Functions of Two Variables Up until this point, we have studied functions that take in a single input and produce a single output. Often times in everyday life, quantities depend on more than one variable. For example, the body mass index (BMI) that is often used to classify whether or not a person is overweight is given by the formula 2Weight (in pounds)BMI 703[Height (in inches)] If we were to let x denote a person’s weight (in pounds), y denote a person’s height (in inches), and z denote their BMI, then we would have 2( , ) 703xzfxyy. In this case, we say that f(x, y) is a function of two inputs (variables). The independent variables are x and y and z is the dependent variable. That is, the output z (in this case, BMI) depends on the inputs x and y (weight and height). A natural question is how to convey the information of such a function. When we had only a single input and a single output, we could make a one-dimensional table of values and we could plot the result in two dimensions. Now, with two inputs, we require a two-dimensional table of values and plot the result in three dimensions. Below is a table of some BMI values. Weight (lbs)110 120 130 140 150 160 170 180 1905'4" 18 20 22 24 25 27 29 31 325'5" 18 20 21 23 25 26 28 30 315'6" 17 19 21 22 24 25 27 29 30Height 5'7" 17 18 20 22 23 25 26 28 29 (in) 5'8" 16 18 19 21 22 24 25 27 285'9" 16 17 19 20 22 23 25 26 285'10" 15 17 18 20 21 2324 25 275'11"1516181921222325266'0" 14 16 17 19 20 21 23 24 25 Table 1: BMI values based upon Height (in) and Weight (lbs) The values inside the table are BMI values. For example, if a person is 5¢5 and weights 140 lbs, their BMI is 23. 1Example 1: Suppose that $1,000 is deposited into a bank account and gains interest at a rate r, compounded continuously. Find a function for A, the amount in the account after t years in terms of both r and t. Solution: Since the interest is compounded continuously, the growth of the account’s balance is akin to that of exponential growth. Thus, A = f(r, t) = 1000ert. Before we continue, we need to introduce three dimensions and how to graph points in 3-space. A typical illustration of 3-space is as follows: Figure 1: Three dimensional coordinate system This is the standard placement of the axes in this class. It follows the right-hand rule. That is, if you look down from the positive z-axis, you will see the standard xy-plane. Note: only the positive directions are shown by the axes. If we need the negative axis for any reason we will put them in as needed. Also note the various points on this sketch. The point P is the general point sitting out in 3-D space at coordinates (x, y, z). From the point P if we drop straight down until we reach a z-coordinate of zero we arrive at the point C. We say that the point C lies in the xy-plane. The xy-plane corresponds to all the points for which z = 0. We can also start at P and move in the other two directions as shown to get points in the xz-plane (this is B where y = 0) and the yz-plane (this is A, where x = 0). 2Collectively, the xy-, xz-, and yz-planes are called the coordinate planes. The point C is often referred to as the projection of P in the xy-plane. Likewise, A is the projection of P in the yz-plane and B is the projection of P in the xz-plane. Graphs of equations in 3-space behave differently than their two dimensional counterparts. To see this, consider the following example. Example 2: Graph y = 3 in 3-space. How does it compare to y = 3 in 2-space? Solution: In two dimensions, this was the equation of a horizontal line. In particular, y was fixed to be the value 3, but x could take on any value. In 3-space, y is again fixed, but now both x and z are allowed to take on any value. This has the effect of creating a “wall” in three dimensions. This “wall” is actually a plane, similar to the coordinate planes we introduced above, but this one is the xz-plane, moved out three. See Figure 2. y=3 Figure 2: The plane y = 3 Example 3: Plot x2 + y2 = 1 in 3-space. How does it compare to x2 + y2 = 1 in 2-space? Solution: 3In 2-space, the equation x2 + y2 = 1 defines a circle of radius 1. In 3-space, however, z is not specified, so it is allowed to take any value. In particular, the result is an infinite number of circles of radius 1 stacked on top of each other. This has the effect of forming a cylinder in 3-space, as depicted in Figure 3. Figure 3: The cylinder x2 + y2 = 1 If we want to find the distance between two points P, (x1, y1, z1), and Q, (x2, y2,z2), we can appeal to the Pythagorean Theorem. In Figure 4, to find the distance between P and Q, we notice that PSQ forms a triangle. In particular, we have that (PS)2 + (SQ)2 = (PQ)2. Figure 4: Pythagorean Theorem in 3-Space But we can apply the Pythagorean Theorem again to determine the value of (PS)2. In particular, we have that (PR)2 + (RS)2 = (PS)2. 4Thus, at the end of the day, we have that (PQ)2 = (PR)2 + (RS)2 + (SQ)2. But since PR is just x2 – x1, RS is y2 – y1 and SQ is z2 – z1. Thus, we have the following formulas: Distance between Two Points The distance, d, between two points (x1, y1) and (x2, y2) in two dimensions is given by 2221 21()(dxx yy) The distance, d, between two points (x1, y1, z1) and (x2, y2, z2) in three dimensions is given by 2221 21 21()()(dxx yy zz2) Example 3: Find an equation for the set of all points that are one unit away from the origin. What does this equation describe? Solution: We are looking for points (x, y, z) that satisfy 221(0)(0)(0)xyz2. Squaring both sides, we have the equation x2 + y2 + z2 = 1. This is the equation of the surface of a sphere with radius 1, centered at the origin. A graph of the sphere appears in Figure 5. Figure 5: The unit sphere