 
Brief User Guide for TI83 Plus and TI84 Business and Economics
INDEX:
To facilitate lookup, the instructions are divided into
the following categories:
I.
Supply, Demand, and Costs  Equilibrium point; Marginal average cost;
Marginal
cost and “Next Item”
cost; Maximum Profit; Price, Revenue, and Quantity at given demand; determining pricedemand
function
from given data,
II. Miscellaneous  Harmonic mean, Geometric mean, Mean of grouped data, How long to double interest, Leontief
inputOutput
problem,
III. Linear Programming  Graphical Method; Graphing Using the Inequality
APP, Simplex Method,
IV. Calculus Applications  Cost,
Average Cost, Marginal Cost, AverageCost Minimization, Cost minimization,
Complementary/Substitute Product Determination,
Equality of Marginal Product of Labor and Capital, Consumers'
surplus,
Suppliers' surplus
IRELEASE DATE: 2/2/10 DATE LAST
REVISED: 2/10/12
© 2003 Frank Kizer
NOTE: See copy restrictions and printing
hints at the end of this document.
General:
Some of these problems can be solved more easily with a little calculus and some
manipulations with paper
and pencil. Nevertheless, I have included some rather
simple problems as examples of how a graphing calculator
can be used.
I. Supply,
Demand, Costs and Related Items:
1.
Equilibrium Point:
A certain
produce has a demand function p = 2x² +80 and a supply function of p= 15x +25.
If x
represents the demand quantity in thousands and p
the unit price in dollars, find the
equilibrium quantity and price.
a) Press Y= and enter the demand
equation opposite Y_{1 }and the supply equation opposite Y_{2 }
.
Use the [x, T, 0, n] key to enter “x” and the x² key
to enter the exponent.
b) Press Zoom, 0 (ZoomFit) to get the graphs on
the screen. Now press WINDOW and set
Xmin and Ymin
=0 and press GRAPH.
c) Press 2^{ND}, CALC, 5(Intersect).
d) From the graphing screen that appears, press ENTER,
ENTER; then move the cursor
approximately to the intersection of the graphs and press ENTER again.
e) The equilibrium point price, y= 67.45, and equilibrium
quantity, x=2.69699, will be displayed. Multiply the xvalue by
1000 to get 2689.99 or 2690
rounded off.
2. Marginal
Average Cost:
The cost function for manufacturing a certain household
item is c(x) = 5000+2x. Find the
marginal average cost for 500 units.
Preliminary: The average cost function is c(x)/c =
(5000+2x)/x.
a) Enter (5000+2x)/x opposite Y_{1}.
b) Set Xmin=0, Xmax=550, Ymin=0, and Ymax=150 and
press GRAPH.
c) Press 2^{ND}, CALC, 6(dy/dx).
d) Press ENTER and the value 0.02 will be displayed.
3. Marginal
Cost and “Next Item” Cost :
The cost of
a certain product is given by c=700+0.03x +.0002x² . Compare the marginal cost
and
the “next item” cost at 150 units.
a) Press Y= and enter 700+0.03x +.0002x² opposite Y_{1
}. Use the [x, T, 0, n] key to enter “x” and
the x² key to enter the exponent.
Press ZOOM, move the cursor to ZoomFit and press ENTER.
b) Press 2^{ND, }CALC, 6(dy/dx). And press
ENTER.
c) On the screen that appears, enter 150 and press ENTER to
get dy/cy=.09 for the marginal
cost.
Now,
find the cost of item 151.
a) Press 2^{ND}, CALC, ENTER, for Value.
b) ENTER 150 and press ENTER to get 709.
c) Press 2^{ND}, CALC, ENTER.
d) Enter 151 and press ENTER to get 709.902.
e) This gives 709.902709=0.0902 for the actual cost of manufacturing
item 151.
4. Finding
Maximum Profit:
The demand
of a certain product is represented by p=40/√q and the cost of producing q items
is given by c= .04x +400. Find the number of items that gives maximum
profit and the price
that yields maximum profit.
We know, of course, that the maximum profit occurs when
the cost of producing another item
(approximated by the marginal cost)) is equal to the revenue and on that
item (marginal
revenue).
P=RC
= xp(x)  C(x)
=x(40/√(x)) –(0.4x +400)
=40√(x) –0.4x 400
a)
Enter 40√(x) –0.4x 400 opposite Y_{1} using the [x,T,0,n] key to enter
“x.”
b) Press WINDOW and set xmin=0, Xmax=5000, Ymin=300,
and Ymax=700. Press GRAPH.
c) Press 2^{ND}, CALC, 4(maximum).
d) Move the cursor somewhat to the left of the maximum of
the curve and press ENTER.
