to integrate the sin(x)/x.

btw, maxima can do many more things than is shown in the commands on top. for example, taylor series isn't in the commands on top, but i google-searched for "maxima taylor series" to find out what command to use.

2. In proving the limit of function, what is the relationship between delta and epsilon?

- that's a very general question. were you able to give an answer? (in general, delta depends on the value of epsilon, the function, and the point x_0 -- for different functions, x_0, and epsilons, the value of delta is different. to prove that a limit exists, one has to find the delta -- to prove that it cannot exist, one has to prove that no such delta can exist.)

sometimes, to understand why the definition is important, it's useful to give a crazy example. here's an example exercise that you can post on the wiki, to see their response.

(Thought-provoking and challenging question for those interested): Consider the function f that is defined as 0 on every irrational number, and 1/q for every rational number of the form p/q. Can you prove that the function has a limit for every irrational number, but no limit for any rational number? 

(it's a crazy function -- think about it -- near every x, there are infinitely many rationals AND irrationals.)

please post this entire discussion on the wiki.

thanks!

-- 

sid.