Error Bounds (Taylor Polynomial Remainder) 

When we approximate a function using a Taylor polynomial, we want to know:

“How accurate is our approximation?”

To answer this, we use the Taylor Remainder (Error Bound).

What Is Error Bound?

When a function $f(x)$ is approximated by its n-degree Taylor polynomial $$P_{n}(x)$$:

$$f(x)=P_{n}(x)+R_{n}(x)$$

Where:

• $$P_{n}(x)$$ = Taylor polynomial of degree $n$

• $$R_{n}(x)$$ = remainder (error term)

The goal is to find an upper bound for $$|R_{n}(x)|$$.

Lagrange Error Bound Formula

The most important error bound formula for AP BC:

$$|R_{n}(x)|=\left|\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}\right|$$​

For some number c between x and the center a.

actually use on AP exam:

Find a maximum value of $$|f^{(n+1)}(c)|$$ on the interval between $x$ and $a$.

Call this maximum value M.

Then:

$$|R_{n}(x)|\leq\frac{M}{(n+1)!}|x-a|^{n+1}$$

This is the Lagrange Error Bound.

Error Bound Process 

Step 1. Identify the function and Taylor degree

Given function f(x).
Taylor polynomial centered at $a$.
You are approximating at $x$.

Step 2. Find $(n+1)$-th derivative of the function

If you’re using $$P_{3}$$​, compute the 4th derivative.

Step 3. Find a bound M for $$|f^{(n+1)}(c)|$$

Find the maximum value of $$|f^{(n+1)}(c)|$$ on interval between interval: x and a

Step 4. Use formula

$$|R_{n}(x)|\leq\frac{M}{(n+1)!}|x-a|^{n+1}$$

Example 1 (Classic BC Exam)

Approximate $$e^{0.2}$$ using the 2nd-degree Maclaurin polynomial.

Then find an error bound.

Step 1: Taylor polynomial degree 2

Maclaurin series for $$e^{x}$$:

$$P_{2}(x)=1+x+\frac{x^{2}}{2}$$​

Approximation:

$$e^{0.2}=1+0.2+\frac{(0.2)^{2}}{2}=1.22$$

Step 2: Use Lagrange error bound

The $(n+1)$th derivative is $$f^{(3)}(x)=e^{x}$$.

Over the interval between 0 and 0.2:

Maximum of $$e^{x}=e^{0.2}$$.

Approximate maximum:

$$e^{0.2}<e^{0.3}<1.35$$

Thus choose

M=$$e^{0.2}\approx1.22$$

Step 3: Apply formula

Here, n=2 , a=0 , x=0.2:

$$|R_{2}(0.2)|$\leq\frac{M}{3!}|(0.2)^{3}|^{3}$$

$$=\frac{1.22}{6}(0.008)=0.00163$$

Final Answer:

$$e^{0.2}$\approx1.22$ with error < 0.00163

Example 2 (Sine Function)

Approximate $$\sin(0.3)$$ using the 3rd-degree Maclaurin polynomial and find an error bound.

Step 1: Maclaurin polynomial $$P_{3}$$​

$$P_{3}(x)=x-\frac{x^{3}}{6}$$​

Plug in:

$$\sin(0.3)\approx0.3-\frac{(0.3)^{3}}{6}=0.3-0.0045=0.2955$$

Step 2: Find a bound for the error

For sine, the 4th derivative is:

$$f^{(4)}(x)=\sin(x)$$

But absolute value of sine is bounded:

$$|\sin(x)|\leq1$$

Thus:

M=1

Step 3: Apply error formula

$$|R_{3}(0.3)|\leq\frac{1}{4!}(0.3)^{4}$$

Final Answer:

$$\sin(0.3)\approx0.2955$$ with error < 0.00034

When Do You Use Error Bounds in BC?

You use error bounds for:

• Taylor polynomial approximations

• Alternating series error estimates

• Proving accuracy of approximations

• FRQs requiring inequality justification