1. Lial/Hungerford/Holcomb/Mullins: Mathematics with Applications 11e Finite Mathematics with Applications 11e
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2. Multivariate Calculus
Chapter 14
Multivariate Calculus
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3. Functions of Several Variables
Section 14.1
Functions of Several Variables
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5. 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
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A saddle
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13. Graph of a function of 2 variables
Find the domain and the range of
Domain:
Range
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15. Exercises: Find the domain and the range of functions
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Section 14.2
Partial Derivatives
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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:
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23. 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.
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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
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Show that the function z = 5xy satisfies Laplace’s equation
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29. Extrema of Functions of Several Variables
Section 14.3
Extrema of Functions of Several Variables
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33. 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.
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34. 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).
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≥ 3
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fx(0, 0) and fy(0, 0) are unfedined
 1
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Saddle point
fx(x, y) = -2x fy(x, y) = 2y
fx(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.
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The test fails
≥ 0
Every point on the x-axis and y-axis yields local (and global) minimum.
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Examine the function for local extrema and saddle point.
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Section 14.4
Lagrange Multipliers
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46. 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:
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47. 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:
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48. 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
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49. 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
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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 , 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?
x = , y = , A = ft2
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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 =
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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
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Example 6:
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Exercises
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Exercises
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