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3 Functions of More Than One Variable 3 Functions of More Than One Variable Learning Outcome: 3. Understand single-valued functions of two or three variables and their derivatives, perform associated computations, and apply understanding and computations to solve problems. 3.1 Introduction During the first and second terms of calculus you worked with functions of one variable, which were usually denoted y = f(x) (for example, y = x2). We could think of such a function as a “machine” named f, with the variable x being the input to the machine and the variable y being the output. We will now investigate functions of more than one input variable; the ones with which we will work will have the form z = f(x, y) or w = f(x, y, z). Examples would be things like f(x, y)= x2 + y2 and w = xyz +3y + z2. Consider the function w. There are three input variables x, y and z, and one output variable, w. We say that w is a function of three variables, the number of input variables. Similarly, f is a function of two variables. OK, so those two examples are bit abstract. Some more concrete (and practical!) examples would be the following: • The perceived temperature (“wind-chill”) depends on the actual air temperature and the speed of the wind. • The temperature in a well depends on the depth of the well and the time of year. • The pressure at a point in a room depends on the location (x, y, z) of the point. • The temperature on a plate of metal depends on the location (x, y) of the point. In the first two semesters of calculus the two main themes were differentiation and integration of functions of one variable. We will now take up differentiation and, a bit later, integration of functions of more than one variable. 3.2 Representing Functions of More Than One Variable Learning Outcome: 3. (a) Determine a function value of a function of more than one variable from an equation, a table of values, or a level curve plot. In the past, when you were given a function of one variable, you often considered several repre- sentations of such a function: an equation of the function, a table of corresponding input and output values, a graph of the function. Each has their own advantages and disadvantages. Exact function values can be obtained from either the equation or the table, but the table has a limited number of values, whereas the equation can be used to get an unlimited number of input-output pairs. Exact input-output values are impossible to obtain from a graph, but the overall behavior of a function can be obtained quickly from looking at a graph. Functions of three or more variables are challenging to represent in any form other than an equation, so we will often focus on functions of two variables, which are more easily represented. There are four common representations of such functions; we will use three of the four regularly. • An equation can be given, from which function values can be obtained by simply inserting input values and obtaining corresponding output values. The two equations f(x, y) = x2 + y2 and w = xyz +3y + z2 are examples of this. An equation can easily be used to describe a function of any number of variables. The following three methods are only possible for functions of two variables. • A table of values can be given. Such a table lists values of one input variable horizontally across the top or bottom of the table and values of the other input variable along one of the sides (usually the left side). The output value is for a given pair of input values is then located in the cell of the table where the row and column of the two input values intersect. An example is the table below, which gives (in the “body” of the table) the wind-chill temperature W (in degrees Fahrenheit) for various actual temperature/wind speed combinations (in that order). Wind speed is in miles per hour and actual temperature is given in degrees Fahrenheit. Actual Temperature (◦F ) T v −10 0 10 20 30 40 5 −22 −11 1 13 25 36 10 −28 −16 −4 9 21 34 15 −32 −19 −7 6 19 32 −35 −22 −9 Wind Speed (mph) 20 4 17 30 • A level curve plot (contour plot) is essentially a “map” of the two input variables, with points having certain fixed output values connected by curves. The one below represents temperatures T on a rectangular sheet of metal. The temperatures are in degrees Fahrenheit, and the distances are in feet (with x-values being on the horizontal axis and y-values on the vertical, as usual). 40 10 90 50 80 60 70 70 60 5 50 90 80 40 30 5 10 • A surface plot, which is a three-dimensional representation of the function, where the input values represent points in the xy-plane and the output values are z-values. When the z-values are plotted above (or below) their corresponding x and y values, a surface is obtained. Such plots are valuable for seeing the behavior of the function, but they are challenging to draw by hand, and almost impossible to get input-output values from. We will rarely (never?) use these. For examples of these, see page 792 1st Ed., page 779 2nd Ed. 3.3 Average Rates of Change Learning Outcome: 3. (b) Determine the average rate of change of a function of two or three variables from one point to another from either a level curve graph or the equation of the function. • Examples: 780: 8, 9 • Exercises: 786: 40, 41, 43, 45 Answer to 786: 40 is −1.67 × 10−4 ppt/◦C Looking at either the table of values or the level curve plot from the previous page, it is clear that the values of the functions (wind chill temperatures for the table, temperatures on the metal plate for the level curves) vary as the input values change. The main focus of our next couple weeks will be the rates at which the function values change as the input values change. We will begin by calculating average rates of change as one or the other (or perhaps both) of the input values change. Suppose that for a function f(x, y) we go from inputs of (x1,y1) to (x2,y2). Then the corresponding outputs go from f(x1,y1) to f(x2,y2). The average rate of change of the function is then the change in output divided by the change in input, which we show symbolically like this: f(x ,y ) − f(x ,y ) Ave ROC = 2 2 1 1 “distance” from (x1,y1) to (x2,y2) x1, y1, x2 and y2 should all be the appropriate numbers, and instead of f you should use the letter for whatever function you are working with. The “distance” from (x1,y1) to (x2,y2) will always be either a subtraction or an application of the Pythagorean Theorem. I’ve put distance in quotes because in many cases it will not actually represent a geometric distance, but simply a distance between numbers. Note that we generally find the difference by taking the final values minus the initial values. Here are a few examples, demonstrating the way that I’d like you to show your calculations: ⋄ Example 3.3(a): For the temperature function with level curves shown on the previous page, find the average rate of change of the temperature from (7, 1) to (7, 5). T (7, 5) − T (7, 1) 65 − 38 Ave ROC = = =6.75◦F/foot ♠ 5 − 1 4 ⋄ Example 3.3(b): For the function given by the table on the previous page, find the rate of change in wind-chill from a actual temperature/wind speed combination of 0◦F and −10 mph to 0◦F and −20 mph. W (0, −20) − W (0, −10) −22 − (−16) Ave ROC = = = −0.6◦F/mph ♠ 60 − 30 10 Note that in this last example the “distance” is really the change in wind speed. In both of the above example we have held one of the input variables fixed while the other changes. In the next example we let both of the input variables vary. This only makes sense to do when both input variables measure the same quantity and have the same units. (In this case the quantity is true distance, measured in feet. ⋄ Example 3.3(c): For the temperature function from Exercise 1 of Assignment 8, find the average rate of change in temperature from (5, 6) to (9, 1). T (9, 1) − T (5, 6) 60 − 75 Ave ROC = = = −2.3◦F/foot ♠ (9 − 5)2 + (1 − 6)2 6.4 p The quantity (9 − 5)2 + (1 − 6)2, is just the distance from (5, 6) to (9, 1), calculated using the distance formulap for two dimensions. ⋄ Example 3.3(d): In some location the temperature T in degrees Fahrenheit of the ground at a depth of d feet and at time t days from some zero date is modeled by T (d, t)=30e−0.2d cos(0.0172t − 0.2d)+60 . Find the average rate of change of temperature at time fifty days, from a depth of 3 feet to a depth of 40 feet. T (40, 50) − T (3, 50) 60.0066 − 75.9110 Ave ROC = = = −0.4282◦F/foot ♠ 40 − 3 37 3.4 Partial Derivatives Learning Outcome: 3. (c) Find and interpret (give location, direction, change in dependent variable per unit of change in independent variable) partial derivatives of a function at a point.
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