Calculus 9.2 day 1

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Find the local linear approximation of f(x) ex
at x 0. Find the local quadratic
approximation of f(x) ex at x 0.
2
we want to find a second degree polynomial of the
form
at
that approximates the behavior of
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4
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Suppose we wanted to find a fourth degree
polynomial of the form
at
that approximates the behavior of
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If we plot both functions, we see that near zero
the functions match very well!
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Our polynomial
has the form
or
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If we want to center the series (and its graph)
at some point other than zero, we get the Taylor
Series
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Find the Taylor Series for
centered at a1
And determine the interval of convergence
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If the limit of the ratio between consecutive
terms is less than one, then the series will
converge.
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If the limit of the ratio between consecutive
terms is less than one, then the series will
converge.
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Write the Taylor Series for f(x) cos x
centered at x 0.
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The more terms we add, the better our
approximation.
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For
use the Ratio Test to determine the interval of
convergence.
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If the limit of the ratio between consecutive
terms is less than one, then the series will
converge.
The interval of convergence is
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When referring to Taylor polynomials, we can talk
about number of terms, order or degree.
This is a polynomial with 3 positive terms.
It is a 4th order Taylor polynomial, because it
was found using the 4th derivative.
It is also a 4th degree polynomial, because x is
raised to the 4th power.
The x3 term drops out when using the third
derivative.
A recent AP exam required the student to know the
difference between order and degree.
This is also the 2nd order polynomial.
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Both sides are even functions.
Cos (0) 1 for both sides.
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Both sides are odd functions.
Sin (0) 0 for both sides.
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What is the interval of convergence for the sine
series?
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Power series for elementary functions
Interval of Convergence