Chapter 3: Applications of Differentiation
Section 3.3: Taylor Polynomials
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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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Solution
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Initialize and define
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Tools≻Load Package: Student Calculus 1
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Loading Student:-Calculus1
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Type , being sure to use the exponential "e" (palette or Command Completion).
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Context Panel: Assign Function
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Invoke the
tutor
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Type
Context Panel: Evaluate and Display Inline
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Context Panel: Tutors≻Taylor Approximation
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Figure 3.3.1(a) contains an image of the Taylor Approximation tutor adjusted for Degree = 3 (Default is 4).
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Figure 3.3.1(b) provides an animation showing the first three Taylor polynomials. The Animate button in the tutor also produces an animation, but that animation shows and the next four approximations.
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Figure 3.3.1(a) Taylor Approximation tutor
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>
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Student:-Calculus1:-TaylorApproximation(exp(x),0,degree=3,view=[-4..4,-5.44..26],output=animation,caption="",title="",order=1..3);
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Figure 3.3.1(b) Animation showing and
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Taylor Polynomials from the Context Panel
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Type
Context Panel: Evaluate and Display Inline
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Context Panel: Series≻Series≻
Series order≻4
Select "Remove order term" (See Figure 3.3.1(c).)
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The Series expansion point is the "" in the Taylor Formula. The Series order is the degree of ; it is one higher than the degree of . Removing the order term deletes the symbol , which stands for .
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Figure 3.3.1(c) Context Panel for Series
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Taylor Polynomials from first principles
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Expression palette: Summation template
Type the mathematical notation for .
Context Panel: Evaluate and Display Inline
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Alternatively, compute each coefficient of separately and assemble the polynomial according to the Taylor Formula.
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Since for integer , .
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