Worksheet 2.1: Introduction to Multivariate Functions (Functions of Two or More Independent Variables)

Transcription

1 Boise State Math 275 (Ultman) Worksheet 2.1: Introduction to Multivariate Functions (Functions of Two or More Independent Variables) From the Toolbox (what you need from previous classes) Know the meaning of domain and range for scalar valued functions of a single variable, y = f (x) (the type of function studied in Calc I/II). Be able to evaluate and graph functions of a single variable. Functions of two variables can be represented graphically using graphs z = f (x, y), traces, and contour diagrams. In this worksheet, you will: Find the domain and range of scalar valued functions of two and three variables. Given a graph of a function z = f (x, y), be able to sketch vertical and horizontal traces. Use a graphing app to generate graphs and level curves/contour maps of functions of two variables. Examine how the contour map of a function relates to the general behavior of the function. For a function z = f (x, y), use a contour map to determine the value of a function, the change in a function s value between two points in the domain, and the average rate of change of a function s value with respect to distance in the function s domain. You will be asked to use graphing apps during this worksheet. You may always use graphing apps for any problem on homework and activities, unless explicitly directed otherwise. Suggestions for online and downloadable apps can be found on the course webpage: General Online Resources (e.g.: Wolfram Alpha, Geogebra): Online Applets: Downloadable Apps:

2 Boise State Math 275 (Ultman) Worksheet 2.1: Introduction to Multivariate Functions 1 Model 1: Domain and Range Diagram 1A: z =x +y 2 Domain: all of R Range: all of R Diagram 1B: z= x +y Range: z 0 Domain: x + y 0 Critical Thinking Questions In this section, you learn about the domain and range of functions of two variables. (Q1) Recall from previous math classes, the domain of a function is the set of all allowable inputs k possible outputs. (Q2) Refer to Diagrams 1A & 1B: the domain of function of two variables z = f (x, y ) is a subset of R (inputs are real numbers z ) k R2 (inputs are points (x, y ) in the plane). (Q3) Recall from previous math classes, the range of a function is the set of all allowable inputs k possible outputs. (Q4) Refer to Diagrams 1A & 1B: the range of function of two variables z = f (x, y ) is a subset of R (outputs are real numbers z) k R2 (outputs are points (x, y ) in the plane.

3 Boise State Math 275 (Ultman) Worksheet 2.1: Introduction to Multivariate Functions 2 (Q5) Refer to Diagrams 1A & 1B, and examine the domains of the two functions z = x + y and z = x + y. Complete the table, indicating whether the points (x, y) given in the top row are in the domain of each function. point (x, y ): (0, 0) (1, 1) ( 1, 1) ( 1, 1) ( 1, 0) (2, 1) ( 2, 1) z = x + y YES z = x + y NO YES (Q6) Refer to Diagrams 1A & 1B, and examine the ranges of the two functions z = x + y and z = x + y. Complete the table, indicating whether the z-values given in the top row are in the range of each function. z value: z = x + y z = x + y YES NO (Q7) In Diagram 1A: The domain of the function z = x + y is the entire plane R 2, because the sum x + y is is not defined for all values of x and y. In Diagram 1B: The domain of the function z = x + y is the part of the plane where x + y 0 x + y = 0 x + y 0. This is the part of the plane above (and including) the line y =. (Q8) In Diagram 1A: The range of the function z = x +y is all real numbers < z < because for any value of z, there is always may or may not be a pair of numbers x and y so that z = x + y. In Diagram 1B: The range of the function z = x + y is all non-negative real numbers z 0. Negative values are not in the range of this function because the output values for the real-valued square-root function is always sometimes never negative. ( Q9) (a) The expressions for the functions f (x, y) = x 2 + y 2 and g(x, y) = ( x 2 + y 2) 1 look very similar, but there is a point in the domain of one that is not in the domain of the other. What is this point, and which function does not include it in its domain? (b) Using an app, graph these functions and compare their behaviors. ( Q10) (a) Change the equation defining the function z = x + y from Diagram 1B so that the range of the new function is all negative numbers (z < 0). (b) Graph your new function, and compare it to the graph of z = x + y. ( Q11) Construct a function z = f (x, y) so that the range of the function is 1 z 1. ( Q12) Construct a function z = g(x, y) so that the domain of the function lies in the first and third quadrants of the xy-plane, and the range is z 1.

