AP Calculus BC : Maclaurin Series

Example Questions

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Example Question #1 : Taylor's Theorem

For which of the following functions can the Maclaurin series representation be expressed in four or fewer non-zero terms?

Explanation:

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 #1 : Maclaurin Series

Let

Find the the first three terms of the Taylor Series for  centered at .

Explanation:

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 : Maclaurin Series

Write the first three terms of the Taylor series for the following function about :

Explanation:

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 #1 : Maclaurin Series

Write out the first four terms of the Taylor series about  for the following function:

Explanation:

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 #2 : Maclaurin Series

Write out the first three terms of the Taylor series about  for the following function:

Explanation:

The general formula for the Taylor series about x=a for a given function is

We must find the zeroth, first, and second derivative of the function (for n=0, 1, and 2). The zeroth derivative is just the function itself.

The derivatives were found using the following rule:

Now, follow the above formula to write out the first three terms:

which simplified becomes

Example Question #3 : Maclaurin Series

Find the Taylor series expansion of  at .

Explanation:

The Taylor series is defined as

where the expression within the summation is the nth term of the Taylor polynomial.

To find the Taylor series expansion of  at , we first need to find an expression for the nth term of the Taylor polynomial.

The nth term of the Taylor polynomial is defined as

For,  and .

We need to find terms in the taylor polynomial until we can determine the pattern for the nth term.

Substituting these values into the taylor polynomial we get

The Taylor polynomial simplifies to

Now that we know the nth term of the Taylor polynomial, we can find the Taylor series.

The Taylor series is defined as

where the expression within the summation is the nth term of the Taylor polynomial.

For this problem, the taylor series is

Use the above to calculate the taylor series expansion of  at ,

Example Question #4 : Maclaurin Series

Use the first 3 terms of the Maclaurin series to approximate

Explanation:

The Maclaurin series for sine is

Plugging in  for x gives:

Example Question #5 : Maclaurin Series

Use the first 4 terms of the Maclaurin series to approximate

Explanation:

The Maclaurin series for  is

Plugging in 0.78 for x gives:

Example Question #6 : Maclaurin Series

Use the first four terms of the Maclaurin series to approximate  to 6 decimal places.

Explanation:

The Maclaurin series for  is

Plugging in x = 0.38 gives: