Taylor Polynomials

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Taylor Polynomials

• Dr. Dillon
• Calculus II
• Southern Polytechnic State University
• Fall 1999

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The Best Functions...
are polynomials.
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Polynomials Are Easy to Evaluate
You just have to multiply and add.
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They Are Easy to Differentiate
The derivative of a polynomial is another
polynomial.
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Polynomials Are Easy to Integrate
then
just another polynomial.
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Its Easy to Find the Limit of a Poly
You plug in
then multiply and add.
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Compare Polys to Other Functions
How do you evaluate?
Ask Fadyn!
He remembers!
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Even Worse Functions
How do you evaluate??
You need a circle and ratios...
Its much more than adding and multiplying.
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And Others...
How would you calculate it?
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If We Could...
approximate these functions with polynomials,
where would we begin?
Begin with the simplest polynomial.
The simplest polynomial is linear.
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Begin With a Line
Whats the best straight line approximation to a
function at a point?
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The Tangent!
Note that you wont have a tangent unless its a
differentiable function.
And, you get different tangents at different
points on the curve.
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Heres one straight line approximation
Heres another
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Linear Approximation at a Point
What is the equation
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To Find It

• First differentiate
• Then evaluate the derivative at the point
• Thats the slope of the line
• A point on the line is the point on the curve
• Thats enough to get the equation for the line

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An Example
The best straight line approximation to
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The Slope
The slope of the tangent is the slope of the
curve.
the slope of the tangent is
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The Equation for the Line
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The General Procedure
The best straight line approximation
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Second Degree Approximations
A line is a degree one polynomial. A degree two
polynomial should give us a better approximation
to our function.
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The first derivative gave us our degree one
approximation.
The second derivative will give us our degree
two approximation.
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The Tangent Line
This is the only line with the same first
derivative as the function, passing through the
designated point.
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The Degree Two Approximation
The only parabola with the same first and second
derivatives as the function, passing through the
designated point.
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Our Example
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How?
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This Gives Us
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Thus
Thats our best second degree polynomial
approximation to
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Compare
The exponential is in red. The second degree
polynomial is in blue.
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Notice
Its hard to tell them apart when you zoom in.
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Implications
For values of x between -0.3 and 0.3
and
describe just about the same function.
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Compare
and
Not exact, but pretty close.
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Take x Closer to Zero
and
The closer x is to zero,
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Better Approximations
To get better approximations, use higher degree
polynomials.
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Definition
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The Taylor polynomials are
the polynomials with the same value
as the given function,
at the given point.
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Why Use Derivatives?
The derivatives of a function
determine its contours.
Functions with the same derivatives
have the same shape.
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The Taylor Polynomial