Maclaurin Series for exponential and trigonometric functions - AP Calculus BC

The Taylor series about x=a for a function is given by

For the first three terms (n=0, 1, 2) we must find the zeroth, first, and second derivative of the function. The zeroth derivative is just the function itself.

The derivatives were found using the following rules:

,

Now, using the above formula, write out the first three terms:

which simplified becomes

The formula for an i-th degree Maclaurin Polynomial is

For the fifth degree polynomial, we must evaluate the function up to its fifth derivative.

The summation becomes

And substituting for the values of the function and the first five derivative values, we get the Maclaurin Polynomial

The Taylor series about x=a for a function is given by

For the first three terms (n=0, 1, 2) we must find the zeroth, first, and second derivative of the function. The zeroth derivative is just the function itself.

The derivatives were found using the following rules:

,

Now, using the above formula, write out the first three terms:

which simplified becomes

The formula for an i-th degree Maclaurin Polynomial is

For the fifth degree polynomial, we must evaluate the function up to its fifth derivative.

The summation becomes

And substituting for the values of the function and the first five derivative values, we get the Maclaurin Polynomial

The Taylor series about x=a for a function is given by

For the first three terms (n=0, 1, 2) we must find the zeroth, first, and second derivative of the function. The zeroth derivative is just the function itself.

The derivatives were found using the following rules:

,

Now, using the above formula, write out the first three terms:

which simplified becomes

The formula for an i-th degree Maclaurin Polynomial is

For the fifth degree polynomial, we must evaluate the function up to its fifth derivative.

The summation becomes

And substituting for the values of the function and the first five derivative values, we get the Maclaurin Polynomial

The Taylor series about x=a for a function is given by

For the first three terms (n=0, 1, 2) we must find the zeroth, first, and second derivative of the function. The zeroth derivative is just the function itself.

The derivatives were found using the following rules:

,

Now, using the above formula, write out the first three terms:

which simplified becomes

The formula for an i-th degree Maclaurin Polynomial is

For the fifth degree polynomial, we must evaluate the function up to its fifth derivative.

The summation becomes

And substituting for the values of the function and the first five derivative values, we get the Maclaurin Polynomial

The Taylor series about x=a for a function is given by

For the first three terms (n=0, 1, 2) we must find the zeroth, first, and second derivative of the function. The zeroth derivative is just the function itself.

The derivatives were found using the following rules:

,

Now, using the above formula, write out the first three terms:

which simplified becomes

The formula for an i-th degree Maclaurin Polynomial is

For the fifth degree polynomial, we must evaluate the function up to its fifth derivative.

The summation becomes

And substituting for the values of the function and the first five derivative values, we get the Maclaurin Polynomial

The Taylor series about x=a for a function is given by

For the first three terms (n=0, 1, 2) we must find the zeroth, first, and second derivative of the function. The zeroth derivative is just the function itself.

The derivatives were found using the following rules:

,

Now, using the above formula, write out the first three terms:

which simplified becomes

The formula for an i-th degree Maclaurin Polynomial is

For the fifth degree polynomial, we must evaluate the function up to its fifth derivative.

The summation becomes

And substituting for the values of the function and the first five derivative values, we get the Maclaurin Polynomial

The Taylor series about x=a for a function is given by

For the first three terms (n=0, 1, 2) we must find the zeroth, first, and second derivative of the function. The zeroth derivative is just the function itself.

The derivatives were found using the following rules:

,

Now, using the above formula, write out the first three terms:

which simplified becomes

The formula for an i-th degree Maclaurin Polynomial is

For the fifth degree polynomial, we must evaluate the function up to its fifth derivative.

The summation becomes

And substituting for the values of the function and the first five derivative values, we get the Maclaurin Polynomial

The Taylor series about x=a for a function is given by

For the first three terms (n=0, 1, 2) we must find the zeroth, first, and second derivative of the function. The zeroth derivative is just the function itself.

The derivatives were found using the following rules:

,

Now, using the above formula, write out the first three terms:

which simplified becomes

The formula for an i-th degree Maclaurin Polynomial is

For the fifth degree polynomial, we must evaluate the function up to its fifth derivative.

The summation becomes

And substituting for the values of the function and the first five derivative values, we get the Maclaurin Polynomial

The Taylor series about x=a for a function is given by

For the first three terms (n=0, 1, 2) we must find the zeroth, first, and second derivative of the function. The zeroth derivative is just the function itself.

The derivatives were found using the following rules:

,

Now, using the above formula, write out the first three terms:

which simplified becomes

The formula for an i-th degree Maclaurin Polynomial is

For the fifth degree polynomial, we must evaluate the function up to its fifth derivative.

The summation becomes

And substituting for the values of the function and the first five derivative values, we get the Maclaurin Polynomial

The Taylor series about x=a for a function is given by

For the first three terms (n=0, 1, 2) we must find the zeroth, first, and second derivative of the function. The zeroth derivative is just the function itself.

The derivatives were found using the following rules:

,

Now, using the above formula, write out the first three terms:

which simplified becomes