Unit 2: Differentiation in the Multivariable Context

This unit covers the theory, techniques, and applications of multivariable functions, limits, continuity, and differentiation. You will learn how to visualize graphs of functions of multiple variables (e.g., f(x,y)=x^2+y^2), how to think about limits and continuity in this setting, and how to extend what we know about derivatives from the single-variable context to this multivariable setting. As we'll find out, doing so will add new insights that were simply not available in the single-variable context. For example, when considering the limit of a single-variable function f(x), there are only two directions of approach: left or right. In the multivariable context, one can approach a point f(a,b) on the graph of a two-variable function f(x,y) via an infinite number of directions. This added richness (and complexity) means that differentiation -- which is defined in terms of limits -- also becomes a richer concept, along with everything that depends on differentiation (e.g., optimization). This unit guides you through all these new insights and generalizations.

Lesson 5 begins by discussing how to graph multivariable functions and, in particular, how to visualize the graph of f(x,y) in the plane by using the mathematician's analogue of a topographical map. The lesson then moves on to discussing limits and continuity.

Lesson 6 builds on that limits discussion and defines perhaps the simplest generalization of f'(x): partial derivatives. These are when we hold constant all but one variable in a multivariable function, effectively reducing the function to a single-variable function, and then differentiate using the derivative rules we already know from single-variable calculus. The lesson then explores this new "partial" world and ends with a discussion of the multivariable Chain Rule.

Lesson 7 generalizes the partial derivatives approach to produce "directional" derivatives, wherein one specifies the direction in which one wants to calculate instantaneous change in. This new concept is tied to, as we'll see in the lesson, the notion of a tangent plane, the generalization of tangent lines from single-variable calculus.

Lesson 8 generalizes the study of local and global extrema to the multivariable context, laying the foundation of optimization theory.

Finally, Lesson 9 discusses optimization theory, with a specific focus on real-world optimization problems. As in single-variable calculus, this is the topic with perhaps the most real-world applicability.

After completing this unit you will have learned how to graph multivariable functions, evaluate their limits, calculate their derivatives, and optimize them. The next unit is focused on integration in the multivariable context, and so will not use much of the context covered in this unit (with the exception of graphing multivariable functions). But we will rely on the differentiation content covered in this unit in Unit 4 when we discuss vector calculus.

This unit will also continue to introduce, reinforce, and emphasize mathematical modeling, particular in Lessons 8-9 in the context of optimization.