Math 20C. Lecture Examples.

Transcription

1 Math 0C. Lecture Eamples. (8/30/08) Section 14.1, Part 1. Functions of two variables Definition 1 A function f of the two variables and is a rule = f(,) that assigns a number denoted f(,), to each point (,) in a portion or all of the -plane. f(,) is the value of the function f at (,), and the set of points where the function is defined is called its domain. The range of the function is the set of its values f(,) for all (,) in its domain. If a function = f(, ) is given b a formula, we assume that its domain consists of all points (,) for which the formula makes sense, unless a different domain is specified. Eample 1 (a) What is the domain of f(,) = +? (b) What are the values f(, 3) and f(, 3) of this function at (, 3) and (, 3)? (c) What is its range? Answer: (a) The domain of f is the entire -plane. (b) f(, 3) = 13 f(, 3) = 13. (c) The range of f is the closed infinite interval [0, ). Definition The graph of = f(,) is the surface = f(, ) formed b the points (,,) in -space with (,) in the domain of the function and = f(,) (Figure 1). = f(,) FIGURE 1 Lecture notes to accompan Section 14.1, Part 1 of Calculus, Earl Transcendentals b Rogawski. 1

2 Math 0C. Lecture Eamples. (8/30/08) Section 14.1, Part 1, p. Fiing or : vertical cross sections of graphs One wa to stud the graph = f(,) of a function of two variables is to stud the graphs of the functions of one variable that are obtained b holding or constant. Eample Determine the shape of the surface = + b studing its cross sections in the planes = c perpendicular to the -ais. Answer: The intersection of the surface = + with the plane = c is a parabola that opens upward and whose verte is at the origin if c = 0 and is c units above the -plane if c 0 Figure Aa The surface has the bowl-like shape in Figure Ab c = c Figure Aa Figure Ab Eample 3 Determine the shape of the surface = + of Eample b studng its cross sections in the planes = c perpendicular to the -ais. Answer: The intersection of the surface = + with the plane = c is parabola that opens upward and whose verte is at the origin if c = 0 and is c units above the -plane if c 0. Figure A3a The surface has the bowl-like shape from Eample. (Figure A3b shows the cross sections from Eamples and 3 together.) c = c Figure A3a Figure A3b

3 Section 14.1, Part 1, p. 3 Math 0C. Lecture Eamples. (8/30/08) Eample 4 Determine the shape of the surface = b studing its cross sections in the planes = c perpendicular to the -ais. Answer: The intersection of the surface = with the plane = c is a parabola that opens upward and whose verte is c units below the -plane. Figure A4a The verte is at the origin for c = 0 and drops below the -plane as c moves awa from ero. The surface has the saddle shape in Figure A4b. c Figure A4a Figure A4b Eample 5 Determine the shape of the surface = of Eample 4 b studing its cross sections in the planes = c perpendicular to the -ais. Answer: The intersection of the surface = with the plane = c is a parabola that opens downward and whose verte is c units above the -plane. Figure A5a The verte is at the origin for c = 0 and rises above the -plane as c moves awa from ero. The surface has the saddle shape from Eample 4. (Figure A5b shows the two sets of cross sections together.) = c Figure A5a Figure A5b

4 Math 0C. Lecture Eamples. (8/30/08) Section 14.1, Part 1, p. 4 Eample 6 Use the curve = in the -plane of Figure to determine the shape of the surface = FIGURE = Answer: One solution: The cross section of the surface in the plane = c has the shape of the curve in Figure if c = 0, is that curve moved down and forward if c > 0 and is that curve moved down and back if c < 0. The surface has the boot-like shape in Figure A6 Another solution: The cross section in the plane = c is a parabola that opens downward and has its verte on the curve in Figure. The surface has the boot-like shape in Figure A6. Figure A6

