Unit 10.12 – Lagrange Error Bound BC ONLY
AP® Calculus BC | Bounding the Error in Taylor Approximations
Why This Matters: The Lagrange Error Bound (also called Taylor's Remainder Theorem) tells us the MAXIMUM possible error when approximating a function with a Taylor polynomial. Unlike the alternating series error bound, this works for ALL functions—not just alternating series! It's the ultimate tool for quantifying accuracy in approximations.
🎯 The Lagrange Error Bound Formula
Lagrange Error Bound (Taylor's Remainder)
If \(P_n(x)\) is the n-th degree Taylor polynomial for \(f(x)\) centered at \(a\), then the error (remainder) satisfies:
where \(M = \max |f^{(n+1)}(c)|\) for \(c\) between \(a\) and \(x\)
Components Explained:
- \(R_n(x)\) = remainder/error at \(x\)
- \(f(x)\) = actual function value
- \(P_n(x)\) = n-th degree Taylor polynomial
- \(M\) = maximum of \(|f^{(n+1)}(c)|\) on interval
- \(n+1\) = one more than polynomial degree
- \(a\) = center of Taylor polynomial
📐 Alternative Forms
for some \(c\) between \(a\) and \(x\)
where \(M = \max_{c \in [a,x]} |f^{(n+1)}(c)|\)
📝 Key Insight: We don't know the exact value of \(c\), so we find the MAXIMUM of \(|f^{(n+1)}(c)|\) on the entire interval to get an upper bound!
📋 How to Apply the Lagrange Error Bound
Step-by-Step Process:
- Identify n: Degree of Taylor polynomial used
- Find \(f^{(n+1)}(x)\): The next derivative
- Find M: Maximum of \(|f^{(n+1)}(c)|\) on interval \([a,x]\)
- Identify interval: From \(a\) (center) to \(x\) (evaluation point)
- Apply formula: \(|R_n(x)| \leq \frac{M|x-a|^{n+1}}{(n+1)!}\)
- Calculate: Get numerical bound
💡 Pro Tip: Finding M is the hardest part! Look for maximum absolute value of \(f^{(n+1)}\) on the interval. Use critical points, endpoints, or known bounds.
🔍 Strategies for Finding M
How to Find the Maximum M
Evaluate \(|f^{(n+1)}(a)|\) and \(|f^{(n+1)}(x)|\), use larger value
If \(f^{(n+1)}\) has max/min in \((a,x)\), check those too
- \(|\sin c| \leq 1\) and \(|\cos c| \leq 1\) for all \(c\)
- \(|e^c| \leq e^{\max(a,x)}\) for \(c \in [a,x]\)
- For increasing functions, max is at right endpoint
- For decreasing functions, max is at left endpoint
If unsure, use a slightly larger M—gives valid (though not tightest) bound
📖 Comprehensive Worked Examples
Example 1: Error Bound for \(e^x\)
Problem: Find the error bound when approximating \(e^{0.5}\) using the 3rd degree Maclaurin polynomial for \(e^x\).
Solution:
Step 1: Identify components
- Function: \(f(x) = e^x\)
- Polynomial degree: \(n = 3\)
- Center: \(a = 0\) (Maclaurin)
- Evaluation point: \(x = 0.5\)
Step 2: Find \(f^{(4)}(x)\)
\(f^{(4)}(x) = e^x\) (all derivatives of \(e^x\) are \(e^x\))
Step 3: Find M on \([0, 0.5]\)
\(|f^{(4)}(c)| = e^c\) is increasing
Maximum occurs at \(c = 0.5\)
\(M = e^{0.5} < e^1 = e \approx 2.718\)
For safety, use \(M = 2\) (or \(M = e\) for exact)
Step 4: Apply error bound formula
Error Bound: \(|R_3(0.5)| \leq 0.0052\)
Example 2: Error Bound for \(\sin x\)
Problem: When approximating \(\sin(0.3)\) using \(P_5(x)\) (5th degree Maclaurin), what's the maximum error?
Setup:
- \(f(x) = \sin x\)
- \(n = 5\), so need \(f^{(6)}(x)\)
- \(a = 0\), \(x = 0.3\)
Find \(f^{(6)}(x)\):
Derivatives of sine cycle: \(\sin, \cos, -\sin, -\cos, \sin, \cos, -\sin, \ldots\)
\(f^{(6)}(x) = -\sin x\)
Find M:
\(|f^{(6)}(c)| = |-\sin c| = |\sin c| \leq 1\) for all \(c\)
So \(M = 1\)
Calculate error bound:
Error: \(|R_5(0.3)| \leq 0.000001\) (extremely accurate!)
Example 3: Finding Degree for Desired Accuracy
Problem: What degree Maclaurin polynomial for \(\cos x\) is needed to approximate \(\cos(0.5)\) with error < 0.001?
Set up inequality:
Need: \(|R_n(0.5)| \leq \frac{M(0.5)^{n+1}}{(n+1)!} < 0.001\)
All derivatives of \(\cos x\) are \(\pm \sin x\) or \(\pm \cos x\), so \(M = 1\)
Try different values of n:
- \(n = 2\): \(\frac{(0.5)^3}{3!} = \frac{0.125}{6} \approx 0.021\) (too big)
- \(n = 4\): \(\frac{(0.5)^5}{5!} = \frac{0.03125}{120} \approx 0.00026\) (works!)
Need \(n = 4\) (4th degree polynomial)
Example 4: Taylor Polynomial Centered at a ≠ 0
Problem: Using \(P_2(x)\) for \(\ln x\) centered at \(a = 1\), find the error bound for \(x = 1.2\).
Find derivatives:
- \(f(x) = \ln x\)
- \(f'(x) = \frac{1}{x}\)
- \(f''(x) = -\frac{1}{x^2}\)
- \(f'''(x) = \frac{2}{x^3}\)
Find M on \([1, 1.2]\):
\(|f'''(c)| = \left|\frac{2}{c^3}\right| = \frac{2}{c^3}\)
This decreases as \(c\) increases
Maximum at \(c = 1\): \(M = \frac{2}{1^3} = 2\)
Calculate:
Example 5: Comparison with Known Value
Problem: The actual value of \(e^{0.5} \approx 1.6487\). Verify the error bound from Example 1.
Calculate \(P_3(0.5)\):
Find actual error:
Compare to bound:
Our bound was 0.0052
Actual error 0.0029 < 0.0052 ✓
The bound works!
📊 Error Bounds for Common Functions
| Function | All Derivatives | Typical M |
|---|---|---|
| \(e^x\) | \(e^x\) | \(e^{\max(a,x)}\) |
| \(\sin x\) | \(\pm \sin x, \pm \cos x\) | \(M = 1\) |
| \(\cos x\) | \(\pm \sin x, \pm \cos x\) | \(M = 1\) |
| \(\ln(1+x)\) | \(\frac{(-1)^{n+1}(n-1)!}{(1+x)^n}\) | Depends on interval |
| \(\frac{1}{1-x}\) | \(\frac{n!}{(1-x)^{n+1}}\) | Depends on interval |
💡 Essential Tips & Strategies
✅ Success Strategies:
- Degree n uses (n+1)-th derivative: Key pattern!
- M is maximum, not value at c: Find max on entire interval
- For sine/cosine: M = 1: Always!
- For \(e^x\): M = e^{\text{larger endpoint}}\)
- Overestimate M if unsure: Still gives valid bound
- Check interval carefully: From \(a\) to \(x\)
- Factorial grows fast: Higher degree = much smaller error
- Show all steps: Find M, apply formula, calculate
🔥 Quick Checklist:
- Identify \(n\) (degree of polynomial)
- Find \(f^{(n+1)}(x)\)
- Determine interval \([a, x]\) or \([x, a]\)
- Find M = max of \(|f^{(n+1)}(c)|\) on interval
- Calculate \(\frac{M|x-a|^{n+1}}{(n+1)!}\)
- State conclusion clearly
❌ Common Mistakes to Avoid
- Mistake 1: Using n-th derivative instead of (n+1)-th
- Mistake 2: Forgetting absolute value on \(f^{(n+1)}(c)\)
- Mistake 3: Using value at specific \(c\) instead of maximum
- Mistake 4: Wrong factorial (\(n!\) instead of \((n+1)!\))
- Mistake 5: Wrong power (\((x-a)^n\) instead of \((x-a)^{n+1}\))
- Mistake 6: Not finding M on correct interval
- Mistake 7: Forgetting to take absolute value of error
- Mistake 8: Arithmetic errors in calculation
- Mistake 9: Using wrong center \(a\)
- Mistake 10: Not simplifying final answer
📝 Practice Problems
Find the error bound:
- Error in \(P_4(0.8)\) for \(e^x\) (Maclaurin)
- Error in \(P_3(0.1)\) for \(\sin x\) (Maclaurin)
- What degree needed for \(\cos(0.4)\) with error < 0.0001?
- Error in \(P_2(1.5)\) for \(\ln x\) centered at \(a=1\)
- For \(e^{-x}\), error in \(P_3(0.5)\) (Maclaurin)
Answers:
- \(|R_4| \leq \frac{e^{0.8}(0.8)^5}{5!} \approx 0.0098\)
- \(|R_3| \leq \frac{1 \cdot (0.1)^4}{4!} \approx 0.0000042\)
- Need \(n = 4\)
- \(|R_2| \leq \frac{2 \cdot (0.5)^3}{3!} = 0.0417\)
- \(|R_3| \leq \frac{1 \cdot (0.5)^4}{4!} \approx 0.0026\)
✏️ AP® Exam Success Tips
What AP® Graders Look For:
- State the formula: Write error bound formula explicitly
- Identify n clearly: "Using degree n polynomial..."
- Show (n+1)-th derivative: Write \(f^{(n+1)}(x) = \ldots\)
- Find and justify M: Show how you found maximum
- State interval: "On \([a, x]\), the maximum is..."
- Show calculation: Substitute into formula step-by-step
- Include absolute values: \(|x-a|^{n+1}\)
- Simplify answer: Give numerical value
💯 Exam Strategy:
- Read problem: identify function, degree, center, evaluation point
- Write: "Using Lagrange error bound..."
- Compute \(f^{(n+1)}(x)\)
- Find M on interval (justify!)
- Write formula: \(|R_n(x)| \leq \frac{M|x-a|^{n+1}}{(n+1)!}\)
- Substitute values
- Calculate (show work)
- State: "Error is at most [value]"
⚡ Quick Reference Guide
LAGRANGE ERROR BOUND ESSENTIALS
The Formula:
where \(M = \max |f^{(n+1)}(c)|\) on \([a,x]\)
Key Components:
- \(n\) = degree of polynomial
- \(n+1\) = derivative order AND power
- \(M\) = maximum of next derivative
- \((n+1)!\) = factorial of (n+1)
Common M Values:
- \(\sin x, \cos x\): \(M = 1\)
- \(e^x\): \(M = e^{\max}\)
- Always use MAX on interval!
Remember:
- Degree n uses (n+1)-th derivative!
- M = maximum, not just any value!
- Works for ALL functions!
Master the Lagrange Error Bound! The Lagrange Error Bound (Taylor's Remainder Theorem) states: when approximating \(f(x)\) with n-th degree Taylor polynomial \(P_n(x)\) centered at \(a\), the error satisfies \(|R_n(x)| = |f(x) - P_n(x)| \leq \frac{M|x-a|^{n+1}}{(n+1)!}\), where \(M = \max |f^{(n+1)}(c)|\) on interval between \(a\) and \(x\). Critical insight: use (n+1)-th derivative (one more than polynomial degree) and find MAXIMUM on interval, not just value at one point. For common functions: sine/cosine have \(M=1\); exponential uses \(M=e^{\text{max}}\). Process: (1) identify n, (2) find \(f^{(n+1)}\), (3) find M on interval, (4) apply formula. Unlike alternating series error (only for alternating), this works for ALL functions! Higher degree polynomials have much smaller errors (factorial in denominator grows fast). Essential for quantifying accuracy—appears on every BC exam! Show how you find M! 🎯✨