Topic 10.11 Notes – Finding Taylor Polynomial Approximations of Functions

What a Taylor polynomial is

A Taylor polynomial is a polynomial that matches a function’s value and derivatives at a specific point $x = a$.

It’s built from the derivatives of $f$ evaluated at $a$.

The general form is:

$$T_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k$$

Written out:

$$f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \dots + \frac{f^{(n)}(a)}{n!}(x-a)^n$$

Lock these in:

• The coefficient of $(x-a)^n$ is $\frac{f^{(n)}(a)}{n!}$.
• $f^{(0)}(a) = f(a)$.
• A Maclaurin polynomial is just centered at $a=0$.
• Higher degree → matches more derivatives → usually better approximation near $a$.

Think of it as forcing a polynomial to have the same value, slope, concavity, and higher-order behavior as the function at one point.

Here’s the visual idea:

Maclaurin polynomials for $\sin x$ (degrees 1, 3, and 5)

Notice how they agree closely near 0 but drift away farther out. Taylor polynomials are local approximations.

How to build a Taylor polynomial

When you’re asked for, say, the 3rd-degree Taylor polynomial centered at $x=a$, the structure is always the same.

Step-by-step

1. Compute derivatives up to the 3rd derivative.
2. Evaluate each at $x=a$.
3. Plug into
   $$\frac{f^{(k)}(a)}{k!}(x-a)^k$$
4. Add terms from $k=0$ to $k=3$.

Quick example

Find the second-degree Taylor polynomial for $f(x)=\ln x$ centered at $a=2$.

Derivatives:

• $f(x)=\ln x$
• $f'(x)=1/x$
• $f''(x)=-1/x^2$

Evaluate at 2:

• $f(2)=\ln 2$
• $f'(2)=1/2$
• $f''(2)=-1/4$

Build the polynomial:

$$T_2(x) = \ln 2 + \frac{1}{2}(x-2) + \frac{-1/4}{2}(x-2)^2$$

$$T_2(x) = \ln 2 + \frac{1}{2}(x-2) - \frac{1}{8}(x-2)^2$$

Every coefficient came directly from $\frac{f^{(k)}(a)}{k!}$. That’s the core skill.

On tests, most errors happen from:

• Forgetting the factorial
• Using $x^n$ instead of $(x-a)^n$
• Stopping at the wrong degree

What improves as degree increases

Each added term forces another derivative to match at $x=a$.

So as $n$ increases:

• The graphs agree more tightly near $a$.
• The polynomial often stays accurate over a slightly larger interval.
• The approximation error typically shrinks near the center.

But this is always near $a$. Far away, even high-degree polynomials can behave wildly.

That idea connects to the larger theme of Unit 10: power series represent functions over certain intervals. A Taylor polynomial is just a finite “snapshot” of that full series.

Using a Taylor polynomial to approximate values

Taylor polynomials are used to approximate $f(b)$ when $b$ is close to $a$.

Example idea: suppose you built a third-degree Maclaurin polynomial for $e^x$:

$$T_3(x)=1+x+\frac{x^2}{2}+\frac{x^3}{6}$$

To approximate $e^{0.2}$, plug in $x=0.2$. That’s it.

Key idea:

• The closer $b$ is to $a$, the better the approximation.
• Higher degree usually improves accuracy.

On the AP exam, this often shows up as:

• “Use the third-degree Taylor polynomial to approximate…”
• Calculator section problems where you evaluate your polynomial numerically.
• FRQs where you must actually write the polynomial first, then substitute.