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 Depdendent Variable

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 Dependent Variable

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Section 3.4

Modeling continuous relationship which is
increasing and concave down

Model: Cost Function

A company manufactures a
quantity q of a product per month.
They invest \$2.3 million in capital
towards technology (machinery,
computers, etc.) and a fixed cost
investment of \$5.4 million. Data
is collected on the cost for varying
quantities q (in 100,000 units).

Determine a model for cost.

Cost Data

Properties of Model

Is the data increasing or decreasing?
Cost increases as production increases

Is the data concave up or concave down?
Data is concave down

Use the Ladder of Powers to determine a
model with these characteristics. Can it be a
polynomial model?

No, polynomial functions are either increasing
concave up or decreasing concave down

Function of form
where p(x) is a polynomial, integer n>1

If n is even, what is the domain of r(x)?
If n is odd, what is the domain of r(x)?
What is the concavity of a radical power
function y = axn when 0 < n < 1?

index 0<n<1

Ladder of Powers to decide if n = ½,
n = 1/3, n = ¼ etc.

Graph and , which is
a better fit to the Cost Data?

Use Derive or a graphing calculator to fit a

Solution:
Cobb-Douglas Production Function

Cost Model

-Algebraic Method

Undo the nth root by using the inverse
function –the nth power

For which nth roots do we need to check for
extraneous roots?
If nth root is even, taking an even power can
result in extraneous roots
If nth root is odd, taking an odd power will not
result in extraneous roots

Isolate one of the radical expressions

Take the square to remove one of the square roots

Isolate the remaining radical and square again

Check for extraneous roots
Solve using algebraic and graphic method

Class Participation Exercise

1. Determine what
type of function
has the given
graph? Is it
polynomial or

2. Solve the following radical equation:

Linearization

make nonlinear polynomial
data linear

Model the linear data

Convert linear model into
polynomial model

Linearization

Use Derive or graphing calculator to
plot data and transformed data

Which of these makes the data linear?

Linearization –Transformed Data

Linearization –Data Plot

Linearization

Which transformed data is linear?

Solution: Cube root, so data is modeled by
a cubic function

Linearized data model: y = 1.3x + 0.77

Actual data model:

Best fit cubic model: