What is the application of Taylor expansion in physics?

[quietrain]

i am very confuse how my profs always use taylor expansion in physics which somehow doesn't follow the general equation of

f(x) = f(a) + f'(a)(x-a) + 1/2! f''(a)(x-a)² and so on...

like for example, what is the taylor expansion of x - kx where k is small

it was given as something like

x - kx f'(x) + (1/2) k² x² f''(x) + ...

is this taylor expansion? but there is no 'about which point, i.e, a=? '

i don't even understand how the first term x is gotten. f(a) = x??

please help thank you!

 

[tiny-tim]

hi quietrain!

… something like …

hmm … but not exactly like!

the (x)s should be a, and the x at the start should be f(a)

(and k = x - a)

 

[I like Serena]

Hi quietrain! :)

Suppose you expand f(a-kx).

You would get: f(a-kx) = f(a) - kx f'(a) + (1/2) k² x² f''(a) + ...

Now replace "a" with "x"...
Apparently you've got a typo in the first term.

EDIT: Oops, ^TM got here first!

 

[Fredrik]

like for example, what is the taylor expansion of x - kx where k is small

TT and ILS have done a great job of guessing what you meant, but you should probably explain what function you really wanted to expand. (The Taylor expansion of x-kx is x-kx, so that's probably not it ).

 

[Ray Vickson]

i am very confuse how my profs always use taylor expansion in physics which somehow doesn't follow the general equation of

f(x) = f(a) + f'(a)(x-a) + 1/2! f''(a)(x-a)² and so on...

like for example, what is the taylor expansion of x - kx where k is small

it was given as something like

x - kx f'(x) + (1/2) k² x² f''(x) + ...

is this taylor expansion? but there is no 'about which point, i.e, a=? '

i don't even understand how the first term x is gotten. f(a) = x??

please help thank you!

Fix x, and let g(k) = f(x-kx). Expand as g(k) = g(0) + k*g'(0) + (k^2/2)*g''(0) + ...
g(0) = f(x), g'(k) = -x*f'(x-kx) --> g'(0) = -x*f'(x), g''(k) = x^2*f''(x-kx) --> g''(0) = x^2*f''(x), etc.

RGV

 

[quietrain]

hi everyone thanks for helping

but the taylor expansion was for x-kx

or perhaps only -kx was expanded?

but anyway it was given as x - kx f'(x) + ...

Hi quietrain! :)

Suppose you expand f(a-kx).

You would get: f(a-kx) = f(a) - kx f'(a) + (1/2) k² x² f''(a) + ...

Now replace "a" with "x"...
Apparently you've got a typo in the first term.

with regards to the above, how did you get that formula?

because i only know this f(x) = f(a) + f'(a)(x-a) + 1/2! f''(a)(x-a)² as the general formula? in any case, does it mean if i want to expand say 1 + kx

then it would be f(1+kx) = f(1) + kx f'(1) + ...

so f(1) is 1 + k(1) = 1+k?
then f'(1) is k? then where do i put my 1 since i don't have x
also does it mean f'' onwards are all 0?

Fix x, and let g(k) = f(x-kx). Expand as g(k) = g(0) + k*g'(0) + (k^2/2)*g''(0) + ...
g(0) = f(x), g'(k) = -x*f'(x-kx) --> g'(0) = -x*f'(x), g''(k) = x^2*f''(x-kx) --> g''(0) = x^2*f''(x), etc.

RGV

YES! this is the one. so it was an expansion about the small value k ? and not x? i see.

but in particular, how did you get g'(k) and g'(k)?

is it through the chain rule differentiation?

thanks!

 

[I like Serena]

with regards to the above, how did you get that formula?

because i only know this f(x) = f(a) + f'(a)(x-a) + 1/2! f''(a)(x-a)² as the general formula? in any case, does it mean if i want to expand say 1 + kx

then it would be f(1+kx) = f(1) + kx f'(1) + ...

so f(1) is 1 + k(1) = 1+k?
then f'(1) is k? then where do i put my 1 since i don't have x
also does it mean f'' onwards are all 0?

Okay, let's start with the formula you have:
f(x) = f(a) + f'(a)(x-a) + 1/2! f''(a)(x-a)² + ...
but let's rewrite it with y instead of x to eliminate the ambiguity between the x's.

So:
f(y) = f(a) + f'(a)(y-a) + 1/2! f''(a)(y-a)² + ...

You want to expand f(x-kx).
To do this, first define y=x-kx, and define a=x.
Then replace all occurrences of y by (x-kx), and replace all occurrences of a by x.

