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 twodimensional 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.
Multiple Level
Not all level curve
are continuous
Contour Curve
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-ofchange 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 Kcontour 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