Today we looked at examples of functions of two or more variables and how to graph them. There are two options for graphing function with two variables. You can either graph the level curves on a regular xy-plane or you can graph the surface \(z = f(x,y)\) using a 3D grapher. A good online 3D graphing calculator is this one: Geogebra 3D Graphing Calculutor.
We discussed the following examples:
Blood alcohol content (BAC) can be estimated using three variables: the number of drinks consumed (D), a person’s weight (W) measured in pounds, and the number of hours (H) since the drinks were consumed. The formula to estimate BAC is: \[BAC = 11.7 \frac{D}{W} - 0.015 H.\] (I got this formula from the internet. I don’t think the 11.7 constant is accurate. I think it should be closer to 4.2 for men and 4.7 for women.)
Body mass index (BMI) is \[BMI = 703 \frac{W}{h^2}\] where \(W\) is a person’s weight in lbs and \(h\) is their height in inches. A BMI less than 18.5 is underweight, over 25 is overweight, and over 30 is obese.
We made a 3D graph of the function \(f(x,y) = 4-x^2 -y^2\) using Geogebra.
We also made a 3D graph and a level curve graph of the distance function \(f(x,y) = \sqrt{x^2+y^2}\).
Make a graph showing three different level curves for \(g(x,y) = x^2 + y\).
We finished by reviewing how to use a constraint to remove variables by substitution from multivariables functions. We reviewed how to do problems 16-18 from HW 6.
Another good constraint practice problem is problem 6 from the midterm 3 review.
If \(f(x,y)\) is a function of \(x\) and \(y\), then the partial derivative of \(f\) with respect to \(y\), \(d\frac{\partial f}{\partial x}\), is the derivative of the function \(f\) with respect to \(x\) with the \(y\)-variable treated as a constant. The partial derivative \(\dfrac{\partial f}{\partial y}\) is defined likewise.
We did the following examples:
Find \(d\frac{\partial f}{\partial x}\) and \(d\frac{\partial f}{\partial y}\) when \(f(x,y) = x^2 + 5y\).
Find the partial derivatives of \(f(x,y) = x^2 y\).
Find the partial derivatives of \(g(x,y) = \dfrac{e^x}{y}\). We did this one two ways: you can move the \(y\) to the numerator first by making it \(y^{-1}\) or you can just use the quotient rule to find both partial derivatives.
Find the partial derivatives of \(xe^{-2xy}\).
Blood alcohol content for men is roughly \[\text{BAC} = \frac{4D}{W}\] where \(D\) is the number of drinks and \(W\) is the man’s weight. Using this formula find both partial derivatives \(\dfrac{\partial \text{BAC}}{\partial D}\) and \(\dfrac{\partial\text{BAC}}{\partial W}\).
Both of the partial derivatives in the last problem mean something. \(\dfrac{\partial \text{BAC}}{\partial D}\) tells us the rate of change that BAC increases if someone has another drink, and \(\dfrac{\partial\text{BAC}}{\partial W}\) is how much lower BAC would be for a person who weighs one pound more.
Calculate the values of the two partial derivatives in the previous problem when \(D = 4\) and \(W=160\). Then explain what the numbers mean in words.
A manufacturer has a factory with \(x\) skilled laborers and \(y\) unskilled laborers. Suppose that the production level of the factory is \(Q = 10x^2y\). Find \(\dfrac{\partial Q}{\partial x}\) and \(\dfrac{\partial Q}{\partial y}\) when \(x = 20\) and \(y =40\), and explain what the two numbers mean.
The two partial derivatives \(\dfrac{\partial Q}{\partial x}\) and \(\dfrac{\partial Q}{\partial y}\) are called the marginal productivity for skilled and unskilled laborers respectively. A company won’t want to pay their workers more than their marginal productivity, otherwise the company would have an incentive to reduce their workforce.
We finished by writing down the formula for a tangent plane. If \(z = f(x,y)\), then the tangent plane at a point \(z_0 = f(x_0,y_0)\) has a formula: \[\Delta z = \frac{\partial f}{\partial x} \Delta x + \frac{\partial f}{\partial y} \Delta y\] where \(\Delta z = z-z_0\), \(\Delta x = x-x_0\) and \(\Delta y = y-y_0\).
We did Problem #8 and graphed the result on the 3D Geogebra Grapher.