# ๐งฎ Lecture 15 Animations

To start an animation, first click the โโฏ๏ธ Stop animationโ button and then click the โโถ๏ธ Start animationโ button.

### Derivatives and Tangent Lines

**Key ideas**:

- \(\frac{df}{dt}(t)\) is the slope of the tangent line to \(f\) at the point \((t, f(t))\).
- When the slope of the tangent line is negative, increasing \(t\) brings you closer to a minimum.
- When the slope of the tangent line is positive, increasing \(t\) brings you further away from a minimum.
- The closer \(t\) is to a minimum, the shallower the slope of the tangent line is โ at a minimum, the slope of the tangent line is 0!

### Gradient Descent

**Key idea**: Depending on (1) the nature of \(f\), (2) the initial guess, and (3) the learning rate, gradient descent may or may not converge to a global minimum!