Matching Derivatives

The Taylor recipe

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Match the value, slope, curvature and higher derivatives at a point.
• Taylor coefficients: $c_n = f^{(n)}(0)/n!$.
• $\cos x \approx 1 - \tfrac{x^2}{2}$ and $\sin x \approx x - \tfrac{x^3}{6}$ near 0.

Steps

1. Add some bend
2. Cosine’s parabola
3. Use it
4. Match the curve
5. A quadratic for eˣ
6. The x³ coefficient