Worked example: estimating e_ using Lagrange error bound | AP Calculus BC | Khan Academy | Summary and Q&A

TL;DR
Using the LaGrange error bound or Taylor's remainder theorem, we can estimate an error lower than 0.001 by using a Taylor polynomial of degree 5.
Key Insights
- ❓ The LaGrange error bound or Taylor's remainder theorem is used to estimate the error when approximating a function with a Taylor polynomial.
- 🪡 An upper bound on the n+1th derivative of the function is crucial in determining the degree of the polynomial needed to achieve a desired error bound.
- 🤶 Interval selection for the function and center of the polynomial is important in finding an appropriate value for M, the upper bound.
Transcript
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Questions & Answers
Q: What is the purpose of using the LaGrange error bound or Taylor's remainder theorem in estimating a Taylor polynomial's error?
The LaGrange error bound or Taylor's remainder theorem helps us determine the degree of the polynomial required to bound the error when approximating a function. It provides a way to ensure that the approximation is within a desired error threshold.
Q: How do we calculate the value of M, the upper bound on the n+1th derivative of the function?
In order to calculate M, we need to find an interval that contains the value we are approximating (x-value) as well as the center of the Taylor polynomial (c-value). We then determine an upper bound for the absolute value of the n+1th derivative within that interval. In this case, we used the fact that the derivative of e^x is always less than or equal to e^2 for values between 0 and 2.
Q: Why is it necessary to find an upper bound on the n+1th derivative of the function?
Finding an upper bound on the n+1th derivative is crucial because it allows us to determine how many terms or the degree of the Taylor polynomial we need to use in order to achieve a desired level of accuracy in approximating the function.
Q: How is the LaGrange error bound formula used to find the least degree of the polynomial that assures an error smaller than 0.001?
By rearranging the LaGrange error bound formula and substituting the values of M, x, and c, we can set an inequality that needs to hold true for the error to be smaller than the given threshold. We then solve this inequality to find the least degree of the polynomial, which in this case is equal to 5.
Summary & Key Takeaways
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The video discusses using the LaGrange error bound or Taylor's remainder theorem to estimate the error of approximating a function with a Taylor polynomial.
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The LaGrange error bound formula is introduced, which includes the absolute value of the remainder, the degree of the polynomial, the value at which the error is evaluated, and an upper bound on the n+1th derivative of the function.
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The video demonstrates the process by estimating e to the 1.45 using a Taylor polynomial centered at x equals 2, and determines that a polynomial of degree 5 is needed to assure an error smaller than 0.001.
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