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Calculus and its applications: 6.2

"Calculus and its applications: 6.2" - Objective: find the partial derivatives of a given function, evaluate partial derivatives, find the four second-order partial derivatives of a function in two variables. | 2012 Pearson Education, Inc. All rights reserved Slide 6.2- Partial Derivatives OBJECTIVE Find the partial derivatives of a given function. Evaluate partial derivatives. Find the four second-order partial derivatives of a function in two variables. 2012 Pearson Education, Inc. All rights reserved Slide 6.2- DEFINITION: For z = f (x, y), the partial derivatives with respect to x and y are 6.2 Partial Derivatives 2012 Pearson Education, Inc. All rights reserved Slide 6.2- Example 1: For find In order to find , we regard x as the variable (biến) and y and z as constants (hằng số) 6.2 Partial Derivatives 2012 Pearson Education, Inc. All rights reserved Slide 6.2- Example 1 (concluded): Similarly, from we get 6.2 Partial Derivatives p. 453, formula 12, there is no “+C” 2012 Pearson Education, Inc. All rights reserved Slide 6.2- 6.2 Partial Derivatives Quick Check 1 2012 Pearson Education, Inc. All rights reserved Slide 6.2- Example 2: For Treating y as a constant Treating x2 and x as a constants 6.2 Partial Derivatives 2012 Pearson Education, Inc. All rights reserved Slide 6.2- 6.2 Partial Derivatives Quick Check 2 2012 Pearson Education, Inc. All rights reserved Slide 6.2- Example 3: For 6.2 Partial Derivatives 2012 Pearson Education, Inc. All rights reserved Slide 6.2- Example 4: A cellular phone company has the following production function for a certain product: where p is the number of units produced with x units of labor and y units of capital. a) Find the number of units produced with 125 units of labor and 64 units of capital. b) Find the marginal productivities. c) Evaluate the marginal productivities at x = 125 and y = 64. 6.2 Partial Derivatives 2012 Pearson Education, Inc. All rights reserved Slide 6.2- Example 4 (continued): 6.2 Partial Derivatives 2012 Pearson Education, Inc. All rights reserved Slide 6.2- Example 4 (continued): 6.2 Partial Derivatives 2012 Pearson Education, Inc. All rights reserved Slide 6.2- Example 4 (continued): 6.2 Partial Derivatives 2012 Pearson Education, Inc. All rights reserved Slide 6.2- Example 4 (concluded): 6.2 Partial Derivatives 2012 Pearson Education, Inc. All rights reserved Slide 6.2- DEFINITION: Second-Order Partial Derivatives 6.2 Partial Derivatives 2012 Pearson Education, Inc. All rights reserved Slide 6.2- DEFINITION (concluded): 6.2 Partial Derivatives 2012 Pearson Education, Inc. All rights reserved Slide 6.2- Example 5: For find the four second-order partial derivatives. 6.2 Partial Derivatives 2012 Pearson Education, Inc. All rights reserved Slide 6.2- Example 5 (concluded): 6.2 Partial Derivatives 2012 Pearson Education, Inc. All rights reserved Slide 6.2- 6.2 Partial Derivatives Quick Check 3 For find the four second-order partial derivatives. 2012 Pearson Education, Inc. All rights reserved Slide 6.2- 6.2 Partial Derivatives Section Summary For the partial derivatives with respect to x and y are, respectively: Simpler notations for partial derivatives are 2012 Pearson Education, Inc. All rights reserved Slide 6.2- 6.2 Partial Derivatives Section Summary Concluded For a surface and a point on this surface, the partial derivative of f with respect to x gives the slope of the tangent line at in the positive x-direction. Similarly, the partial derivative of f with respect to y gives the slope of the tangent line at in the positive y-direction. For the second-order partial derivatives are Often (but not always),