4.1

Chapter 4: Partial Derivatives

Introduction to Functions of Several Variables

  • Objective: Extend differential calculus concepts to functions of multiple variables.

  • Key Questions Addressed:

    • How to calculate limits of multivariable functions?

    • Does the derivative definition involve limits for multivariable functions?

    • Are differentiation rules applicable in multiple variable contexts?

    • Can we find relative extrema of multivariable functions?

4.1 Functions of Several Variables

  • Real-Valued Function:

    • Definition: A function ( f ) defined on domain ( D ) assigns a unique real number ( w = f(x_1, x_2, \ldots, x_n) ) for each element in ( D ).

    • Components:

      • ( D ): Domain of the function.

      • ( w ): Dependent variable of ( f ).

      • ( x_1, x_2, \ldots, x_n ): Independent variables/input variables.

Domain and Ranges of Functions

  • Example 1: Functions of Two Variables

    • Restrictions can affect the domains to maintain real values for dependent variable ( z ).

  • Example 2: Functions of Three Variables

    • Specific restrictions can apply to the domains to preserve real outputs.

Functions of Two Variables

  • Definition:

    • A function of two variables ( z = f(x, y) ) correlates each ordered pair ( (x, y) ) in a subset domain ( D ) of the real plane ( \mathbb{R}^2 ) to a real number ( z ).

    • Domain: Set ( D ), where ( (x, y) ) pairs are valid.

    • Range: Set of real numbers ( z ) for which there exists at least one pair ( (x, y) ) such that ( f(x, y) = z ).

Graphing Functions of Two Variables

  • Graph Definition:

    • The graph of ( f(x, y) ) consists of points ( (x, y, z) ) in ( \mathbb{R}^3 ) such that ( z = f(x, y) ) and ( (x, y) ) belongs to domain ( D ).

    • This graph forms a surface ( S ).

Sketching Graphs of Functions

  • Example 3:

    • Sketch ( f(x, y) = 6 + 3x + 2y ) in the first octant.

  • Example 4:

    • Sketch ( g(x,y) = x^2 + y^2 ).

Level Curves and Contours

  • Definitions:

    • Level Curve: The set of points where ( f(x, y) ) has a constant value ( f(x, y) = c ).

    • Contour Map: The graphical representation formed by these level curves.

  • Example 5:

    • Graph ( f(x, y) = 100 - x^2 - y^2 ) and plot level curves for ( f(x,y) = 0, 51, 75 ).

Functions of Three Variables and Level Surfaces

  • Definition:

    • A level surface is the set of points ( (x, y, z) ) where ( f(x, y, z) = c ) remains constant.

  • Example 6:

    • Identify level surfaces for ( f(x, y, z) = \sqrt{x^2 + y^2 + z^2} ).