Welcome to MTH-114
APPLIED BUSINESS
Chapter 1.1
Calculus is the study of the rate of change of values of interest. Example:
A credit card company charges a daily finance charge of 0.062% on unpaid balances. At what
rate is the balance increasing
Solution:
The balance is increasing at a
rate of:
$1.25 per day after 10 days $1.32 per day after 100 days $1.55 per day after 365 days
Example:
The US Census Bureau has
developed a mathematical model to estimate the percentage, p, of the US population 65 years and older,
where x is the number of years since 2000. In what year is this
Solution:
The percentage of the US
population 65 years old or older is increasing most rapidly in the
year 2022. The rate of increase
is 16.4% per year.
Recall that a function is a
mathematical expression, or rule, which quantifies the relationship between:
• an input or independent
variable, and
• an output or dependent
If x is the input to a function f, then the output is written f(x).
This is functional notation.
Other variable names may be
associated with both the input to, and output from, a function.
For example, if g is the input to a function C, then the output is written C(g).
Recall that when we graph a function on a Cartesian
coordinate system:
• Input values are graphed on the horizontal axis and are increasing from left to right
• Output values are graphed on the vertical axis and are
The graph indicates, for example, that • an input value of -1
A function assigns exactly one output to each input.
It is easiest to identify a function
when it is displayed graphically.
This is because any vertical line will intersect a function only
It may be possible to determine or estimate the output from a
function in the following ways: • Numerically, for example with
data in a table
• Algebraically, with a function • Graphically
Following is a function represented
numerically in the form of a table:
input, t output, g(t)
2 18
3 54
4 156
The function may be represented
graphically as:
graphically,
The same function may be
represented algebraically as:
Therefore:
When units are associated with the input to, and output from, a function, then they must be
included in any representation of the function.
Here the units are displayed on the axes of this graphical representation of the function. • Input units: hours since
dosage
• Output units: mg/mL
(micrograms medication per milliliter of blood)
When a function is used to approximate the relationship between inputs and
outputs in a real-world application, the
Example:
The resident US population
between 1900 and 2000 can be modeled by the following:
P(t) = 80(1.013t)
Input: t (years) since end of 1900 Output:P(t) (millions of people)
The model, implies:
• the resident US population at the end of 1900 was 80 million, and
• The population increased by 1.3% (.013) per year between 1900 and 2000.
This is an example of an
exponential model, one we will be
Two general types of modeling problems are of interest:
1. given an input value, what is the
associated output value,
2. given an output value, what is the
In the following question
• we know the input, t, and • are interested in the
associated output, P(t)
According to the model, what was the resident US population at the end of 1945? Since, for this model, t is in years since the end of 1900, we are
At the end of 1945 the US population would have been 143.1 million, according to this model.
In the following question
• we know the output, P(t), and • are interested in the
associated input, t
According to the model, in what year will the population be 250 million? That is, for what value of t is P(t)=250?
Solving for t:
250
80
=
(
1.013
�)
��
(
250
80
)
=
��
(
1.013
�
)
��
(
3.125
)
=
� ��
(
1.013
)
1.14
=
�
(
.0129
)
where
�
=
1.14
.0129
�
=
88.2
�����
Since 88 years would mark the end of 1988, the population was 250 million in the beginning of 1989, according to this model.
1.14
=
�
(
.0129
)
In the last example we sought the input value, t, which would produce a specific output value, P(t) = 250. We were able to
mathematically solve for t, that
is, isolate t on one side of the equation.
We will encounter other functions for which it will not be
mathematically possible to
isolate the input variable on one side of the equation. In fact, we have already encountered such a function:
What input value, t, is associated with an output value g(t) = 300?
We cannot solve this equation algebraically. Rather we will
later see how to solve this using the Equation Solver application of the TI-84 calculator.
Here, we now consider some
other characteristics of functions. An increasing function is a
function for which the output values are increasing as the input values are increasing.
A decreasing function is a function for which the output values are decreasing as the input values are increasing.
A decreasing function falls from
A constant function is a
function for which the output
values are the same for all input values.
A constant function graphs as a
Some functions may have
• intervals over which they are increasing,
• other intervals over which they are decreasing, and
• other intervals still over which they are constant.
The function below increases for input values between 0 and about 0.8 and decreases for input values greater than about 0.8 .
The curvature of a function
relates to whether the graph of the function is opening upwards or opening downwards.
• A concave up function opens upwards. A concave up
function can “hold water”. • A concave down function
opens downwards. A concave up function cannot “hold
Some functions may have
• intervals over which they are concave up and
• other intervals over which they are concave down.
