Unit 10.11 – Finding Taylor Polynomial Approximations BC ONLY

AP® Calculus BC | Approximating Functions with Polynomials

Why This Matters: Taylor polynomials are one of the most powerful tools in calculus! They let us approximate complicated functions (like \(e^x\), \(\sin x\), \(\ln(1+x)\)) using simple polynomials. This is how calculators compute transcendental functions and how physicists approximate complex systems. Master this and you unlock the bridge between calculus and series!

🎯 Taylor Polynomial Formula

Taylor Polynomial (General Form)

THE FORMULA:

The n-th degree Taylor polynomial for \(f(x)\) centered at \(x = a\) is:

\[ P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n \]

Compact form:

\[ P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k \]

Components:

\(a\) = center point of approximation
\(n\) = degree of polynomial
\(f^{(k)}(a)\) = k-th derivative evaluated at \(a\)
\(k!\) = k factorial
\((x-a)^k\) = powers of \((x-a)\)

⭐ Maclaurin Polynomial (Special Case)

Maclaurin Polynomial (Centered at a = 0)

WHEN a = 0:

\[ P_n(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots + \frac{f^{(n)}(0)}{n!}x^n \]

Compact form:

\[ P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(0)}{k!}x^k \]

📝 Key Point: Maclaurin polynomials are just Taylor polynomials centered at 0. They're much simpler because \((x-a)^k\) becomes \(x^k\)!

📋 How to Find Taylor Polynomials

Step-by-Step Process:

Find derivatives: Compute \(f(x), f'(x), f''(x), \ldots, f^{(n)}(x)\)
Evaluate at center: Find \(f(a), f'(a), f''(a), \ldots, f^{(n)}(a)\)
Build polynomial: Use formula with factorials
Simplify: Combine like terms if needed

💡 Pro Tip: For Maclaurin (a=0), just evaluate all derivatives at 0. For other centers, substitute x=a into each derivative.

📚 Common Maclaurin Series (MEMORIZE!)

Essential Series to Know

1. Exponential Function:

\[ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!} \]

2. Sine Function:

\[ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} \]

(Only odd powers!)

3. Cosine Function:

\[ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} \]

(Only even powers!)

4. Geometric Series:

\[ \frac{1}{1-x} = 1 + x + x^2 + x^3 + x^4 + \cdots = \sum_{n=0}^{\infty} x^n \quad (|x| < 1) \]

5. Natural Logarithm:

\[ \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n} \quad (|x| < 1) \]

6. Binomial Series (for any r):

\[ (1+x)^r = 1 + rx + \frac{r(r-1)}{2!}x^2 + \frac{r(r-1)(r-2)}{3!}x^3 + \cdots \]

📖 Comprehensive Worked Examples

Example 1: Maclaurin Polynomial for \(e^x\)

Problem: Find the 4th degree Maclaurin polynomial for \(f(x) = e^x\).

Solution:

Step 1: Find derivatives

\(f(x) = e^x\)
\(f'(x) = e^x\)
\(f''(x) = e^x\)
\(f'''(x) = e^x\)
\(f^{(4)}(x) = e^x\)

Step 2: Evaluate at a = 0

\(f(0) = e^0 = 1\)
\(f'(0) = 1\)
\(f''(0) = 1\)
\(f'''(0) = 1\)
\(f^{(4)}(0) = 1\)

Step 3: Build polynomial

\[ P_4(x) = 1 + x + \frac{1}{2!}x^2 + \frac{1}{3!}x^3 + \frac{1}{4!}x^4 \]

\[ = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} \]

Example 2: Maclaurin Polynomial for \(\sin x\)

Problem: Find the 5th degree Maclaurin polynomial for \(f(x) = \sin x\).

