2011 Calculus BC free response #6a  AP Calculus BC  Khan Academy  Summary and Q&A
TL;DR
The video explains how to find the first four nonzero terms of the Taylor series for sine of x and sine of x squared.
Questions & Answers
Q: What is the purpose of a Taylor series?
A Taylor series is used to approximate a function using a polynomial with an infinite number of terms.
Q: How do you find the first nonzero terms of a Taylor series for sine of x?
By taking the derivatives of the function, evaluating them at zero, and dividing by the factorial of the power term.
Q: Why is it more efficient to find the Taylor series for sine of x squared by substituting x squared for x?
It avoids the complexity of taking higherorder derivatives and simplifies the calculations.
Q: How do you find the first nonzero terms of a Taylor series for sine of x squared?
Substitute x squared for x in the Taylor series for sine of x and simplify the terms.
Summary & Key Takeaways

The Taylor series is a polynomial approximation of a function.

The first four nonzero terms of the Taylor series for sine of x are: 0, x, 0, x^3/3!.

The first four nonzero terms of the Taylor series for sine of x squared are: x^2, x^6/3!, x^10/5!, x^14/7!.