A point on a continuous function at which the concavity changes
• For input values less than ~1.8 the graph is concave down.
• For input values greater than ~1.8 the graph is concave up .
The function below has an inflection point near an input value of about 1.8.
The end behavior of a function refers to the value to which the output approaches as the input value increases or decreases without bound.
That is, we are interested in what
happens to the function as the input value approaches
• very large positive numbers
(input +), or
• very small negative numbers
There are 3 possibilities for the end behavior of a function. The
output value may:
1. equal, or approach, a finite
value (that is a finite limit)
2. increase or decrease without bound,(that is, increase or
decrease infinitely) or
• As t approaches +,
v(t), the output, approaches 20.
• As t approaches -,
v(t) approaches -3.
v(t) approaches finite output values as t
g(x) does not approach finite output values as x increases and decreases without
j(x) oscillates. It neither approaches a finite limit nor increases or decreases infinitely. Functions that
behave like j(x)
display sinusoidal
behavior.
The output values of such functions repeat on a
The function f(x) does not approach a finite value as x decreases without bound but
We can numerically estimate end behavior of a function by substituting increasingly larger and increasingly smaller input values into an algebraic expression of a function.
This is an opportune time to turn our
Most keys have three functions:
1. The function identified on the key.
2. The function
accessed via the 2nd key to which it is color coded.
3. The function
accessed via the ALPHA key to which it is color coded.
Note:
Subtraction is performed using this key.
Negative values are entered using
Note: Be sure the 2nd entry on this screen has FLOAT highlighted.
We will be making extensive use of the LISTS on the calculators to enter input and output data to be modeled.
Here we make sure the Lists are set up properly. Before you do anything select STAT and then EDIT Edit.
Otherwise, to set up the six lists, L1 through L6, select STAT and then EDIT SetUpEditor.
Prepare the calculator for graphing. Select FORMAT (2ND ZOOM). Be sure all the items in the left column, as indicated below are highlighted. To toggle between two choices (for example between RectGC and
PolarGC in the first entry), use the cursor arrows to move to the item you desire, and hit ENTER. The item you desire should now be highlighted, that is, have a black background.
Perform some sample calculations as indicated below. The screen shots show the correct results.
Perform some sample calculations as indicated below. The screen shots show the correct results.
Sometimes you enter a fairly complicated expression to evaluate and then later you need to make a slight modification to that expression and
re-evaluate it. To do so, you can use ENTRY (2ND ENTER). Every time you select ENTRY, the calculator displays the next prior expression you typed in.
Suppose you execute an expression and then you need to, say, multiply it by a value. Just select the X (multiply key) and the calculator will display ANS, indicating your last result. Then enter the number that you want to multiply the previous result by.
Suppose now we want the same function to be evaluated at a different input value:
Using the TABLE (2nd GRAPH) function:
When a function has already been entered into memory, say as Y1, and we wish to evaluate it at several input values, it is
easiest to do so by creating a Table by using the TABLE command.
But first we need to set up the table. Select TBLSET, immediately below the screen by selecting 2nd WINDOW.
Under the entry for independent variable
(Indpnt) toggle to the right to select Ask. This will permit you to enter values into the table at will without depending upon using a table starting value (TblStart) and a Dt value (DTbl).
Now access TABLE immediately below the screen by selecting 2nd GRAPH.
The 1st column should have a heading of X.
If there is no heading in the 2nd column, go
back to the list of equation using Y=. Make sure the equal sign in the appropriate Y equation is highlighted. That is, make sure there is a solid black block behind the equal sign in the equation. If not, toggle to the equal sign and select ENTER.
Enter 4 in the first row under X. The calculated functional value, 155.79, will appear in the Y1
column.
Enter 5 in the second row under X. The calculated functional value, 435.24, will appear in the second row of the Y1 column.
Enter more values in the X column if you like.
You can see more decimal digits for any value in the table by resting the cursor on a table value, and then viewing the value at the bottom of the screen.
Here, for example, by
resting the cursor on Y1(4) we see the output, shown as 155.79 in the table, actually appears with 9
decimal digits at the bottom of the screen.
Earlier in class we sought the value of the input variable, t, which would produce an output value of g(t) = 300 for the function
Here we pursue this solution using the Equation Solver.
NOTE: To access the Equation Solver, select MATH and then scroll up to select Solver.
The following will appear:
Equation Solver eqn: 0 =
All Equation Solver equations must have 0 on one side of the equation.
So, for example, if you wish to solve
the equation in the Equation Solver must be written as:
�
1
=
300
0
=
�
1
−
300
This value should be 300, not 30.