Find derivatives and evaluate at 0:

\(f(x) = \sin x\), \(f(0) = 0\)
\(f'(x) = \cos x\), \(f'(0) = 1\)
\(f''(x) = -\sin x\), \(f''(0) = 0\)
\(f'''(x) = -\cos x\), \(f'''(0) = -1\)
\(f^{(4)}(x) = \sin x\), \(f^{(4)}(0) = 0\)
\(f^{(5)}(x) = \cos x\), \(f^{(5)}(0) = 1\)

Build polynomial:

\[ P_5(x) = 0 + 1 \cdot x + 0 + \frac{-1}{3!}x^3 + 0 + \frac{1}{5!}x^5 \]

\[ = x - \frac{x^3}{6} + \frac{x^5}{120} \]

📝 Pattern: Notice only odd powers appear! Even derivatives at 0 are all zero.

Example 3: Taylor Polynomial Centered at a ≠ 0

Problem: Find the 3rd degree Taylor polynomial for \(f(x) = \ln x\) centered at \(a = 1\).

Find derivatives:

\(f(x) = \ln x\), \(f(1) = 0\)
\(f'(x) = \frac{1}{x}\), \(f'(1) = 1\)
\(f''(x) = -\frac{1}{x^2}\), \(f''(1) = -1\)
\(f'''(x) = \frac{2}{x^3}\), \(f'''(1) = 2\)

Build polynomial (note: centered at a=1):

\[ P_3(x) = 0 + 1(x-1) + \frac{-1}{2!}(x-1)^2 + \frac{2}{3!}(x-1)^3 \]

\[ = (x-1) - \frac{(x-1)^2}{2} + \frac{(x-1)^3}{3} \]

Example 4: Using Known Series

Problem: Find the 4th degree Maclaurin polynomial for \(f(x) = e^{2x}\).

Use known series for \(e^x\):

\[ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots \]

Substitute \(2x\) for \(x\):

\[ e^{2x} = 1 + (2x) + \frac{(2x)^2}{2!} + \frac{(2x)^3}{3!} + \frac{(2x)^4}{4!} \]

\[ = 1 + 2x + \frac{4x^2}{2} + \frac{8x^3}{6} + \frac{16x^4}{24} \]

\[ = 1 + 2x + 2x^2 + \frac{4x^3}{3} + \frac{2x^4}{3} \]

Example 5: Combining Series

Problem: Find the 4th degree Maclaurin polynomial for \(f(x) = e^x \cos x\).

Use known series:

\(e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \cdots\)

\(\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} + \cdots\)

Multiply (keep terms up to \(x^4\)):

\[ e^x \cos x = \left(1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24}\right)\left(1 - \frac{x^2}{2} + \frac{x^4}{24}\right) \]

After multiplication and collecting terms:

\[ = 1 + x + 0 \cdot x^2 - \frac{x^3}{3} - \frac{x^4}{6} \]

\[ = 1 + x - \frac{x^3}{3} - \frac{x^4}{6} \]

📊 Quick Reference Table

Common Taylor/Maclaurin Polynomials
Function	Center	Series (first few terms)
\(e^x\)	0	\(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\)
\(\sin x\)	0	\(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\)
\(\cos x\)	0	\(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\)
\(\frac{1}{1-x}\)	0	\(1 + x + x^2 + x^3 + \cdots\)
\(\ln(1+x)\)	0	\(x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots\)

💡 Essential Tips & Strategies

✅ Success Strategies:

Memorize common series: \(e^x\), \(\sin x\), \(\cos x\), geometric, \(\ln(1+x)\)
Use substitution: For \(e^{2x}\), substitute into \(e^x\) series
Recognize patterns: Sine has odd powers, cosine has even
Check factorials: Denominator is always \(n!\)
Center matters: \((x-a)\) for general, \(x\) for Maclaurin
Degree = highest power: 3rd degree means up to \(x^3\)
Show work: Write out derivatives step by step
Simplify fractions: \(\frac{1}{2!} = \frac{1}{2}\), not just "2!"

