Question

Math HL IA topic suggestions???

I am in math hl and I really need a good math IA topic. If anyone has an interesting IA suggestion--I would really appreciate it!!
Our teacher just gave us a whole list of IA topics but I don't know what to pick
Here is the site:
https://www.fairviewhs.org/staff/emily-silverman/classes/ib-math-hl/files/30763

Answer 1

Applications of power series are very interesting.

Explanation:

I may be a bit slanted because of my affinity for calculus, but I strongly recommend that you consider examining the various applications of power series.

In particular, how a power series can be used as a means for approximating a diverse class of functions systems, such as logarithms, the inverse tangent, and variations of these.

I.E., one could show logarithmic approximation:

`f(x)=ln(1+x)`

`f(x)=intdx/(1+x)`

`f(x)=intdx/(1-(-x))`

`f(x)=intsum_(n=0)^oo(-1)^nx^ndx`

`f(x)=C+sum_(n=0)^oo(-1)^nx^(n+1)/(n+1),` converges for `|x|<1`

`ln(1+x)=C+sum_(n=0)^oo(-1)^nx^(n+1)/(n+1)`

`ln(1+0)=C+sum_(n=0)^oo(-1)^n0^(n+1)/(n+1)`

`C=ln(1)=0`

`ln(1+x)=sum_(n=0)^oo(-1)^nx^(n+1)/(n+1)`

So long as you plug in any `-1<x<1,` the above can be used to approximate a logarithm, IE, let `x=-1/2:`

`ln(1/2)=sum_(n=0)^oo(-1)^n(-1/2)^(n+1)/(n+1)`

`ln(1/2)=ln(1)-ln2=-ln2`

`-ln2=sum_(n=0)^oo(-1)cancel((-1)^n(-1)^n)(1/2)^n/(n+1)`

`cancel-ln2=cancel-sum_(n=0)^oo(1/2)^n/(n+1)`

`ln2approx1+1/4+1/12+1/32+...`

And there are many other possibilities, such as how Taylor Polynomials could be used to show how `sintheta approx theta` when `theta` is very small is a good approximation.