10.11 Finding Taylor Polynomial Approximations of Functions

Welcome to AP Calc 10.11! In this lesson, you’ll learn how to approximate a function over at a point.

📈 Taylor Approximations Theorem

This theorem states that for a function $f(x)$, it’s Taylor series approximation at $x=a$ is…

This can be rewritten as…

where $f^{(n)}(a)$ is the $n^{\text{th}}$ deriviative of the function and $f^{(0)}(a)=f(x)$. The $n^{\text{th}}$-order Taylor polynomial is the $n^{\text{th}}$ partial sum of the infinite series.

Taylor series centered at $x=0$ are common and are called Maclaurin series.

🧱 Breaking Down the Theorem

Taylor series look very daunting when you first approach them. Let’s define each portion and build a table that will help you tackle problems of this type!

Now, let’s try a practice problem using this table to walk through it step by step.

✏️ Applying the Theorem

Find the third-degree Maclaurin polynomial for $e^{5x}$.

Solution: First, let’s build our table. Remember that a Maclaurin series is just a Taylor series where $a=0$!

Now, we just put the terms in our final column together as a full formula. The third-degree Maclaurin polynomial for $e^{5x}$ is:

📝 Practice

Now it’s your turn to apply what you’ve learned!

❓Problems

1. Find the fifth-degree Maclaurin polynomial for $f(x)=\text{cos}(x)$.
2. Find the third-degree Taylor polynomial for $f(x)=\text{ln}(x)$ about $x=1$.
3. Find the fourth-degree Taylor polynomial about $x=2$ for $f(x)=\sqrt{x}$.

💡 Solution for Question 1

Start by building your table and filling in the values:

Putting it all together, we get that the fifth-degree Maclaurin polynomial for $f(x)=\text{cos}(x)$ is

💡 Solution for Question 2

Keep on building your tables! This time, our $(x-a)^n$ column will be a bit more complicated.

We then find the polynomial to be equal to:

💡 Solution for Question 3

One more table!

If we put this all together, we get:

We can simplify a few of these terms a bit more using exponent rules to get:

This is the fourth-degree Taylor polynomial centered at $x=2$ for $\sqrt{2}$.

💫 Closing

Great work! Taylor polynomials may seem daunting at first, but when in doubt, break it down with a table and you’ll be sure to master them!