4 Boise State Math 275 (Ultman) Worksheet 2.1: Introduction to Multivariate Functions 3 Model 2: Graphs and Traces Diagram 2: Traces on the Cone z = x 2 + y 2 Diagram 2A Graph of the Cone Equation of the Cone: z = x 2 + y 2 Diagram 2B Horizontal Traces: z = constant Equation of the Horizontal Trace in the Plane z = c: c = x 2 + y 2 for c = 1, 2, 3, 4, 5 Diagram 2C Vertical Traces: x = constant Equation of the Vertical Trace in the Plane x = c: z = c 2 + y 2 for c = 4, 2, 0, 2, 4 Diagram 2D Vertical Traces: y = constant Equation of the Vertical Trace in the Plane y = c: z = x 2 + c 2 for c = 4, 2, 0, 2, 4 Critical Thinking Questions In this section, you will look at horizontal and vertical traces of functions of two variables. A trace of a function z = f (x, y) is a curve in the graph of the function generated by holding one of the three variables x, y, or z constant.

5 Boise State Math 275 (Ultman) Worksheet 2.1: Introduction to Multivariate Functions 4 (Q13) Diagram 2A shows a graph of the cone z = x 2 + y 2. Using an app that will graph multiple equations simultaneously, graph this cone and the five horizontal planes z = 1, z = 2, z = 3, z = 4, and z = 5. Observe: The intersection of each plane with the cone is a curve. These curves are shown in Diagram 2B. Since the planes you graphed are parallel to the xy-coordinate plane, the curves of intersection of the graph with these planes are also parallel to the xy-plane xz-plane y z-plane. (Q14) Diagram 2B shows the curves of intersection from (Q13). These curves are called horizontal traces. On a horizontal trace, x y z is constant. On Diagram 2B, label two of these horizontal traces: the one corresponding to holding z constant at z = 1, and the one corresponding to holding z constant at z = 4. (The circles in the xy-plane are the curves below the horizontal traces. They are called level curves or contour lines. They will be the main topic in Model 3.) (Q15) Diagrams 2C and 2D show vertical traces. In Diagram 2C, the vertical traces are generated in the graph by holding x constant: z = c 2 + y 2 for different values of c. On Diagram 2C, label the vertical trace corresponding to x = 0. In Diagram 2D, the vertical traces are generated in the graph by holding y constant: z = x 2 + c 2 for different values of c. On Diagram 2D, label the vertical trace corresponding to y = 2. (Q16) Horizontal traces (Diagram 2B) are curves of intersection of the graph of the cone z = x 2 + y 2 with planes that are parallel perpendicular to the xy-plane. Vertical traces (Diagrams 2C & 2D) are curves of intersection of the graph of the cone z = x 2 + y 2 with planes that are parallel perpendicular to the xy-plane. (Q17) If you were an ant walking on the cone, and you wanted to follow a curve so that the x- coordinate of your path did not change, you would walk along a: Horizontal trace with z held constant (one of the curves in Diagram 2B). Vertical trace with x held constant (one of the curves in Diagram 2C). Vertical trace with y held constant (one of the curves in Diagram 2D). (Q18) If you were an ant walking on the cone, and you wanted to maintain a constant height above the xy-plane, you would walk along a: Horizontal trace with z held constant (one of the curves in Diagram 2B). Vertical trace with x held constant (one of the curves in Diagram 2C). Vertical trace with y held constant (one of the curves in Diagram 2D).

6 Boise State Math 275 (Ultman) Worksheet 2.1: Introduction to Multivariate Functions 5 (Q19) The following three diagrams represent vertical and horizontal traces for the function z = 0.5x 2 + y 3. Identify the one that represents horizontal traces. ( Q20) Sketch the paraboloid: z = x 2 + y 2. Include the x-, y-, and z-axes label them! Add the vertical traces at x = 1 and y = 1, and the horizontal trace at z = 1. ( Q21) Surfaces of Revolution are surfaces that have rotational symmetry about a coordinate axis. The graph of a function is a surface of revolution if its horizontal traces are circles centered about a coordinate axis. (a) Graph the following. Which ones are surfaces of revolution? z = x + y surface of revolution not a surface of revolution z = x 2 + y 2 surface of revolution not a surface of revolution z = 4 (x 2 + y 2 ) surface of revolution not a surface of revolution z = x 2 y 2 surface of revolution not a surface of revolution z = e x 2 y 2 surface of revolution not a surface of revolution (b) What characteristic of do the equations that define surfaces of revolution have in common? (c) Try to answer first without graphing (then graph to check your answer): is the graph of h(y, z) = y 2 + z 2 a surface of revolution? (d) Try to answer first without graphing (then graph to check your answer): is the graph of h(x, y) = x 2 + y 3 a surface of revolution?

