Question

How do you write the first four non-zero terms of the Maclaurin series for `sin(x^2)`?

Answer 1

`sin (x^2) ~~ x^2 - x^6/3! + x^10/5! - x^14/7! + ...`

Explanation:

Recall that the Maclaurin series for `sin x` is given by:

`sin (x) ~~ x - x^3/3! + x^5/5! - x^7/7! + ...`

Note: This is a common Maclaurin series and many exams require you to know this (which is why I directly referred to it). If you are not familiar with deriving Maclaurin series of any function (like `y = sin x`) I recommend that you read this

Hence, observing from the above approximation, we can replace `x = x^2` in the formula to obtain:

`sin (x^2) ~~ x^2 - x^6/3! + x^10/5! - x^14/7! + ...`

Hope this helps! Comment below or PM me if you have any doubts!

All the best!