**3.1.1. Functions of several variables**

An introduction to **multivariate functions** and **level curves** \(\mathscr{L}_k = \{(x,y) \mid f(x,y)=k\}\)

You should be able to

- understand what a multivariate function is
- evaluate a multivariate function at a point
- draw level curves of a function of two variables
- compute first order partial derivatives
- compute higher order direct and mixed partial derivatives
- understand how many partial derivatives of each order a multivariate function has

Our recommendation is to:

**Test your ability by trying the diagnostic quiz**. You can reattempt it as many times as you want and can leave it part-way through.**Identify which topics need more work.**Make a note of any areas that you are unable to complete in the diagnostic quiz, or areas that you don't feel comfortable with.**Watch the tutorials for these topics,**you can find these below. Then try some practice questions from the mini quiz for that specific topic. If you have questions, please ask on the forum.**Re-try the diagnostic quiz.**And repeat the above steps as necessary until you can pass the diagnostic test. A pass grade is 80%.

Optionally, also see if you can **apply these skills to an economic application **or look at the **additional resources** and **advanced quizzes **at the bottom of the page,

An introduction to **multivariate functions** and **level curves** \(\mathscr{L}_k = \{(x,y) \mid f(x,y)=k\}\)

\( \frac{\partial f}{\partial x} (a,b) =\left. \frac{\partial }{\partial x} (f) \right|_{(x,y)=(a,b)} = f'_x(a,b) =\left.f_x(x,y)\right|_{(a,b)} = f'_1(a,b) \)

**direct****: **\( \frac{\partial^2 f}{\partial x^2} =\frac{\partial }{\partial x} \left(\frac{\partial }{\partial x} (f) \right)= f''_{xx}\), **mixed: **\(\frac{\partial^2 f}{\partial y\partial x} =\frac{\partial }{\partial y} \left(\frac{\partial }{\partial x} (f) \right)= f''_{xy} \)

In economics, production technology is usually modelled by the means of a **production function** - a (usually multivariate) function describing how quantities of *inputs *used are mapped into *output*. This application
introduces production functions as a modelling device, as well as the important concept of **isoquants **(contours of the production function) and **marginal products** (partial derivatives of the production function).

The following quiz allows you to test your understanding of production functions, isoquants and marginal products. In addition, the second question introduces two important classes of contour lines in the context of consumer theory -** indifference curves** and **budget lines**.

You can find **practice quizzes on each topic **in the links above the tutorial videos, or you can take the **diagnostic quiz **as many times as you like to. If you want to test yourself further, then try the **advanced quiz** linked below.