What do you get?Edit: Btw, an alternative formula for Taylor expansion is: f(x+h)=f(x) + h f'(x) + 1/2! h² f''(x) + ...

 

[Fredrik]

in any case, does it mean if i want to expand say 1 + kx

then it would be f(1+kx) = f(1) + kx f'(1) + ...

Your terminology is inaccurate. What you're doing here is to expand $f$ around the point 1 in its domain. The expansion of 1+kx (i.e. the function $x\mapsto 1+kx$) around 0 (the point I'd assume you have in mind unless you specify another one) is just 1+kx.

 

[quietrain]

Okay, let's start with the formula you have:
f(x) = f(a) + f'(a)(x-a) + 1/2! f''(a)(x-a)² + ...
but let's rewrite it with y instead of x to eliminate the ambiguity between the x's.

So:
f(y) = f(a) + f'(a)(y-a) + 1/2! f''(a)(y-a)² + ...

You want to expand f(x-kx).
To do this, first define y=x-kx, and define a=x.
Then replace all occurrences of y by (x-kx), and replace all occurrences of a by x.

What do you get?


Edit: Btw, an alternative formula for Taylor expansion is: f(x+h)=f(x) + h f'(x) + 1/2! h² f''(x) + ...

oh, we can just define a as x?

does it mean i can define a as -kx or -k also?

but i thought a is the point that we evaluate the taylor expansion on?

am i right to say if i taylor expand on a point say x =1 for a curve graph. then as my orders of taylor expansion become greater, the approximation is better too?

 

[quietrain]

Your terminology is inaccurate. What you're doing here is to expand $f$ around the point 1 in its domain. The expansion of 1+kx (i.e. the function $x\mapsto 1+kx$) around 0 (the point I'd assume you have in mind unless you specify another one) is just 1+kx.

oh... do you mean since 1+kx is a straight line, so at the point 0, it is just 1 + kx ?

so if i had a curve, would taylor expansion ( approximation? ) make more sense here?

that means the first term of the expansion gives me a straight line, the 2nd makes it more curve, the 3rd makes it even more like the original curve function?

does it then mean i cannot taylor expand things like 1+kx? i have to expand only curves?

 

[I like Serena]

oh, we can just define a as x?

does it mean i can define a as -kx or -k also?

but i thought a is the point that we evaluate the taylor expansion on?

am i right to say if i taylor expand on a point say x =1 for a curve graph. then as my orders of taylor expansion become greater, the approximation is better too?

"x" and "a" are just letters.
You can replace them by anything you want, as long as you do so consistently and do not mix letters up.
(Note that "x" has 2 different meanings in your problem statement. That's why I introduced "y" - to eliminate one of the two.)

In your original formula f(x)=f(a)+f'(a)(x-a)+...
the expansion is around "a".

In the formula f(x-kx)=f(x)+f'(x)((x-kx)-x)+...= f(x) - kx f'(x) + ...
the expansion is around "x".

 

[I like Serena]

does it then mean i cannot taylor expand things like 1+kx? i have to expand only curves?

If you expand 1+kx around x=0, you are effectively defining f(x)=1+kx, and a=0.
So using your formula to expand it, you get:

f(x)=1+kx
f'(x)=k
f''(x)=0
f'''(x)=0

So:
f(x)=f(a)+f'(a)(x-a)+1/2! f''(a)(x-a)²+...
f(x)=f(0)+f'(0)(x-0)+1/2! f''(0)(x-0)²+...
f(x)=1+k(x-0)+1/2! 0.(x-0)²+ 0 + ...
f(x)=1+kx

Hey! But you already had that! :)

 

[Ray Vickson]

i am very confuse how my profs always use taylor expansion in physics which somehow doesn't follow the general equation of

f(x) = f(a) + f'(a)(x-a) + 1/2! f''(a)(x-a)² and so on...

like for example, what is the taylor expansion of x - kx where k is small

it was given as something like

x - kx f'(x) + (1/2) k² x² f''(x) + ...

is this taylor expansion? but there is no 'about which point, i.e, a=? '

i don't even understand how the first term x is gotten. f(a) = x??

please help thank you!

I think your expression was copied wrong, or had a misprint: you should have written
f(x) - kx f'(x) + (1/2) k² x² f''(x) + ... [with f(x) as the first term, not just x]. You did say the expression was "something like...", which says to me that you were not sure.

RGV

 

[quietrain]

alright thanks everyone!