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Lial/Hungerford/Holcomb/Mullins: Mathematics with Applications 11e Finite Mathematics with Applications 11e Copyright ©2015 Pearson Education, Inc. All right reserved. Chapter 14 Multivariate Calculus Copyright ©2015 Pearson Education, Inc. All right reserved. Section 14.1 Functions of Several Variables Copyright ©2015 Pearson Education, Inc. All right reserved. Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 4 Example: 9 Let f ( x, y ) 4 x 2 xy , and find each of the given y quantities. 2 (a) f (1,3) Solution: Replace x with −1 and y with 3: f (1, 3) 4(1)2 2(1)(3) 9 4 6 3 1. 3 (b) The domain of f Solution: Because of the quotient 9/y, it is not possible to replace y with zero. So, the domain of the function f consists of all ordered pairs ( x, y) such that y 0. Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 5 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 6 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 7 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 8 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 9 A saddle Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 10 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 11 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 12 Graph of a function of 2 variables Find the domain and the range of Domain: Range Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 13 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 14 Exercises: Find the domain and the range of functions Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 15 Section 14.2 Partial Derivatives Copyright ©2015 Pearson Education, Inc. All right reserved. Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 17 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 18 Example: Solution: Let f ( x, y) ln( x 2 y). Find fx and fy. Recall the formula for the derivative of the natural logarithmic function. If y ln( g ( x)), then y g( x) / g ( x). Using this formula and treating y as a constant, we obtain Similarly, treating x as a constant leads to the following result: Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 19 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 20 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 21 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 22 Marginal Productivity A company that manufactures computers has determined that its production function is given by: where x is the size of labor force (in work hours per week) and y is the amount of capital invested (in units of $1000 ). Find the marginal productivity of labor and the marginal productivity of capital when x = 50 and y = 20, and interpret the results. Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 23 Surface Area of Human The surface area of human (in m2) is approximated by: A(M, H) = .202M.425H.725 where M is the mass of the person (in kg) and H is the height (in meters). Find the approximate change in surface area under the given condition: (a) The mass changes from 72kg to 73kg, while the height remains 1.8m 0.0112 (b) The mass remains stable at 70kg, while the height changs from 1.6m to 1.7m. 0.0783 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 24 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 25 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 26 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 27 Show that the function z = 5xy satisfies Laplace’s equation Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 28 Section 14.3 Extrema of Functions of Several Variables Copyright ©2015 Pearson Education, Inc. All right reserved. Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 30 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 31 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 32 Example: Find all critical points for Solution: Since the partial derivatives always exist, we must find all points (a, b) such that f x (a, b) 0 and f y (a, b) 0. f ( x, y) 6 x2 6 y 2 6 xy 36 x 54 y 5. Here, f x 12 x 6 y 36 and f y 12 y 6 x 54. Set each of these two partial derivatives equal to 0: 12 x 6 y 36 0 and 12 y 6 x 54 0. These two equations form a system of linear equations that we can rewrite as 12 x 6 y 36 6 x 12 y 54. To solve this system by elimination, multiply the first equation by −2 and then add the equations. Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 33 Example: Find all critical points for f ( x, y) 6 x 2 6 y 2 6 xy 36 x 54 y 5. Solution: Substituting x 7 in the first equation of the system, we have Therefore, (−7, 8) is the solution of the system. Since this is the only solution, (−7, 8) is the only critical point for the given function. By the previous theorem, if the function has a local extremum, it must occur at (−7, 8). Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 34 ≥3 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 35 fx(0, 0) and fy(0, 0) are unfedined 1 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 36 Saddle point fx(x, y) = -2x fx(0, 0) = 0 fy(x, y) = 2y fy(0, 0) = 0 Around (0,0) the function takes negative values along the x-axis and positive values along the y-axis. The point (0, 0) can not be a local extremum. Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 37 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 38 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 39 The test fails ≥0 Every point on the x-axis and y-axis yields local (and global) minimum. Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 40 Examine the function for local extrema and saddle point. Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 41 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 42 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 43 Section 14.4 Lagrange Multipliers Copyright ©2015 Pearson Education, Inc. All right reserved. Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 45 Example 1: Solution: Use Lagrange’s method to find the minimum value of f ( x, y) x2 y 2 , subject to the constraint x y 4. First, rewrite the constraint in the form g ( x, y ) 0 : x y 4 0. Then, follow the steps in the preceding slide. Step 1 As we saw previously, the Lagrange function is Step 2 Find the partial derivatives of F: Step 3 Set each partial derivative equal to 0 and solve the resulting system: Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 46 Example: Use Lagrange’s method to find the minimum value of Solution: Since this is a system of linear equations in x, y, and λ, it could be solved by the matrix techniques. However, we shall use a different technique, one that can be used even when the equations of the system are not all linear. Begin by solving the first two equations for λ: f ( x, y) x2 y 2 , subject to the constraint x y 4. Set the two expressions for λ equal to obtain Now make the substitution y = x in the third equation of the original system: Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 47 Example: Use Lagrange’s method to find the minimum value of Solution: Since y x and 2 x we see that the only solution of the original system is f ( x, y) x2 y 2 , subject to the constraint x y 4. Graphical considerations show that the original problem has a solution, so we conclude that the minimum value of f ( x, y) x 2 y 2 , subject to the constraint x y 4, occurs when x 2 and y 2. The minimum value is f (2,2) 22 22 8. Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 48 Example 2: Find a rectangle of maximum area that is inscribed in the elipse Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 49 Example 3: A builder plans to construct a 3-story building with a rectangular floor plan. The cost of the building is given by: xy + 30x + 20y + 474000, where x and y are the length and thw width of the rectangular floors. What length and width should be used if the building is to cost $500000 and have maximum area on each floor? x = 113.17, y = 169.75, A = 19211 ft2 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 50 Example 4: Find three positive numbers x, y and z whose sum is 50 and such that xyz2 is as large as possible. x = 12.5, y = 12.5, z = 25, xyz2 = 97656.25 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 51 Example 5: Find the dimensions of the closed rectangular box of maximum volume that can be constructed from 6 ft2 of material. x = 1, y = 1, z = 1, Volume = 1 ft3 Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 52 Example 6: Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 53 Exercises Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 54 Exercises Copyright ©2015 Pearson Education, Inc. All right reserved. Slide 1 - 55