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Example Question #1 : Maclaurin Series
For which of the following functions can the Maclaurin series representation be expressed in four or fewer non-zero terms?
Recall the Maclaurin series formula:
Despite being a 5th degree polynomial recall that the Maclaurin series for any polynomial is just the polynomial itself, so this function's Taylor series is identical to itself with two non-zero terms.
The only function that has four or fewer terms is as its Maclaurin series is.
Example Question #2 : Maclaurin Series
Find the the first three terms of the Taylor Series for centered at .
Using the formula of a binomial series centered at 0:
where we replace with and , we get:
for the first 3 terms.
Then, we find the terms where,
Example Question #1 : Taylor Series
Determine the convergence of the Taylor Series for at where .
Does not exist.
By the ratio test, the series converges absolutely:
Example Question #3 : Applications Of Derivatives
Find the interval of convergence for of the Taylor Series .
Using the root test
herefore, the series only converges when it is equal to zero.
This occurs when x=5.
Example Question #4 : Applications Of Derivatives
Suppose that the derivative of a function, denoted , can be approximated by the third degree Taylor polynomial, centered at :
If , find the third degree Taylor polynomial for centered at .
To get , we need to find the antiderivative of by integrating the third degree polynomial term by term.
We only want up to a third degree polynomial, so we can disregard the fourth order term:
Since , substitute for the final .
Example Question #2 : Taylor Series
Consider the following series:
What is the evaluation of this series at ?
Recall the important McLaurin series for :
The series in this question is the same, only replacing x with .
So if we have , we need only evaluate at , and thus we obtain .
Example Question #1 : Maclaurin Series
Write the first three terms of the Taylor series for the following function about :
The general form for the Taylor series (of a function f(x)) about x=a is the following:
Because we only want the first three terms, we simply plug in a=1, and then n=0, 1, and 2 for the first three terms (starting at n=0).
The hardest part, then, is finding the zeroth, first, and second derivative of the function given:
The derivative was found using the following rules:
Then, simply plug in the remaining information and write out the terms:
Example Question #3 : Taylor Series
Write out the first four terms of the Taylor series about for the following function:
The Taylor series about x=a of any function is given by the following:
So, we must find the zeroth, first, second, and third derivatives of the function (for n=0, 1, 2, and 3 which makes the first four terms):
The derivatives were found using the following rule:
Now, evaluated at x=a=1, and plugging in the correct n where appropriate, we get the following:
which when simplified is equal to
Example Question #11 : Applications Of Derivatives
Find the first two terms of the Taylor series about for the following function:
The general formula for the Taylor series about x=a for a function is
First, we must find the zeroth and first derivative of the function.
The zeroth derivative of a function is just the function itself, so we only have to find the first derivative:
The derivative was found using the following rule:
Now, write the first two terms of the sequence (n=0 and n=1):
Example Question #3 : Taylor's Theorem
Write out the first three terms of the Taylor series for the following function about :
The general formula for the Taylor series of a given function about x=a is
We were asked to find the first three terms, which correspond to n=0, 1, and 2. So first, we need to find the zeroth, first, and second derivative of the given function. The zeroth derivative is just the function itself.
The derivatives were found using the following rules:
Now use the above formula to write out the first three terms:
Simplified, this becomes