Download Chapter 4 – Mathematical Modeling

Chapter 4 – Mathematical Modeling
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Definition of a Model
o A mathematical model is an equation (or equations) that behaves like a real
world system.
o We use variables to represent the quantities that we are studying.
 Some variables are classified as independent. The values of these
variables are determined by the researcher. These are called input
variables. (in mathematics they are called domain variables)
 Some variables are classified as dependent. The values of these
variables depend upon the independent variables. These are called
output variables. (In mathematics they are called range variables)
Definition of a Function
o A function is a method used to assign outputs to inputs. It has the following
requirements:
 Assigns exactly one output to each input. (For example if we have an
input of x = 4, we do not want to have y = 6, and y = 10 be its outputs.
We would not be able to predict which one is the proper output.)
 Example: input – social security number, output – person’s
name. (bummer if there are two with the same ssn)
 Every input value is assigned. (This is a real problem with ssn. If
someone dies and that ssn is not eliminated or reassigned then the ssn
could be used fraudulently.)
o The names of variables do not matter.
 Example y = x2 has the same meaning as w = t2 .
First Models
o Modeling a real life situation means find an equation that behaves like the real
world situation.
 Example: Don Hill pays me $10 dollars an hour when I work for him.
Determine a model for the amount of money I make.
 Label your variables
 List several inputs and outputs
 Draw a graph
 Determine an equation.
 Example: When I rent a truck from Rent-A-Wreck it costs $25 to rent
the truck, plus 20 cents per mile. Determine a model for the amount of
money it costs to rent the truck.
 Label your variables
 List several inputs and outputs
 Draw a graph
 Determine an equation
Linear Models
o A linear function has the form y = mx + b, where m and b are numbers, and x
and y are variables.
o Which of the following are linear functions:
 y  4x 1
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4x – 5y = 6
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y  x 2 1
x2
 11
3
 4 y  7 x  12
Worksheet Part 1: Identify Linear Functions
Slope Of A Line
o For the line equation y = mx + b, the slope is m.
y rise
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o Slope is defined as m 
x run
o Given any two points P(x1, y1) and Q(x2, y2) the slope of the line going
y 2  y1
through these two points is m 
. (Note: x1≠x2)
x 2  x1
 If x1 = x2 then the slope is undefined. (a vertical line)
o Example: y = 2x – 3. What is the slope?
o Example: Find the slope of the line through the points P(5, 8) and Q(-2, 4).
o Draw lines with slope m = 1, m = 4, m = ¼, m = -1, m = -4, m = -1/4
The y-intercept
o For the line y = mx + b, the y-axis intercept is b.
o If you know the equation of the line, then the y-axis intercept can be found by
letting x = 0.
o If you know a point on the line and the slope you can find the y-intercept.
 Substitute in the point for x and y, and the slope for m, and solve.
The Equation of a Line
o Example: Find the equation of the line through (1, 2) and (3, 4).
o Example: Ron’s Widget Repair Shop hasn’t been open long. In January, the
shop showed a net loss of $500. In May, however, it showed a profit of
$1000. If a linear trend continues, what profit can Ron expect in November?
o Example: The depth of water in a rectangular tank depends on how long it has
been filling. When it has been filling for four minutes, the water is seven
inches deep. Write an equation that gives the depth d of water after the tank
has been filling for t minutes. If the tank is 2 feet tall, how long does it take to
fill completely?
Work Part 2: Line Equations
Quadratic Models
o A quadratic model has the form ax 2  bx  c , where a, b, c are real numbers
and a ≠ 0
o The graph of a parabola has a U shape. If a > 0, the U opens up. If a < 0, the
U opens down.
 Show several graphs on the calculator (y = ax2)
o As |a| increases, the steepness of U increases.
 Show several different values of a for (y = ax2)
o The point at which the graph turns around is called the vertex.
 If V(x, y) is the vertex then:
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y  3x 
b
2a
 y is found by substituting the above x value into ax 2  bx  c .
o Example: For y = x 2  6 x  4
 Does the graph open up or down?
 Draw the graph.
 What is the vertex?
 Use the calculator to construct a Tchart of x, y values.
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o Example: For y = 3 – x2
 Does the graph open up or down?
 Draw the graph.
 What is the vertex?
 Use the calculator to construct a Tchart of x, y values.
Work Part 3: Quadratic Models
Quadratic: Finding the intercepts.
o The y-axis intercept is found by setting x = 0.
o The x-axis intercept(s) are found by setting y = 0.
o Example: For y = x 2  6 x  4 find the intercepts.
o Example: For y = 3 – x2 , find the intercepts.
Work Part 4: Intercepts
Quadratic Problems:
o Many things behave according to a quadratic model. Anything that is in
motion, and is affected by gravity, is modeled by a quadratic equation.
o Example: the path of a kicked football can be modeled by
y  0.333x 2  1.42 x where the ball is kicked at point (0, 0). Here x is the
horizontal distance that the ball has traveled since it was kicked, in yards, and
y is the height of the ball above the ground, also in yards.
 Where does the ball hit the ground?
 What is the maximum height of the ball during its flight?
 What is the height of the ball when its horizontal distance from where
it hits the ground is 15 yards?
o Example: A cannon ball is fired in such a way that its height above ground is
given by y  16t 2  600t  3 , where t is time in seconds, and y is the height
of the cannon ball in feet.
 What is the height of the cannonball 3 seconds after it is fired?
 When does the cannonball reach its maximum height?
 What is the maximum height of the cannonball?
 When does the cannonball hit the ground?
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Suppose that the horizontal distance that the cannonball travels is
given by d = 600t. How far does the cannonball travel horizontally
before it hits the ground?
Work part 5: Quadratic Problems
Exponential Models
o An exponential model is modeled by an equation of the form y = bx, where the
variable x is an exponent.
o Exponential models are very important in many scientific fields.
o The graph of an exponential
 Determine the horizontal asymptote. (For y = bx, this is y = 0). Draw
a dashed line.
 Solve y = bo = 1. Plot this point.
 Create a t-chart with 3 values below 0, and 4 values above 0. Plot
these points.
 Connect the 8 points with a smooth curve and extend both ends of the
curve off toward infinity. (The graph has an appearance of a hokey
stick and is sometimes called the hokey stick graph)
 Example: Graph y = 2x. (increasing exponential)
 Example: Graph y = 3-x. (decreasing exponential)
 Example: Graph y = 0.75x. (which is it?)
 Example: Graph y = 0.25-x
 Note: the difference between the graphs (the shape of the hokey stick)
in relation to the exponent or the value of the base, where they
approach the horizontal asymptote and where they diverge to infinity.
Work part 6: The Graph of Exponentials
o So far the exponential functions have a horizontal asymptote at y = 0. This
can be changed.
 Example: y  3  2 x . The horizontal asymptote is at y = 3.
 Example: y  4  3 x . The horizontal asymptote is at y = -4.
o Because it appears so often in solutions to mathematical problems and
surprisingly in nature, the natural exponential has a symbol representing it,
just like π. Its symbol is e, and has an approximate value of 2.7182818….
Like π it is an irrational number.
 Example: A company has determined that the profit p for a fashion
item it has had on the market for t years is given by
p  5000  30000e 0.1t . What profit can be expected in the tenth year?
 In 1970, the population of the U.S. was 205,052,000, the birth rate was
18.4 per 1000 population, and the death rate was 9.5 per 1000.
Determine the number of people in the U.S. in 1991, assuming that
these rates remained constant. ( P  P0 e kt )
Work part 7: Exponential Problems
o Using your calculator, find each of the following:
 e / 2.
 e2
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