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.

Transcript

Problem number six. Let f of x is equal to sine of x squared plus cosine of x. The graph of y is equal to the absolute value of the fifth derivative of f at x is shown above. And I haven't shown it here just so we have some space. I'll show it when we need to show it. I think we have to show it in part D. So first, let's do part A right over here. ... Read More

Key Insights

• 🔨 A Taylor series is a useful tool for approximating functions with polynomials.
• ☺️ The first non-zero terms of the Taylor series for sine of x are 0, x, 0, -x^3/3!.
• ☺️ Substituting x squared for x gives the first non-zero terms of the Taylor series for sine of x squared: x^2, -x^6/3!, x^10/5!, -x^14/7!.
• ❓ Taylor series can be used to approximate functions to any desired degree of accuracy.
• 🍉 Taking more terms in the Taylor series improves the approximation of the original function.
• ❓ The Taylor series for a function centered at 0 is defined by its derivatives at 0.
• 👨‍💼 The Taylor series for sine and cosine have repeating patterns in their derivatives.

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 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!.