Local linear approximation!

katekate · #1 $Old$ November 2nd, 2009, 07:19 PM

Find the local linear approximation of the foloowing functions at the specified points:

a) f(x) = e^(5x) * cos(4x) at the point x=0

b) g(x) = e^(-5x) * ln(7x) at the point x=1

I know the basic idea of linear approx, where its f(x) + f'(x)*(x-a) but i have no idea what's what... for example is x or a equal to 0 in part a?

Ruun · #2 $Old$ November 3rd, 2009, 11:14 AM

Hi.

First, you know what $f(x)$ does really mean? It's that kind of question really important that maybe nobody told you, and you don't realize (and this is a problem, but it is not embarrasing).

So, suppose you have studied some physical problem, finding some rule that tells you what's going to happen in some situation, as if you were given some imput number, your rule will tell you what to do with this number in order to predict or confirm something.

In particular, there is a situation in wich, you can predict something just by taking the square of a number, i.e. if you are given $1$ , with the number $1^2=1$ you predict something. if you are given $2$ with the number $2^2=4$ , and, in general for some arbritary (at least in our case, in general the numbers you can use as input form a set known as the domain of the function), but fixed $x$ , you can predict something with the number (pay attention, this is the point) $f(x)=x^2$ .

And why I am boring you with all this stuff? Because once you will understand that $f(x)$ really means a rule named $f$ applied to some number in the domain, absolutely arbritary but fidex $x$ , and check the definition of a Taylor series, you will be able to know what are your "a".

Hope it helps.

If you need the answer and don't care for the above text, then I'm sorry for you, but ask if you actually need your answers instead of thinking a bit.