Ingredients of Multivariable Change: Models, Graphs, Rates

1
Ingredients of Multivariable Change Models,
Graphs, Rates

• 7.1
• Multivariable Functions and
• Contour Graphs

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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 manufacturers 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.

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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).

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Problem 2 page 532
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Multivariable FunctionsGraphically

• Multivariable functions with two input variables
  can be graphed using either contour curves or as
  a three-dimensional graph

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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.

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Contour Curve
Multiple Level
Not all level curve are continuous
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Contour Curves from Data
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Interpreting a Contour Curve Sketched on a Table
of Data. Problem 14 (page 535)
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Problem 20 Page 537
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Change and Percentage Change in Output
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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.

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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?

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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.

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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.

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Problem 26-page 538
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Formulas for Contour Curves

• Problem 24 page 537

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Ingredients of Multivariable Change Models,
Graphs, Rates

• 7.2
• Cross-Sectional Models and
• Rates of Change

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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.

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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).
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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.

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Quick example
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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.

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Problem 2- Page 547
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Rates of Change of Cross-Sectional Models
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Problem 4, 8, 14 pages 548 - 550
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Ingredients of Multivariable Change Models,
Graphs, Rates

• 7.3
• Partial Rates of Change

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• 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.

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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.

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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.

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Second Partial Derivatives

• A partial derivative of a partial-derivative
  function is called a second partial derivative.

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Second Partial Derivatives
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Problem 10, 12, 14, 18, 20, 22, 24
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Ingredients of Multivariable Change Models,
Graphs, Rates

• 7.4
• Compensating for Change

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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.

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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.

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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.

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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.

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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.

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Problem 2, 10, 18