Ingredients of Multivariable Change: Models, Graphs, Rates

7.1 Multivariable Functions and

Contour Graphs

Multivariable Function• Many of the functions that describe everyday

situations are multivariable functions. • These are functions with a single output variable

that depends on two or more input variables. • For example, – a manufacturer’s profit depends on several variables,

including sales, market price, and costs. – The volume of a tree is a function of its height and

diameter. – Crop yield is a function of variables such as

temperature, rainfall, and amount of fertilizer.

Multivariable Function Notation

• A rule f that relates one output variable to several input variables x1, x2, …, xn is called a multivariable function if for each input (x1, x2, …, xn) there is exactly one output f(x1, x2, …, xn).

Problem 2 – page 532

Multivariable Functions—Graphically• Multivariable functions with two input

variables can be graphed using either contour curves or as a three-dimensional graph

Contour Curve• contour curve is similar to a topographical map, a two-

dimensional map that shows terrain by outlining different elevations.

• Each curve on a topographical map represents a constant elevation and is known as a contour curve.

• In general, a contour curve for a function with two input variables is the collection of all points for which , where K is a constant.

• The contour curve for a specific value of K is sometimes referred to as the K-contour curve or a level curve.

Contour CurveMultiple Level

Not all level curve are continuous

Contour Curves from Data

Interpreting a Contour Curve Sketched on a Table of Data. Problem 14 (page 535)

Problem 20 – Page 537

Change and Percentage Change in Output

Direction and Steepness

• If the input variables of a multivariable function can be compared, the idea of steeper descent can be discussed.

• When the constants K used for the K-contour curves are equally spaced, the steepness of the three-dimensional graph at different points (or in different directions) can be compared by noting the closeness (frequency) of the contour curves.

• If the contour curves are close together near a point, the surface is steeper in that region than in a portion of the graph where the contour curves are spaced farther apart.

consider the elevation of the tract of Missouri farmland with the contour graph

• Starting at (0.4, 1) , will a hiker be going downhill or uphill if he walks 0.4 mile north? south? east? west?

consider the elevation of the tract of Missouri farmland with the contour graph

• Starting at (1.0, 0.6), which direction results in the steeper descent:

• east 0.4 mile or north 0.4 mile? Explain.

Contour Graphs for Functions on Two Variables

• Data tables do not show every possible value for the input and output values of a multivariable function.

• When sketching contour curves on tables, assume that the multivariable function is continuous over the entire input intervals and that the contour curve will be continuous and relatively smooth.

Problem 26-page 538

Formulas for Contour Curves

• Problem 24 –page 537

Ingredients of Multivariable Change: Models, Graphs, Rates

7.2 Cross-Sectional Models and

Rates of Change

Cross-sectional modeling

• Cross-sectional modeling is a simple extension of the data-modeling techniques from Chapter 1.

• Cross sections can be used to understand the behavior of data sets having two input variables.

Illustration of Cross Sections

• The number of jobs held by the average American depends on several variables, including his or her age and level of education, as shown in Table 7.6.

The cross section of the population who received high school diplomas but did not have post-high-school education is represented by the row of data with 4 years of education (highlighted in Table 7.6).

Cross Sections from Three Perspectives

• A cross section of a multivariable function is a relation with one less dimension (variable) than the original multivariable function.

Quick example

Cross-Sectional Models from Data

• When data is given in a table with two input variables and one output variable, modeling the data in one row (or one column) results in a cross-sectional model.

• A cross-sectional model is a model of a subset of multivariable data obtained by holding all but one input variable constant and modeling the output variable with respect to that one input variable.

Problem 2- Page 547

Rates of Change of Cross-Sectional Models

Problem 4, 8, 14 pages 548 - 550

Ingredients of Multivariable Change: Models, Graphs, Rates

7.3 Partial Rates of Change

• Derivatives of cross-sectional functions were discussed in Section 7.2.

• In Section 7.3, the discussion of derivatives is expanded to include derivatives of multivariable functions.

• These partial derivative functions give rate-of-change formulas for all simple cross sections of a multivariable function.

Partial Derivatives• Derivatives describe change in the output value of a function caused

when one input variable is changing. • Derivatives of multivariable function are called partial derivatives

because they describe change in only one input direction, so they give only a partial picture of change.

Partial Derivatives as Multivariable Functions

• Partial derivatives of a multivariable function can be used to find rates of change (with respect to a particular input variable) at any point on the function.

• Partial-derivative functions are multivariable functions with the same number of variables as the original functions.

Second Partial Derivatives• A partial derivative of a partial-derivative

function is called a second partial derivative.

Second Partial Derivatives

Problem: 10, 12, 14, 18, 20, 22, 24

Ingredients of Multivariable Change: Models, Graphs, Rates

7.4 Compensating for Change

Compensating for Change

• When the output of a function depends on two input variables and must remain fixed at some constant level, a change in one of the input variables must be compensated for by a change in the other input variable.

• Tangent lines and partial derivatives are used to answer a questions dealing with compensating for change.

Rates of Change in Three Directions

• A rate of change of the output of a multivariable function with respect to one of the input variables can be found as a partial derivative of the function.

• It is also possible to determine the rate of change of one of the input variables with respect to another input variable.

• For functions on two input variables, such a rate of change is represented graphically as a line tangent to a contour graph.

Lines Tangent to Contour Curves

• On a function f with two input variables x and y, if the output is constant at level K, the rate of change of one input variable with respect to the other input variable at a point on the K-contour curve is the slope of the line (in the f = k plane) tangent to the curve at that point.

The Slope at a Point on a Contour Curve• For a function f with input variables x and y, the

slope of a line tangent to a constant contour level can be computed using partial derivatives of f.

Compensation of Input Variables• The change needed in one input variable to compensate for

a change in the other input variable to maintain a constant function output is approximated using a line tangent to a contour curve. The slope of the tangent line can be calculated either directly from an algebraic formula, giving one input variable in terms of the other variable, or indirectly by using partial derivatives of the function.

Problem: 2, 10, 18