Chapter 3: Applications of Differentiation
Section 3.3: Taylor Polynomials
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Essentials
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Taylor's formula states that a suitably well-behaved function can be represented as a sum of a polynomial and a "remainder" term. This is made precise in Theorem 3.3.1.
Theorem 3.3.1 Taylor's Formula
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1.
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The function and (its first derivatives) are continuous in
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⇒
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There exists a value in for which the following representation of holds.
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In Theorem 3.3.1, is a polynomial of degree , called a Taylor polynomial.
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The symbol represents the kth derivative of evaluated at .
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By convention, is taken to mean "no derivative," that is, the function itself.
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The expression for the remainder is the Lagrange form, but there are at least two other forms, one of which is attributed to Cauchy.
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In particular, the first-degree Taylor polynomial is , and the equation is the equation of the tangent line at . Thus, any tool that generates the first-degree Taylor polynomial is a tool that likewise returns the equation of the tangent line!
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Examples
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Example 3.3.1
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a)
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For at , obtain Taylor polynomials of degree 1, 2, and 3.
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b)
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Graphically compare these polynomials on .
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c)
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Use to estimate the largest difference between and on .
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d)
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Find the actual value of the largest difference between and on .
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Example 3.3.2
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a)
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With and as in Example 3.3.1, obtain and use it to estimate the largest difference between and on . Hence, obtain the maximum value of = , which in turn requires finding the maximal value of for .
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b)
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Find the actual value of the largest difference between and on .
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Example 3.3.3
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For with in Taylor's Formula, obtain the general form of .
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Example 3.3.4
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At , obtain the equation of the line tangent to the graph of .
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