5 Section 14.1, Part 1, p. 5 Math 0C. Lecture Eamples. (8/30/08) Horiontal cross sections and level curves Definition 3 The level curves (contour curves) of = f(, ) are the curves in the -plane where the function is constant. The have the equations f(,) = c with constants c (Figure 3). FIGURE 3 Eample 7 Describe the level curves of the function f(,) = + from Eamples and 3. Answer: Figure A7a shows horiontal cross sections of the graph of f and Figure A7b shows the corresponding level curves. The level curve f = c is the circle of radius c with its center at the origin if c > 0, is the origin if = 0, and is empt if c < 0. (The surface is called a circular paraboloid. ) Figure A7a Figure A7b

6 Math 0C. Lecture Eamples. (8/30/08) Section 14.1, Part 1, p. 6 Eample 8 Describe the level curves of g(, ) = from Eamples 4 and 5. Answer: Figures A8a and A8b The level curves g = c is a hperbola with the equation = c. (The surface is a hperbolic paraboloid. ) Level curves of g(, ) = Figure A8a Figure A8b Eample 9 Figures 4 and 5 show horiontal cross sections of the graph of h(,) = from Eample 6 and the corresponding level curves of the function. Describe how the surface can be reconstructed from the level curves. h = 1 6 h = 5 h = 0 6 h = 5 h = 0 h = 1 FIGURE 4 FIGURE 5 Answer: Leave the two parts of the level curve h = 0 on the plane. Raise the two parts of the curve labeled h = 1 one unit on the toe and leg of the boot. Lower the curves above the upper part of h = 1 to form the sides of the boot. Raise the curves below the lower part of h = 1 to form the more of the leg of the boot.

7 Section 14.1, Part 1, p. 7 Math 0C. Lecture Eamples. (8/30/08) Rotating aes The surfaces = k with nonero constants k are important in the stud of maima and minima of functions with two variables. Their shapes can be determined b introducing new -coordinates b rotating the - and -aes 45 counterclockwise as in Figure 4. = 1 ( ), = 1 ( + ). (1) 45 FIGURE 4 Eample 10 Use -coordinates as in Figure 4 to anale the surface =. Answer: The graph is the surface = of Figure A4b rotated 45 as in Figure A10. (Notice that the - and -aes are on the surface.) Figure A10 Interactive Eamples Work the following Interactive Eamples on Shenk s web page, http// ashenk/: Section 14.1: Eamples 1 6 The chapter and section numbers on Shenk s web site refer to his calculus manuscript and not to the chapters and sections of the tetbook for the course.

8 Math 0C. Lecture Eamples. (8/30/08) Section 14.1, Part 1, p. 8 Eercises 1. Add aes to Figure 5 so it is the graph of f(,) =. FIGURE 5 FIGURE 6. What are the values of L(, ) = + on its three level curves in Figure 6? 3. Draw the graph of the function = Draw and label the level curves of N(,) = 1 where it has the values c = 0, ±1, and ±. 5. Draw and label the level curves of S(,) = sin where it has the values 0, ±, ±4. 6. (a) Eplain wh the horiontal cross sections of = ln( + 1 ) and of = are circles. + (b) Match the surfaces in Figures 7 and 8 to their equations in part (a). Eplain our choices. FIGURE 7 FIGURE 8

9 Section 14.1, Part 1, p. 9 Math 0C. Lecture Eamples. (8/30/08) 7. Match the functions (a) = sin, (b) = sin sin, (c) = sin + 1, and (d) = 3e /5 sin to their graphs in Figures 9 through 1. FIGURE 9 FIGURE 10 FIGURE 11 FIGURE 1 8. Match the functions of Eercise 7 with their level curves in Figures 13 through 16. FIGURE 13 FIGURE 14

10 Math 0C. Lecture Eamples. (8/30/08) Section 14.1, Part 1, p. 10 Selected answers FIGURE 15 FIGURE Figure A1 Figure A1 Figure A3. L = 3 on the outer square L = on the middle square L = 1 on the inner square 3. Figure A3 4. Figure A Figure A4