Chapter 7 Functions of Several Variables Copyright 2014

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 Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Section 7. 2 Partial Derivatives Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Section Outline • Partial Derivatives • Computing Partial Derivatives • Evaluating Partial Derivatives at a Point • Local Approximation of f (x, y) • Demand Equations • Second Partial Derivative Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Partial Derivatives Definition Example Partial Derivative of f (x, y) If 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. Copyright © 2014, 2010, 2007 Pearson Education, Inc. , then Slide 5

Interpret Partial Derivatives as Slopes Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide

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) = x 2 and g(x) = e 3 x. To compute , we only differentiate factors (or terms) that contain y and we interpret x to be a constant. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 7

Computing Partial Derivatives CONTINUED This is the given function. Differentiate ln y. Copyright ©

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. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 9

Evaluating Partial Derivatives at a Point EXAMPLE Let Evaluate at (x, y, z) =

Local Approximation of f (x, y) We can generalize the interpretations of following general fact: to yield the 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. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 11

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 12 h. That is, Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 12

Demand Equations EXAMPLE The demand for a certain gas-guzzling car is given by f (p 1, p 2), where p 1 is the price of the car and p 2 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). Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 13

Higher Order Partial Derivatives Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 14

Second Partial Derivative EXAMPLE Let . Find SOLUTION We first note that This means that to compute must take the partial derivative of with respect to x. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 15 , we