Lecture5bc - Welcome to Math 122 Lecture 5 Survey of...

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Unformatted text preview: Welcome to Math 122 Lecture 5 Survey of Calculus and its Applications I1 Instructor: Quanlei Fang Dept. of Math, University at Buffalo, Spring 2009 Chapter 7 Functions of Several Variables Last time Examples of Functions of Several Variables § 7.2 Partial Derivatives Section Outline Partial Derivatives Computing Partial Derivatives Evaluating Partial Derivatives at a Point Local Approximation of f (x, y) Demand Equations Second Partial Derivative Partial Derivatives Partial Definition Example Partial Derivative of f (x, y) If f ( x, y ) = 2 x 3 y 4, then with respect to x: written ∂f , ∂f ∂x = and the derivative of f (x, y), ∂x where y is treated as a constant and f (x, y) is ∂f = considered as a function of x ∂y alone Computing Partial Derivatives EXAMPLE Compute SOLUTION ∂f ∂f for f ( x, y ) = x 2 e 3 x ln y. and ∂x ∂y Computing Partial Derivatives Computing EXAMPLE Compute ∂f ∂L for f (L, K ) = 3 LK . SOLUTION Evaluating Partial Derivatives at a Point EXAMPLE Let f (x, y, z ) = xy 2 z + 5. Evaluate ∂f at (x, y, z) = (2, -1, 3). ∂y SOLUTION Local Approximation of f (x, y) Local Local Approximation of f (x, y) EXAMPLE Let f (x, y, z ) = xy 2 z + 5. Interpret the result SOLUTION ∂f (2,−1,3) = −12. ∂y Demand Equations Demand 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 ∂f < 0. ∂p2 ∂f < 0 and ∂p1 SOLUTION Second Partial Derivative EXAMPLE Le f (x, y ) = xe y + x 4 y + . 3. Find y SOLUTION ∂2 f . ∂x∂y ...
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This note was uploaded on 02/20/2009 for the course MTH 122 taught by Professor Buettgens during the Spring '08 term at SUNY Buffalo.

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Lecture5bc - Welcome to Math 122 Lecture 5 Survey of...

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