# 2011 Calculus BC free response #6a | AP Calculus BC | Khan Academy | Summary and Q&A

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September 13, 2011
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2011 Calculus BC free response #6a | AP Calculus BC | Khan Academy

## TL;DR

The video explains how to find the first four non-zero terms of the Taylor series for sine of x and sine of x squared.

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### 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 non-zero 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 higher-order derivatives and simplifies the calculations.

### Q: How do you find the first non-zero 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 non-zero terms of the Taylor series for sine of x are: 0, x, 0, -x^3/3!.

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