2011 Calculus BC free response #6a | AP Calculus BC | Khan Academy | Summary and Q&A
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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.
Key Insights
- π¨ A Taylor series is a useful tool for approximating functions with polynomials.
- βΊοΈ The first non-zero terms of the Taylor series for sine of x are 0, x, 0, -x^3/3!.
- βΊοΈ 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!.
- β Taylor series can be used to approximate functions to any desired degree of accuracy.
- π Taking more terms in the Taylor series improves the approximation of the original function.
- β The Taylor series for a function centered at 0 is defined by its derivatives at 0.
- π¨βπΌ The Taylor series for sine and cosine have repeating patterns in their derivatives.
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
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!.
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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!.
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