05 Functions of multiple variables, part 1

Exercise 1: A real estate function

The sale price, \(s\), and annual property tax, \(t\), (both measured in thousands of dollars) are jointly a function \((s,t) = f(q,b,a)\) of three variables:

  • \(q\), the square footage of the house in hundreds of feet,
  • \(b\), the number of bedrooms,
  • \(a\), the age of the house in years.

The formula for the function \(f\) has been determined to be

\[ f(q,b,a) = (100 + 2q+10b-a, 1+ 0.01q + 0.05). \]

  1. What is the sale price and annual property tax for a house with \(2000\) square feet, \(3\) bedrooms, and \(5\) years of age?

  2. Assuming (unrealisticly) that \(q\), \(b\), and \(a\) can take on any values, express the domain and codomain of the function \(f\) in the form \(f: \mathbb{R}^m \to \mathbb{R}^n\).

  3. If you could draw the graph of \(f\), in how many ambient dimensions would it be drawn?

Exercise 2: A manufacturing function

The number of units produced, \(u\), the total cost of production, \(c\) (in dollars), and the totel energy consumed, \(k\) (in kWH), are jointly a function \((u,c,k) = g(r,h,m,e)\) of four variables:

  • \(r\), the production rate in units per hour,
  • \(h\), the number of hours worked,
  • \(m\), the amount of raw materials used in kilograms,
  • \(e\), the equipment maintenance time in hours.

The formula for the function \(g\) has been determined to be

\[ g(r,h,m,e) = (r(h-e), 20h+5m+50e, 10h+5e). \]

  1. What is the number of units produced, total cost of production, and total energy consumed for a production rate of \(100\) units per hour, \(8\) hours worked, \(50\) kilograms of raw materials used, and \(2\) hours of equipment maintenance time?

  2. Assuming (unrealisticly) that \(r\), \(h\), \(m\), and \(e\) can take on any values, express the domain and codomain of the function \(g\) in the form \(g: \mathbb{R}^m \to \mathbb{R}^n\).

  3. If you could draw the graph of \(g\), in how many ambient dimensions would it be drawn?

Exercise 3: A pure mathematical function

Consider the function \(h:\mathbb{R}^4 \to \mathbb{R}^3\) defined by

\[ h(x,y,z,w) = \big(x^2y - \sin{z}, e^w - 15y, (z+w)^2\big). \]

  1. What is \(h(-1,0,2,3)\)?
  2. If you could draw the graph of \(h\), in how many ambient dimensions would it be drawn?

Functions of multiple variables

Functions of multiple variables

A multivariable function is a function of the form \(f: \mathbb{R}^m \to \mathbb{R}^n\) for some positive integers \(m\) and \(n\).

  • The functions you are familiar with are the simple ones of the form \(f: \mathbb{R} \to \mathbb{R}\). One input, one output.

  • Now we may have \(m\) inputs and \(n\) outputs.

  • Functions of the form \(f: \mathbb{R}^m \to \mathbb{R}\) are often called real-valued functions.

  • The graph of a function \(f: \mathbb{R}^m \to \mathbb{R}^n\) lives inside of \(\mathbb{R}^{m+n}\). We cannot see this graph unless \(m+n \leq 3\):

    • We can see the graphs of functions \(f:\mathbb{R} \to \mathbb{R}\). (Duh.)
    • We can see the graphs of functions \(f:\mathbb{R}^2 \to \mathbb{R}\).
    • We can see the graphs of functions \(f:\mathbb{R} \to \mathbb{R}^2\).

How to graph functions

How to plot the graph of a function \(f:\mathbb{R} \to \mathbb{R}\)

Suppose the function is \(y=f(x)\). To plot the graph of \(f\), we:

  1. Choose a value of \(x\).
  2. Compute \(y=f(x)\).
  3. Plot the point \((x,y)\) in the \(xy\)-plane (i.e., \(\mathbb{R}^2\)).
  4. Repeat for many values of \(x\). Connect the points with a curve.

How to plot the graph of a function \(f:\mathbb{R}^2 \to \mathbb{R}\)

Suppose the function is \(z=f(x,y)\). To plot the graph of \(f\), we:

  1. Choose a value of \(x\) and a value of \(y\).
  2. Compute \(z=f(x,y)\).
  3. Plot the point \((x,y,z)\) in \(xyz\)-space (i.e., \(\mathbb{R}^3\)).
  4. Repeat for many values of \(x\) and \(y\). Connect the points with a surface.

How to plot the graph of a function \(f:\mathbb{R} \to \mathbb{R}^2\)

Suppose the function is \((y,z)=f(x)\). To plot the graph of \(f\), we:

  1. Choose a value of \(x\).
  2. Compute \((y,z)=f(x)\).
  3. Plot the point \((x,y,z)\) in \(xyz\)-space (i.e., \(\mathbb{R}^3\)).
  4. Repeat for many values of \(x\). Connect the points with a curve.

  • This makes it sound easy. It is very often not.

  • Often you resort to other methods to visualize graphs, such as:

    • Plotting level curves for functions \(f:\mathbb{R}^2 \to \mathbb{R}\).
    • Plotting cross sections for functions \(f:\mathbb{R}^2 \to \mathbb{R}\).
    • Using Desmos or GeoGebra.

Exercise 3: Plotting functions

a. Graphing functions of the form \(\mathbb{R}^2 \to \mathbb{R}\)

Graph the following functions of the form \(\mathbb{R}^2 \to \mathbb{R}\):

  1. \(f(x,y) = 2x + 3y + 5\)
  2. \(g(x,y) = x^2 + y^2\)
  3. \(h(x,y) = \sin{x} + \cos{y}\)
  4. \(k(x,y) = \sin(x^2+y^2)\)

b. Graphing functions of the form \(\mathbb{R} \to \mathbb{R}^2\)

Graph the following functions of the form \(\mathbb{R} \to \mathbb{R}^2\):

  1. \(s(x) = (x-1, -x+1)\)
  2. \(t(x) = (\sin{x}, \cos{x})\)
  3. \(p(x) = (e^x, e^{-x})\)
  4. \(u(x) = (\sin{x}, \sin{2x})\)