Unit 2 Introduction

In this unit we will learn about derivatives of functions of several variables. Conceptually these derivatives are similar to those for functions of a single variable.

1. They measure rates of change.
2. They are used in approximation formulas.
3. They help identify local maxima and minima.

As you learn about partial derivatives you should keep the first point, that all derivatives measure rates of change, firmly in mind. Said differently, derivatives are limits of ratios. For example,

Of course, we’ll explain what the pieces of each of these ratios represent.

Although conceptually similar to derivatives of a single variable, the uses, rules and equations for multivariable derivatives can be more complicated. To help us understand and organize everything our two main tools will be the tangent approximation formula and the gradient vector.

Our main application in this unit will be solving optimization problems, that is, solving problems about finding maxima and minima. We will do this in both unconstrained and constrained settings.

Part A

We start this unit by learning to visualize functions of several variables using graphs and level curves. Following this we will study partial derivatives. These will be used in the tangent approximation formula, which is one of the keys to multivariable calculus. It ties together the geometric and algebraic sides of the subject and is the higher dimensional analog of the equation for the tangent line found in single variable calculus. We will use it in part B to develop the chain rule.

We will apply our understanding of partial derivatives to solving unconstrained optimization problems. (In part C we will solve constrained optimization problems.)

» Session 24: Functions of Two Variables: Graphs
» Session 25: Level Curves and Contour Plots
» Session 26: Partial Derivatives
» Session 27: Approximation Formula
» Session 28: Optimization Problems
» Session 29: Least Squares
» Session 30: Second Derivative Test
» Session 31: Example
» Problem Set 4