1. Functions of Several Variables
Chapter 7
Functions of Several Variables
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Chapter Outline
Examples of Functions of Several Variables
Partial Derivatives
Maxima and Minima of Functions of Several Variables
Lagrange Multipliers and Constrained Optimization
The Method of Least Squares
Double Integrals
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Section 7.2
Partial Derivatives
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Section Outline
Partial Derivatives
Computing Partial Derivatives
Evaluating Partial Derivatives at a Point
Local Approximation of f (x, y)
Demand Equations
Second Partial Derivative
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Partial Derivatives
If , then
Partial Derivative of f (x, y)
with respect to x: written , is the derivative of f (x, y), where y is treated as a constant and f (x, y) is considered as a function of x alone. The partial derivative of f(x, y) with respect to y, written , is the derivative of f(x,y), where x is treated as a constant.
Example
Definition
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Interpret Partial Derivatives as Slopes
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Computing Partial Derivatives
EXAMPLE
Compute for
SOLUTION
To compute , we only differentiate factors (or terms) that contain x and
we interpret y to be a constant.
This is the given function.
Use the product rule where f (x) = x2 and g(x) = e3x.
To compute , we only differentiate factors (or terms) that contain y and
we interpret x to be a constant.
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Computing Partial Derivatives
CONTINUED
This is the given function.
Differentiate ln y.
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Computing Partial Derivatives
EXAMPLE
Compute for
SOLUTION
To compute , we treat every variable other than L as a constant. Therefore
This is the given function.
Rewrite as an exponent.
Bring exponent inside parentheses.
Note that K is a constant.
Differentiate.
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Evaluating Partial Derivatives at a Point
EXAMPLE
Let Evaluate at (x, y, z) = (2, -1, 3).
SOLUTION
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Local Approximation of f (x, y)
We can generalize the interpretations of to yield the following general fact:
Partial derivatives can be computed for functions of any number of variables. When taking the partial derivative with respect to one variable, we treat the other variables as constant.
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Local Approximation of f (x, y)
EXAMPLE
Let Interpret the result
SOLUTION
We showed in the last example that
This means that if x and z are kept constant and y is allowed to vary near -1, then f (x, y, z) changes at a rate 12 times the change in y (but in a negative direction). That is, if y increases by one small unit, then f (x, y, z) decreases by approximately 12 units. If y increases by h units (where h is small), then f (x, y, z) decreases by approximately 12h. That is,
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Demand Equations
EXAMPLE
The demand for a certain gas-guzzling car is given by f (p1, p2), where p1 is
the price of the car and p2 is the price of gasoline. Explain why
SOLUTION
is the rate at which demand for the car changes as the price of the car
changes. This partial derivative is always less than zero since, as the price of the car increases, the demand for the car will decrease (and visa versa).
is the rate at which demand for the car changes as the price of gasoline
changes. This partial derivative is always less than zero since, as the price of gasoline increases, the demand for the car will decrease (and visa versa).
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Higher Order Partial Derivatives
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Second Partial Derivative
EXAMPLE
Let Find
SOLUTION
We first note that This means that to compute , we
must take the partial derivative of with respect to x.
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Incorporating Technology
Example 4
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