Short on time? This page is the recap. For the derivation, the graph and worked examples, use the full study guide. BC-only material. Prerequisites (higher derivatives 3.6, local linearity 4.6, factorials) are on the Calculus BC roadmap.
Recap
- A Taylor polynomial about x = a matches f and its first n derivatives at a.
- The tangent line is P₁. Each higher degree matches one more derivative, so the fit near a improves.
- Differentiating (x − a)ⁿ n times gives n!, which is why each coefficient divides by n!.
- Use Pₙ to estimate f(x) for x near a. Far from a, the estimate can be poor.
- In many cases Pₙ approaches f as n grows, but only on some interval (Topic 10.13).
Key relationships
| Idea | Formula or fact | Note |
|---|---|---|
| Coefficient | cₙ = f⁽ⁿ⁾(a)/n! | 0! = 1, 2! = 2, 3! = 6, 4! = 24 |
| Taylor polynomial | Pₙ(x) = Σ from k = 0 to n of f⁽ᵏ⁾(a)/k! · (x − a)ᵏ | Powers of (x − a), not x |
| Maclaurin polynomial | Centre a = 0 | Powers of x |
| Reading backwards | f⁽ᵏ⁾(a) = k! × coefficient of (x − a)ᵏ | Coefficient 4 on (x − a)² means f″(a) = 8 |
| cos x about 0 | P₄(x) = 1 − x²/2 + x⁴/24 | Odd coefficients are 0 |
| sin x about 0 | P₃(x) = x − x³/6 = P₄(x) | Degree 3 but only two nonzero terms |
Assumptions
- f has the derivatives you need at x = a.
- Trig functions use radians.
- “nth-degree” means the highest power is at most n; “first n nonzero terms” can need a higher degree.
Mistakes to avoid
- No factorial: writing f‴(a)(x − a)³ instead of f‴(a)/6 · (x − a)³.
- Wrong centre: using xᵏ when the centre is a ≠ 0, or reading (x + 3) as centre 3.
- Derivatives at the wrong point: always evaluate at the centre a, not at the x you are estimating.
- Coefficient read as derivative when working backwards.
- Trusting the estimate far from a.
Quick self-check
- Find the second-degree Maclaurin polynomial for e^(3x). (1 + 3x + (9/2)x²)
- Find the second-degree Taylor polynomial for sin x about x = π/2. (1 − ½(x − π/2)²)
- If f⁽⁴⁾(1) = 48, what is the coefficient of (x − 1)⁴? (48/24 = 2)
Next: practice questions.