The Taylor polynomial T_n(x) for the function arcsin x at a = 0 and n = 3 is derived by calculating the function's value and its derivatives at the point a = 0. The derivatives of arcsin x are more complex than those of sine, which adds difficulty to the process. The key steps involve finding f(0), f'(0), and subsequent derivatives, then substituting these values into the Taylor series formula. The final polynomial accurately represents arcsin x near x = 0.
PREREQUISITESStudents studying calculus, particularly those focusing on series expansions, and anyone seeking to understand the complexities of derivatives for inverse trigonometric functions.