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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.

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Chapter 14 Multivariate Calculus

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Copyright ©2015 Pearson Education, Inc. All right reserved. Section 14.1 Functions of Several Variables

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Let and find each of the given quantities. Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Replace x with −1 and y with 3: (b) The domain of f (a) 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 such that

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. A saddle

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Slide Graph of a function of 2 variables Copyright ©2015 Pearson Education, Inc. All right reserved. Find the domain and the range of Range Domain:

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Slide Exercises: Find the domain and the range of functions Copyright ©2015 Pearson Education, Inc. All right reserved.

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Section 14.2 Partial Derivatives

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Let Find f x and f y. Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Recall the formula for the derivative of the natural logarithmic function. If Using this formula and treating y as a constant, we obtain Similarly, treating x as a constant leads to the following result:

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide 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.

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Slide Surface Area of Human The surface area of human (in m 2 ) is approximated by: A(M, H) =.202M.425 H.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 (b) The mass remains stable at 70kg, while the height changs from 1.6m to 1.7m. Copyright ©2015 Pearson Education, Inc. All right reserved

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Show that the function z = 5xy satisfies Laplace’s equation

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Copyright ©2015 Pearson Education, Inc. All right reserved. Section 14.3 Extrema of Functions of Several Variables

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Find all critical points for Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Since the partial derivatives always exist, we must find all points (a, b) such that Set each of these two partial derivatives equal to 0: Here, These two equations form a system of linear equations that we can rewrite as To solve this system by elimination, multiply the first equation by −2 and then add the equations.

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Slide Find all critical points for Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Substituting 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).

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. ≥ 3

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. 1 f x (0, 0) and f y (0, 0) are unfedined

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Slide Saddle point Copyright ©2015 Pearson Education, Inc. All right reserved. Around (0,0) the function takes negative values along the x-axis and positive values along the y-axis. f x (x, y) = -2x f y (x, y) = 2y f x (0, 0) = 0 f y (0, 0) = 0 The point (0, 0) can not be a local extremum.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. The test fails ≥ 0 Every point on the x-axis and y-axis yields local (and global) minimum.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Examine the function for local extrema and saddle point.

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Section 14.4 Lagrange Multipliers

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Use Lagrange’s method to find the minimum value of Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example 1: First, rewrite the constraint in the form 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:

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Slide Use Lagrange’s method to find the minimum value of Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: 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: Set the two expressions for λ equal to obtain Now make the substitution y = x in the third equation of the original system:

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Slide Use Lagrange’s method to find the minimum value of Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Since we see that the only solution of the original system is Graphical considerations show that the original problem has a solution, so we conclude that the minimum value of subject to the constraint occurs when The minimum value is

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Example 2: Find a rectangle of maximum area that is inscribed in the elipse

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Slide 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 , 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 $ and have maximum area on each floor? Copyright ©2015 Pearson Education, Inc. All right reserved. Example 3: x = , y = , A = ft 2

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Slide Find three positive numbers x, y and z whose sum is 50 and such that xyz 2 is as large as possible. Copyright ©2015 Pearson Education, Inc. All right reserved. Example 4: x = 12.5, y = 12.5, z = 25, xyz 2 =

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Slide Find the dimensions of the closed rectangular box of maximum volume that can be constructed from 6 ft 2 of material. Copyright ©2015 Pearson Education, Inc. All right reserved. Example 5: x = 1, y = 1, z = 1, Volume = 1 ft 3

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Example 6:

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Exercises

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Exercises

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