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Lecture 31 – Approximating Functions. Consider the following:. Now, use the reciprocal function and tangent line to get an approximation. 3. 1. 2. 2.01. 2. First derivative gave us more information about the function (in particular, the direction).
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Lecture 31 – Approximating Functions Consider the following: Now, use the reciprocal function and tangent line to get an approximation. 3 1 2
2.01 2
First derivative gave us more information about the function (in particular, the direction). For values of x near a the linear approximation given by the tangent line should be better than the constant approximation. Second derivative will give us more information (curvature). For values of x near a the quadratic approximation should be better than the linear approximation.
What quadratic is used as the approximation? Key idea: Need to have quadratic match up with the function and its first and second derivatives at x = a.
What higher degree polynomial is appropriate? Key idea: Need to have nth degree polynomial match up with the function and all of its derivatives at x = a.
The coefficients, ck, for the nth degree Taylor polynomial approximating the function f(x) at x = a have the form:
Lecture 32 – Taylor Polynomials Def: The Taylor polynomial of order n for function f at x = a: The remainder term for using this polynomial: for some c betweenx and a. whereM provides a bound on how big the n+1st derivative could possibly be.
Estimatethe maximum error in approximating the reciprocal function at x = 2 with an 8th order Taylor polynomial on the interval [2, 3].
What is the actual maximum error in approximating the reciprocal function at x = 2 with an 8th order Taylor polynomial on the interval [2, 3]?
What nth degree polynomial would you need in order to keep the error below .0001?
Lecture 33 – Taylor Series The Taylor series centered at x = a: is a power series with The Taylor series centered at x = 0 is called a Maclaurin series:
Example 1 Find the Maclaurin series for f (x) = sin x.
Example 2 Find the Maclaurin series for f (x) = ex.
For what values of x will the last two series converge? Ratio Test: Series converges for Series converges for
Example 3 Find the Maclaurin series for f (x) = ln(1 + x).
Example 4 Creating new series for:
Lecture 34 – More Taylor Series Create and use other Taylor series like was done with power series.