Lial/Hungerford/Holcomb/Mullins: Mathematics with Applications 11e Finite Mathematics with Applications 11e Copyright ©2015 Pearson Education, Inc. All right reserved.

Multivariate Calculus Chapter 14 Multivariate Calculus Copyright ©2015 Pearson Education, Inc. All right reserved.

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

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

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

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

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

Copyright ©2015 Pearson Education, Inc. All right reserved. Section 14.2 Partial Derivatives Copyright ©2015 Pearson Education, Inc. All right reserved.

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Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Let Find fx and fy. Solution: 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: Copyright ©2015 Pearson Education, Inc. All right reserved.

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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.

Copyright ©2015 Pearson Education, Inc. All right reserved. 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: The mass changes from 72kg to 73kg, while the height remains 1.8m The mass remains stable at 70kg, while the height changs from 1.6m to 1.7m. 0.0112 0.0783 Copyright ©2015 Pearson Education, Inc. All right reserved.

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

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

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Find all critical points for Example: Find all critical points for Solution: Since the partial derivatives always exist, we must find all points (a, b) such that Here, Set each of these two partial derivatives equal to 0: 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. Copyright ©2015 Pearson Education, Inc. All right reserved.

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

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

Copyright ©2015 Pearson Education, Inc. All right reserved. fx(0, 0) and fy(0, 0) are unfedined  1 Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Saddle point fx(x, y) = -2x fy(x, y) = 2y fx(0, 0) = 0 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.

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

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

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Copyright ©2015 Pearson Education, Inc. All right reserved. Section 14.4 Lagrange Multipliers Copyright ©2015 Pearson Education, Inc. All right reserved.

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

Use Lagrange’s method to find the minimum value of 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: 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.

Use Lagrange’s method to find the minimum value of Example: Use Lagrange’s method to find the minimum value of Solution: 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 Copyright ©2015 Pearson Education, Inc. All right reserved.

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

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

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

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

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

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

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