A.5 Summary

Things you should know

• A multivariable function takes a vector of inputs and returns a real number as output.
• A level set for a multivariable function is the set of all inputs that yields the same output: for example, ${(x,y) | f(x,y) = \overline z}$ for some constant $z$.
• A contour map for a multivariable function illustrates a series of level sets
• A partial derivative for a multivariable function describes how the value of the function changes as one of the input variables increases, holding all other input variables constant.
• The chain rule states that if $h(x) = f(g(x))$, \(\frac{dh}{dx} = \frac{df}{dg} \times \frac{dg}{dx}\) This is important in economics for chains of causality: i.e., situations where $x$ affects $y$, which in turn affects $z$.
• The total derivative of a function $f(x,y)$ when moving along a path described by $y(x)$ is given by \(\left.{\partial f \over \partial x}\right|_{y = y(x)} = {\partial f \over \partial x} + {\partial f \over \partial y} \times {dy \over dx}\)
• The implicit function theorem states that the slope of a level set $f(x,y) = \overline z$ is given by \(\left.{dy \over dx}\right|_{f(x,y) = z} = - {\partial f/\partial x \over \partial f/\partial y}\)

Things you should be able to do

For any multivariable function of two variables $f(x,y)$, you should be able to:

• Given a value $z$, plot the level curve $f(x,y) = z$
• Create a contour map of the function by plotting the level curves for various values of $z$
• Calculate the partial derivatives with respect to each of the input variables, $\partial f/\partial x$ and $\partial f/\partial y$
• Calculate the slope along a level set, in general and at a specific point