7 Boise State Math 275 (Ultman) Worksheet 2.1: Introduction to Multivariate Functions Model 3: Level Curves and Contour Maps Diagram 3A: Contour Map of Mt. Rainier ( Diagram 3B: Graph & Contour Map of Paraboloid f ( x, y ) = x 2 + y 2 Diagram 3C: Graph & Contour Map of Cone g((x, y ) = p x2 + y2 6

8 Boise State Math 275 (Ultman) Worksheet 2.1: Introduction to Multivariate Functions 7 Critical Thinking Questions In this section, you will explore contour maps. (Q22) Diagram 3A is a contour map (or contour diagram) of the region around the peak of Mount Rainier. The curves are called contour lines (or level curves). Along these curves, the elevation remains constant. Curves are labeled with their elevation. To make this contour map, imagine slicing the mountain with equally-spaced horizontal planes, then projecting the resulting curves down onto a plane at sea level. On Diagram 3A, on the contour map of Mt. Rainier, mark the path you would follow if you wanted to walk all the way around the mountain while maintaining a constant elevation of 3,000 meters. (Q23) The contour interval z is the difference in elevation between adjacent curves on a contour map. What is the contour interval on the contour map of Mt. Rainier in Diagram 3A? z = (Q24) If there were another curve on the contour map of of Mt. Rainier in Diagram 3A, inside the contour line marked 4000, what would be the elevation of the mountain be along this curve? (Q25) Diagrams 3B and 3C are contour maps (or contour diagrams) of two functions f (x, y) and g(x, y). The curves are called contour lines (or level curves). They are the projections of horizontal traces (see Model 2) onto the xy-plane. Along the level curves, the function values remain constant. The labels c = 1 and c = 2 give you the z-values of the functions on two adjacent curves. (a) On Diagrams 3B and 3C, mark the paths you would follow in the domain in order to maintain a constant function value z = 1. (b) The difference in function values between adjacent curves (called the contour interval z) is constant. What is the contour interval for the contour maps in Diagrams 3B and 3C? z = (c) On Diagrams 3B and 3C, mark the contour lines where f (x, y) = 4 and g(x, y) = 4. (Q26) (a) On Diagram 3B, the level curves are closer together farther apart evenly spaced where the graph of the function is steeper, and closer together farther apart evenly spaced where the graph is less steep. (b) On Diagram 3C, the level curves are closer together farther apart evenly spaced, because the function is increasing at a constant rate.

9 Boise State Math 275 (Ultman) Worksheet 2.1: Introduction to Multivariate Functions 8 ( Q27) The contour map below belongs to a function T (x, y), representing the temperature on a flat metal plate in C. You are an ant walking on the plate. (a) Mark the curve along which you would walk in order to maintain a constant temperature of 31 C (b) On the diagram, sketch the curve you would walk around in order to experience a constant temperature of 30 C. (c) What is the change in temperature from the point P to the point Q? (d) What is the average rate of change in temperature from the point P to the point Q, with respect to distance in the xy-plane? (Assume distance is measured in meters.) (e) At the point Q, sketch arrows indicating the two directions you would walk in to experience a constant temperature of 31 C immediately upon leaving the point Q. (f) At the point Q, sketch arrows indicating the direction you would walk in to experience the greatest immediate increase in temperature, and greatest immediate decrease in temperature.

10 Boise State Math 275 (Ultman) Worksheet 2.1: Introduction to Multivariate Functions 9 ( Q28) Use an app to graph the function f (x, y) = x 2 y 2, and use the graph to sketch a contour map. ( Q29) Construct a contour map for a function whose level curves are circles centered about the origin, and whose graph becomes steeper as you move towards from the origin. Can you find an equation defining this function? If you think you have one, graph it and check!