Lagrange Error Bound Practice
Practice problems for understanding and applying the Lagrange Error Bound in Taylor series approximations.
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Lagrange Error Bound Practice
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Read each question carefully and show all your work. Use the Lagrange Error Bound to justify your answers where appropriate.
1. The Lagrange Error Bound gives an for the maximum possible error when approximating a function with a Taylor polynomial.
2. The formula for the Lagrange Error Bound is |Rₙ(x)| ≤ , where M is the maximum value of the (n+1)th derivative of f on the interval between c and x.
3. If the (n+1)th derivative of a function is bounded by M on an interval, then the error in using the nth Taylor polynomial to approximate the function at x is at most .
4. Explain in your own words what the Lagrange Error Bound tells us.
5. Which of the following is true about the Lagrange Error Bound?
It provides the exact error of the approximation.
It gives an upper bound for the absolute value of the error.
It is only applicable for Maclaurin series.
It requires knowing the actual value of the function.
6. Consider the function f(x) = e^x. Find the maximum error if the third-degree Taylor polynomial centered at x=0 is used to approximate e^0.5.
7. For the function f(x) = sin(x), use the Lagrange Error Bound to determine the maximum error when approximating sin(0.1) using a second-degree Taylor polynomial centered at x=0.
8. The value 'c' in the Lagrange Error Bound formula always lies between 0 and x.
True
False