8.3 Partial Derivatives

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Lecture 28
Functions of Several Variables
Chapter 8
8.3 Partial Derivatives
Ex.
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Partial Derivatives
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Partial Derivatives

• The partial derivative of f with respect to x is
  the derivative of f with respect to x, when all
  other variables are treated as constants.
• Similarly, the partial derivative of f with
  respect to y is the derivative of f with respect
  to y, when all other variables are treated as
  constants.
• The partial derivatives are written

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Partial Derivatives
Ex.
Ex.
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Partial Derivatives
Ex.
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Geometric Interpretation of Partial Derivatives
is the slope of the tangent line at the
point P(a,b, f (a,b)) along the slice through y
b.
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Second-Order Partial Derivatives
Ex.
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Notation for Partial Derivatives
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Example
Marginal Cost Linear Model ? Suppose you own a
company that makes two models of speakers, the
Ultra Mini and the Big Stack. Your total monthly
cost (in dollars) to make x Ultra Minis and y Big
Stacks is given by
What is the significance ?C/?x and ?C/?y?
Solution
The cost is increasing at a rate of 20 per
additional Ultra Mini (if productions of Big
Stacks is held constant).
The cost is increasing at a rate of 40 per
additional Big Stack (if productions of Ultra
Mini is held constant).
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Example
Marginal Cost Interaction Model ? Another
possibility for the cost function in the previous
example is the interaction model
a. What are the marginal costs of the two models
of speakers?
b. What is the marginal cost of manufacturing Big
Stacks at a production level of 100 Ultra Minis
and 50 Big Stacks per month?
Solution
The marginal cost of manufacturing Ultra Minis
increases by 0.1 for each Big Stack that is
manufactured.
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The marginal cost of manufacturing Big Stack
increases by 0.1 for each Ultra Minis that is
manufactured.
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Exercise (Waner, Problem 43, Section 8.3)
Market Share (Cars and Light Trucks) ? Based on
data from 1980-1998, the relationship between the
domestic market shares of three major U.S.
manufacturers of cars and light trucks is
where x1, x2, and x3 are, respectively, the
fraction of the market held by Chrysler, Ford,
and General Motors. Calculate ?x3/?x1 and
?x1/?x3. What do they signify, and how are they
related to each other?
Solution
General Motors market share decreases by 2.2
per 1 increase in Chryslers market share if
Fords share is unchanged.
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Chryslers market share decreases by 1 per 2.2
increase in General Motors market share if
Fords share is unchanged.
That is, the two partial derivatives are
reciprocals of each other.