Lagrange Remainder.

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Presentation transcript:

Lagrange Remainder

Difference Between Lagrange Error and Error in Alternating Series Lagrange Remainder can be used with any Taylor series, it does not matter if the series is alternating and decreasing. The error in the nth degree Taylor series can be found if it is alternating and decreasing With Lagrange Remainder, the error can be bound Where c is between x and x0 To use Lagrange Remainder, the maximum value of the (n+1)th derivative of f(x) at point c, where c is between x and x0 must be known.

Not Using Lagrange Remainder Use the first three terms of this series to approximate sin(3) and find the maximum error in the approximation. Is series alternating and decreasing? We may use formula for error in alternating series: Yes. 2.26x10-10 is the maximum error in our approximation.

Using Lagrange Remainder Given: The third-degree Taylor polynomial of the four-time differentiable function f(x) around x = 2 is And for all x in the interval [1.5, 2] a)Use this series to approximate f(1.5) and find the maximum error in the approximation. Is series alternating and decreasing? Not alternating for sure: second term is negative too. No. We must use Lagrange remainder, not error in alt. series: 0.008 is the maximum error in our approximation.

You will be explicitly told the maximum value of on the interval

Using Lagrange remainder

2. You can be given a graph of and you need to find the maximum value

Using Lagrange Remainder Find the maximum error in the approximation of f(1.5) when you use a third-degree Taylor series to approximate it. The maximum of 4th derivative on the interval [1.5,2] is clearly 3 Continue with calculation with known maximum 3 0.008 is the maximum error in our approximation.

3. You can be given a table of values of and you need to find the maximum value Note: It will be mentioned that the is always decreasing or increasing as only some values are given

Using Lagrange Remainder Find the maximum error in the approximation of f(1.5) when you use a third-degree Taylor series to approximate it. The maximum is clearly 3 on the interval [1.5,2]. Continue with calculation with known maximum 0.008 is the maximum error in our approximation.

4. If the function is sin(x) or cos(x) and then you can use the amplitude of as the maximum value.

Using Lagrange remainder Taylor polynomial: Series is not alternating and decreasing, so use Lagrange Error because

Using Lagrange remainder Try different n values: