Taylor polynomials centered at a ≠ 0

a. Find the nth-order Taylor polynomials for the given function centered at the given point a for n = 0, 1, and 2.

b. Graph the Taylor polynomials and the Junction.

f(x)= cos x, a = π/6

Solution 28E

Step 1:

In this problem we have to find the nth-order Taylor polynomials for cos x centered at the point for n = 0, 1, and 2

Taylor series is given by

Let us first find order Taylor polynomial.

In our case,

That is

So, what we need to do to get desired polynomial is to calculate derivatives, evaluate them at the given point and plug results into given formula.

Then

Now use calculated values in to get the polynomial.

Thus Taylor polynomial of upto with center is

Step 2:

Now let us first find order Taylor polynomial.

In our case,

That is

So, what we need to do to get desired polynomial is to calculate derivatives, evaluate them at the given point and plug results into given formula.

Then

Then

Now use calculated values in to get the polynomial.

Thus Taylor polynomial of upto with center is

Step 2:

Now let us first find order Taylor polynomial.

In our case,

That is

So, what we need to do to get desired polynomial is to calculate derivatives, evaluate them at the given point and plug results into given formula.

Then

Then

Then

Now use calculated values in to get the polynomial.

Thus Taylor polynomial of upto with center is

Step 3:

b. Graph the Taylor polynomials and the function.

The taylor polynomial of upto with center is and its graph is given below