Multivariable calculus: Exercises

05 Functions of multiple variables, part 1

Exercise 1: Multivariable functions in context

  1. A car rental company charges \(\$40\) a day and \(15\) cents a mile for its cars.
    1. Write a formula for the cost, \(C\), of renting a car as a function, \(f\), of the number of days, \(d\), and the number of miles driven, \(m\).
    2. If \(C= f(d,m)\), find \(f(5,300)\) and interpret it.
    3. Draw the graph of \(f\).
  2. Give a formula for the function \(m= f(b,t)\) where \(m\) is the amount of money in a bank account \(t\) years after an initial investment of \(b\) dollars, if interest is accrued at a rate of \(1.2\%\) per year compounded annually. (Hint: Annual compounding means that \(m\) increases by a factor of \(1.012\) each year.)
  3. The concentration \(C\) (mg/L) of a drug in the blood is modeled by \[ C= f(x,t) = te^{−t(5−x)}, \] where \(x\) is the dose (mg) and \(t\) is time (hours) since injection. Here \(0 \leq x \leq 4\) and \(t \geq 0\).
    1. Find \(f(3,2)\). Give units and interpret your answer.
    2. Graph the function \(C = f(4,t)\) in the \(tC\)-plane, and explain its significance.
    3. Graph the function \(C = f(x,1)\) in the \(xC\)-plane, and explain its significance.
    1. \(C = f(d,m) = 40d + 0.15m\)
    2. \(f(5,300) = 245\), which means that the cost of renting a car for \(5\) days and driving \(300\) miles is \(\$245\).
    3. The graph is a plane in \(dmC\)-space, going through the origin, with a slope of \(40\) in the \(d\)-direction and a slope of \(0.15\) in the \(m\)-direction.

  2. \(m = f(b,t) = b(1.012)^t\)

    1. \(f(3,2) \approx 0.0366\) mg/L, which means that the concentration of the drug in the blood is approximately \(0.0366\) mg/L two hours after an injection of \(3\) mg.
    2. For the graph, use Desmos. The significance of the graph is that it shows how the concentration of the drug changes over time for a fixed dose of \(4\) mg.
    3. For the graph, use Desmos. The significance of the graph is that it shows how the concentration of the drug changes with different doses at a fixed time of \(1\) hour.

Exercise 2: Identifying function types

Each of the following real-world scenarios can be modeled by a function of the form \(f: \mathbb{R}^m \to \mathbb{R}^n\) for some \(m\) and \(n\). For each part, identify the appropriate values of \(m\) and \(n\).

  1. Wind velocity as a function of position in \(3\)-dimensional space. (Remember, velocity is a vector, so it has three components.)
  2. RGB color values as a function of position on a computer screen. (RGB color values consist of a triple of numbers, each representing the intensity of red, green, and blue light, respectively.)
  3. Trajectory of a baseball (position in space as a function of time).
  4. Profit and revenue as functions of price and advertising budget.

  1. \(f: \mathbb{R}^3 \to \mathbb{R}^3\). Position in space has three components, and velocity has three components.

  2. \(f: \mathbb{R}^2 \to \mathbb{R}^3\). Position on a computer screen has two components (horizontal and vertical), and RGB color values have three components.

  3. \(f: \mathbb{R} \to \mathbb{R}^3\). Time has one component, and position in space has three components.

  4. \(f: \mathbb{R}^2 \to \mathbb{R}^2\). Price and advertising budget have two components, and profit and revenue have two components.