# Local linearity

Learning objectives

• Use differentials to make linear approximations:
• $\Delta f \approx f_x \Delta x + f_y \Delta y$ for small $\Delta x$ and $\Delta y$
• Use gradients to make linear approximations:
• $\Delta f \approx (\hbox{grad }f) \cdot \Delta \vec r$ for small $\Delta \vec r = \Delta x \vec i + \Delta y \vec j$
• Use the gradient to compute directional derivatives
• Understand the geometry of the gradient

Before class

• Read Hughes-Hallett sections 14.3, 14.4, 14.5.
• Ponder Hughes-Hallett problems: 14.3: 7, 17, 18, 24, 26; 14.4: 1, 2, 7, 17, 23, 32-3; 14.5: 33, 49.

In class