Maths 4.5Lesson outlineLesson 5 of 9 in Module 4
Curvature: Second Derivatives, Taylor and Newton
Use the second derivative to see how a function bends, and step straight to the bottom of a local quadratic.
Video in production.
Learn
In this lesson11 parts
- 01Which way, but how far?
- 02The second derivative
- 03The Hessian
- 04Taylor polynomials
- 05Newton's step
- 06Where Newton fails
- 07Checkpoint: curvature so far
- 08The used car lot in one step
- 09Where this is used
- 10Newton's method in Python
- 11Recap, challenge, and next: integrals
Key terms
The words this lesson introduces, each in one line. The module’s glossary collects them all.
- second derivative
- the derivative of the derivative,
f″(x)|x³ − 3x → 6x - curvature
- how sharply a curve bends; its sign says up or down |
+6at 1,−6at−1 - Hessian
- the matrix of all second partial derivatives |
diag(2, 6)forx² + 3y² - Taylor approximation
- a polynomial built from a function's derivatives at one point, matching it nearby |
1 + x + 1.5x²forx² + eˣat 0 - Newton's method
- repeatedly jump to the minimum of the local parabola,
x − f′/f″| three steps to−0.351734
Quiz
Your first pick on each question is the one that counts, and a right one earns a coin. Getting one wrong here is how the lesson sticks.