Question

Can someone please explain what it means to approximate a function. Google gives me many ways to do approximations but doesn't tell me what it is. I'm learning Taylor polynomials to prepare myself for Numerical Methods and it crops up there

Answer 1

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Answer 2

The taylor expansion of sine about \(x=0\) is\[\sin x = x-x^3/3!+x^5/5!+\dots\]

for values close to \(x=0\)

\[\sin x = x-x^3/3!+\textrm O(x^5)\]

Lets approximate sine of 0.2 using this method

\[\sin (0.2) = (0.2)-(0.2)^3/3!+\textrm O(0.2^5)\\
=0.2-0.008/6+\textrm O(0.00032)\\
\approx0.2-0.00133\\
=0.198666\]

a calculator gives sin(0.2) = 0.198669

Answer 3

notice how the first term in the expansion is the most important, and since 0.2 is close to 0 we expect the approximation to be good

Answer 4

the error in the approximation is equal to the absolute value of the difference between the approximated value and the true value
here the error is 0.000003

the relative error is equal to this error, divided by the true value

0.000003/0.198669 = 0.000015


and the percentage error is this relative error times 100%

0.000015 x 100% = 0.0015%

Which confirms that our approximation was very good

Answer 5

do you get the taylor series (like how you are using more and more derivatives to approximate the value at a point)

Answer 6

Sorry, that doesn't really help. I'm just trying to get an understanding of what it means to approximate a function. I'm working my way through taylor on sosmath and as far as I can tell they calculate functions approximate to another function at a point. Is it as simple as saying "an approximation function" of function f(x) at point a is a function which is similar to f(x) and goes through a"???

I'm still trying to understand taylor series. I only did 1st year calculus and I understand taylor series is 2nd year?

Answer 7

Or does an approximation have to give a value, i.e. a value which is not the exact value, but close enough?

Answer 8

suppose you only know the value of \(\sin (0) \) and want to find a nearby value : \(\large \sin(0.2)\)

the approximation function is just any EASILY workable function that gets you the nearby values with some accuracy