Lagrange Error Bound

Part 1 of 7 — The Formula

Taylor's Theorem with Remainder

$f(x) = P_n(x) + R_n(x)$

Lagrange Error Bound

|R_n(x)| leq rac{M}{(n+1)!}|x - a|^{n+1}

where $M = max|f^{(n+1)}(c)|$ for $c$ between $a$ and $x$.

This tells you: how good is your Taylor polynomial approximation?

Lagrange Error 🎯

Key Takeaways — Part 1

$|R_n| \leq \frac{M|x-a|^{n+1}}{(n+1)!}$ where $M$ bounds $|f^{(n+1)}|$.

Lagrange Error Bound

Part 2 of 7 — Finding $M$

Strategy for Bounding $M$

$M = max_{c ext{ between } a ext{ and } x} |f^{(n+1)}(c)|$

For common functions:

Finding $M$ 🎯

Key Takeaways — Part 2

$\sin x$ and $\cos x$ always have $M = 1$. This makes error bounds very clean.

Lagrange Error Bound

Part 3 of 7 — How Many Terms Do We Need?

Determining Degree for Desired Accuracy

"How many terms of $e^x$ centered at $0$ are needed for $|R_n(1)| < 0.001$?"

Need: rac{M}{(n+1)!} < 0.001 where $M leq 3$ (bound for $e^c$ on $[0,1]$).

rac{3}{(n+1)!} < 0.001

$(n+1)! > 3000$

$7! = 5040 > 3000$ ✓

So $n = 6$ (use $P_6$, i.e., 7 terms).

How Many Terms? 🎯

Key Takeaways — Part 3

Solve $(n+1)! > M/\epsilon$ to find the required degree.

Lagrange Error

Part 4 of 7 — Alternating Series Error vs Lagrange

When to Use Which?

Key Insight

For alternating Taylor series (like $\sin x$, $\cos x$, $e^{-x}$), the alternating series error bound is often easier and tighter.

AS vs Lagrange 🎯

Key Takeaways — Part 4

AS bound is simpler for alternating Taylor series. Lagrange works for all Taylor polys.

Lagrange Error

Part 5 of 7 — AP FRQ Practice

Typical AP Question Format

"Let $f$ be a function with $|f^{(4)}(x)| \leq 6$ for all $x$ in $[-1, 1]$. Show that the error of $P_3(x)$ at $x = 0.5$ is less than $0.02$."

Solution: $|R_3(0.5)| \leq \frac{6}{4!}(0.5)^4 = \frac{6}{24} \cdot \frac{1}{16} = \frac{1}{64} \approx 0.0156 < 0.02$ ✓

AP Practice 🎯

Key Takeaways — Part 5

Show your formula, substitute values, simplify. State conclusion clearly.

Lagrange Error

Part 6 of 7 — Practice Workshop

Error Bound Practice 🎯

Workshop Complete!

Lagrange Error Bound — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Lagrange Error Bound — Complete! ✅