NOTE: The cursor MUST BE ON “X =“ for the Equation Solver to solve the equation.
Note: The Plot1, Plot2 and Plot3 headings at the top must not be highlighted. If any of them has a black background (highlighted), then use the arrow keys to access them, and press ENTER to toggle them off.
Recall that we initially turned our attention to the calculator so that we could
numerically estimate end behavior of a function. We can do so by
• entering the function using the Y= key, and then
• entering increasingly larger and
increasingly smaller input values into the calculator’s table.
The best way to determine endpoint behavior is by using the Table.
Select Y= immediately below the
screen. Make sure only the Y variable which you used to write the function is highlighted (black background behind equal sign in the equation).
Select Table (2nd GRAPH) and enter
increasingly larger positive numbers (10, 100, 1000, etc.), then increasingly smaller negative numbers (-10, -100, -1000, etc.) into the X column.
The functional values will be
immediately displayed in the appropriate Y column.
x f(x) 10 0.9090 100 0.9901 1000 0.9990 10,000 0.9999 � (� )= �
� +1
lim
� � ( � )=1
x f(x)
-10 1.1111
-100 1.0101
-1000 1.0010
-10,000 1.0001
� (� )= �
� +1
In the next example, which uses real-world data, we will explore some of the concepts we have just learned.
Based upon actual and projected data available from the Statistical Abstract,
2009, the number of credit card holders in the US can be modeled by the following function. Note: we will later see this
function is derived by combining a
logistic function with a constant
� ( � )= 29
1+18 �− 0.43� +158
where
• t is the number of years since the end of 2000, and
• C(t) is the number of cardholders in units of millions.
C(t) is graphed on the next slide over the interval 0 < t < 15.
� (� ) = 29
1+18 �− 0.43� +158
Is C(t) increasing or decreasing
over the interval 0 < t < 15?
Is C(t) increasing or decreasing
over the interval 0 < t < 15?
C(t) is increasing over the interval 0 < t < 15.
Describe the
concavity of C(t) over the interval 0 < t < a?
Describe the
concavity of C(t) over the interval 0 < t < a?
Over the interval 0 < t < a, C(t) is concave up.
Describe the
concavity of C(t) over the interval a < t < 15?
Describe the
concavity of C(t) over the interval a < t < 15?
Over the interval a < t < 15, C(t) is concave down.
What is the mathematical significance of
What is the mathematical significance of
the input value a? The input value a is an inflection
point. C(t) changes
concavity at an input value of a.
Numerically
estimate the end behavior of C(t).
� (�)= 29
1+18 �−0.43� +158
Numerically
estimate the end behavior of C(t).
t C(t) 10 181.31 100 187.00 1000 187.00 10,000 187.00 lim
� � (� )=187
� (�)= 29
1+18 �−0.43� +158
Some applications involve the
construction of more complicated functions from simpler functions.
For example, simple functions can be combined by multiplying, dividing, and so forth.
Our main purpose here is to explore some basic business principles.
Business & Economics Terms:
• Fixed costs are those which are constant regardless of the number of units produced.
• Variable costs are those which
are dependent upon the number of units produced.
• Total cost is the sum of fixed and variable costs.
• Average cost is total cost divided by number of items produced.
Business & Economics Terms (cont): • Revenue is the product of selling
price per unit and the number of units sold.
• Profit is revenue minus cost. • The Break-Even Point is the
number of units for which:
Total Cost = Revenue Profit is $0 at the break-even point.
Example: Producing Moonwalks
A toy company produces inflatable moonwalks for children’s events. Monthly fixed costs to run the
operation are $90,000 while the cost to produce each unit is $600. The product sells for $1800 each.
Example: Producing Moonwalks • Fixed costs:
• Variable costs:
x is number of moonwalks produced
• Total costs:
Example:
Example: Producing Moonwalks • Average cost:
´
�
(
�)
= �(
�)
� =
$ 90,000+$ 600 �
�
• Revenue: R
(The average cost function is
constructed by dividing a cost
´
�
(
�)
= �(
�)
� =
$ 90,000+$ 600 �
� Ex: tttttttttttt ttttt $000’s/moonwalks produced
Problem:
1. Use the TI-84 Solver to find the
break-even point for the moonwalk example. Enter C(x) for one
variable, say Y1, and R(x) for
another, say Y2, and use the Solver to find out when they are equal.
2. What is the revenue, and cost, at the break-even point?
R
(
�
)
=
$
1800
�
R
(
�
)
=
�
(
�
)
