Module 05 / §4.1–4.7
Calculus & optimization
Understand gradients, curvature, the chain rule, and local minima.
8 lessons · Visual explanations · Guided experiments
05.1 / Explore → understand → practice
Where are you allowed to look for a minimum?
Distinguish feasible, local, global, and strict minima.
05.2 / Explore → understand → practice
Which direction changes the value fastest?
Read the gradient as a column of partial derivatives and a local direction of steepest change.
05.3 / Explore → understand → practice
What linear map describes a nonlinear change nearby?
Match Jacobian rows to outputs and columns to inputs.
05.4 / Explore → understand → practice
How does curvature change a step?
Use the Hessian to measure directional curvature and compare gradient and Newton steps.
05.5 / Explore → understand → practice
Why does a quadratic gradient contain a transpose?
Derive linear and quadratic gradients, including nonsymmetric matrices.
05.6 / Explore → understand → practice
How does a change travel through a composition?
Multiply Jacobians in the correct order and propagate a scalar gradient.
05.7 / Explore → understand → practice
How accurate is the local picture?
Distinguish first-order, integral-gradient, and second-order Taylor formulas.
05.8 / Explore → understand → practice
When does a stationary point really minimize?
Prove necessary and sufficient local-minimum conditions and test their limits.