🔥 Substitution Tricks:

For \(e^{-x}\): Replace \(x\) with \(-x\) in \(e^x\) series
For \(\sin(2x)\): Replace \(x\) with \(2x\) in \(\sin x\) series
For \(\cos(x^2)\): Replace \(x\) with \(x^2\) in \(\cos x\) series
For \(\frac{1}{1+x}\): Replace \(x\) with \(-x\) in \(\frac{1}{1-x}\)

❌ Common Mistakes to Avoid

Mistake 1: Forgetting factorial in denominator
Mistake 2: Wrong center (using \(x\) instead of \((x-a)\))
Mistake 3: Not simplifying factorials (\(\frac{1}{2!} = \frac{1}{2}\))
Mistake 4: Mixing up sine (odd) and cosine (even) patterns
Mistake 5: Wrong signs in alternating series
Mistake 6: Computing too few or too many terms
Mistake 7: Not showing derivative calculations
Mistake 8: Evaluating derivative at wrong point
Mistake 9: Arithmetic errors when substituting
Mistake 10: Forgetting \(0! = 1\) for constant term

📝 Practice Problems

Find the indicated Taylor/Maclaurin polynomial:

3rd degree Maclaurin polynomial for \(f(x) = e^{-x}\)
4th degree Maclaurin polynomial for \(f(x) = \cos(2x)\)
3rd degree Taylor polynomial for \(f(x) = \sqrt{x}\) at \(a=4\)
5th degree Maclaurin polynomial for \(f(x) = \ln(1-x)\)
4th degree Maclaurin polynomial for \(f(x) = \frac{x}{1+x^2}\)

Answers:

\(1 - x + \frac{x^2}{2} - \frac{x^3}{6}\)
\(1 - 2x^2 + \frac{2x^4}{3}\)
\(2 + \frac{1}{4}(x-4) - \frac{1}{64}(x-4)^2 + \frac{1}{512}(x-4)^3\)
\(-x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} - \frac{x^5}{5}\)
\(x - x^3\)

✏️ AP® Exam Success Tips

What AP® Graders Look For:

Show derivatives: Write \(f(x), f'(x), f''(x),\) etc.
Evaluate at center: Show \(f(a), f'(a), f''(a),\) etc.
Write formula explicitly: Use factorial notation
Include all terms: Up to requested degree
Simplify: Reduce fractions
Use correct notation: \((x-a)^n\) for center \(a\)
State degree: "3rd degree polynomial" or "\(P_3(x)\)"
Box final answer: Make it clear

💯 Exam Strategy:

Identify if Maclaurin (a=0) or Taylor (a≠0)
Check if you can use a known series
If yes: substitute and collect terms
If no: compute derivatives and evaluate
Build polynomial term by term
Simplify each term
Check degree matches requirement
State final answer clearly

⚡ Quick Reference Guide

TAYLOR POLYNOMIAL ESSENTIALS

General Taylor Polynomial:

\[ P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k \]

Maclaurin (a = 0):

\[ P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(0)}{k!}x^k \]

Must Know Series:

\(e^x = \sum \frac{x^n}{n!}\)
\(\sin x = \sum \frac{(-1)^n x^{2n+1}}{(2n+1)!}\)
\(\cos x = \sum \frac{(-1)^n x^{2n}}{(2n)!}\)
\(\frac{1}{1-x} = \sum x^n\)

Remember:

Factorial denominators: \(k!\)
Powers of \((x-a)\) for center \(a\)!
Degree \(n\) = highest power!

Master Taylor Polynomials! The Taylor polynomial of degree n for \(f(x)\) at \(x=a\) is \(P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k\). Maclaurin is special case with \(a=0\). Process: compute derivatives, evaluate at center, build polynomial with factorials. Must memorize: \(e^x = \sum \frac{x^n}{n!}\); \(\sin x\) (odd powers only); \(\cos x\) (even powers only); \(\frac{1}{1-x} = \sum x^n\); \(\ln(1+x) = \sum \frac{(-1)^{n+1}x^n}{n}\). Key tricks: use substitution for \(e^{2x}\), \(\sin(x^2)\), etc.; multiply series for products; sine has odd powers, cosine has even. Common errors: forgetting factorials, wrong center \((x-a)\) vs \(x\), sign errors in alternating. Each term is \(\frac{f^{(k)}(a)}{k!}(x-a)^k\)—show all work! This is THE foundation for power series—master it! Appears heavily on BC exams! 🎯✨