Taylor development I don't understand

  • Thread starter JulienB
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  • #1
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Homework Statement



Hi everybody! In the middle of an exercise, our teacher suddenly wrote:

[tex]
sin(\frac{x}{y} sin y) = \frac{x}{y} sin y - \frac{1}{2} sin θ (\frac{x}{y} sin y)^2
[/tex]

I don't get where does that come from? The closest I've managed to reach is:

[tex]
sin(\frac{x}{y} sin y) = \frac{x}{y} sin y - (\frac{x}{y} sin y)^2 \frac{x sin y}{6y} + O(x^5)
[/tex]

What is this θ? It doesn't show up neither before nor after that line :/


Thanks in advance for your answers.


Julien.
 

Answers and Replies

  • #2
Ray Vickson
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Homework Statement



Hi everybody! In the middle of an exercise, our teacher suddenly wrote:

[tex]
sin(\frac{x}{y} sin y) = \frac{x}{y} sin y - \frac{1}{2} sin θ (\frac{x}{y} sin y)^2
[/tex]

I don't get where does that come from? The closest I've managed to reach is:

[tex]
sin(\frac{x}{y} sin y) = \frac{x}{y} sin y - (\frac{x}{y} sin y)^2 \frac{x sin y}{6y} + O(x^5)
[/tex]

What is this θ? It doesn't show up neither before nor after that line :/


Thanks in advance for your answers.


Julien.
When using TeX/LaTeX, please try to remember to use "\sin" instead of "sin", because using "sin" looks ugly and hard to read, like this ##sin x##, while using "\sin" produced pleasing and easy-to-read results like this: ##\sin x##. Same for cos, tan, csc, sec, cot, (plus their inverses), exp, sinh, cosh, tanh, log, ln, lim, max, min---basically, all the standard functions.

Anyway, he/she is using the standard form of remainder in a Taylor series:
[tex] f(x) = f(a) + (x-a) f'(a) + \int_a^x f''(t) (x-t) \, dt [/tex]
If ##f''## does not change sign in ##[a,x]## the last term can be expressed as
[tex] \frac{1}{2!} (x-a)^2 f''(\theta), \; \theta \in [a,x] [/tex]
using the Mean Value Theorem.
 
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