Move the cursor somewhat to the right of the maximum of the curve
and press ENTER.
Finally, move the cursor approximately to the peak and press ENTER
again.
e) The terms displayed will indicate x=2500 and y=600. So, the maximum
profit occurs at 2500
items and the maximum profit is $600.
f) From the price equation, do the following
calculation:
p= 40/√x
p= 40/√2500
p=0.80
5. Price at
a Given Demand:
The
relation between price, p, and quantity demanded, q, is given by the equation
p² =120002x. Find the price when the demand is 3900 and 4100.
First solve for the price as a function of x.
p² =120002x
p=√(120002x)
Ex 1:
a) Press Y= and enter √(120002x)
opposite Y_{1}. Use the [x,T,0,n] key to enter “x.”
b) Press WINDOW and set Xmin=0, Xmax=6500, Ymin=0, and
Ymax=110. Press GRAPH to
display the graph.
c) Press 2^{ND}, CALC, ENTER and
enter 3900 opposite X.
d) Press ENTER and the answer 64.807… will be displayed.
e) After you have recorded the last answer, enter 4100 and
press ENTER to get 61.644…
Ex 2:
For the pricedemand relation of Ex. 1 above, find the maximum
revenue and the quantity
that gives the maximum revenue.
a) Enter x√(120002x), the expression
for R(x), opposite Y_{2}.
b) Move the cursor to the = sign at Y_{1} and press
ENTER to disable the graph for
revenue. Press ZOOM, 0(ZoomFit) to display the graph.
c) Press 2^{ND}, CALC 4(maximum).
d) When the graph appears, move the cursor
somewhat to the left of the peak and
press ENTER. Move the cursor somewhat to the right of the peak
and press
ENTER. Move the cursor approximately to the peak and press
ENTER again.
e) A quantity of x=4000 and revenue of
y=252982.21 will be displayed.
Ex. 3:
Find the price at maximum revenue.
a) Disable Y_{2} by moving the cursor to the = sign and
pressing ENTER. Enable Y_{1} by moving the cursor
to the = sign opposite Y_{1} and
pressing ENTER.
b) Press ZOOM, 0(ZoomFit) to display the graph.
c) Press 2^{ND}, CALC, ENTER.
d) From the graph screen, enter 4000 opposite x and press ENTER. The
price
63.245… will be displayed.
6. PriceDemand
Function from Given Data:
I will give the method for doing this problem totally with the calculator; then
I will give a method for doing
detailed calculations to find the constants “a” and “b” in
the bestfit equation
A company determines the pricedemand relationship
of one of its products is related as given in the table
below. Determine the pricedemand function,
the price at 8000 units, the maximum revenue and the value at
at which maximum revenue occurs.
Quantity (x 10^{3 }) 
Price 
1000 
72 
4000 
63 
9000 
48 
14000 
33 
20000 
15 
a) Press STAT, ENTER and enter the
quantities in list L_{1 }and the prices in L_{2}.
b) Press STAT, move the cursor to CALC and
press 4 (linReg (ax+b)).
c) Press VARS, move the cursor to YVARS,
and press ENTER, ENTER. LinReg(a+bx) Y_{1}
should be displayed
on the home screen.
Note that if the data are not in the default lists, L_{1} and L_{2},
you will need to enter the lists
separated by
commas before doing the steps to enter Y_{1}.
d) Press ENTER and the values for "a," 0.003,
and "b, " 75, will be displayed along with other results. So, the
demand function is P=
0.003X+75.
Now, we want to determine the price for at 8000 units
from the graph of the demand function.
e) Press Y= and observe that
.003X+75 is displayed
opposite Y_{1}.
f) Press WINDOW and set Xmin=0, Xmax=8500,
andYmin=0. Press ZOOM, 0(ZoomFit), and the graph will be
displayed.
g) Press 2ND, CALC, ENTER to implement the
Value function.
h) Enter 8000 opposite "X" and press ENTER
to obtain 51 for the price.
Now, find the maximum revenue.
a) Press Y=, disable Y_{1} by
moving the cursor to the = sign opposite Y_{1} and pressing ENTER.
b) Opposite Y_{2}, enter "x."
Use the [X,T,0,n] key to enter "x."
c) With the cursor opposite Y_{2} ,
press VARS, move the cursor to YVARS and press ENTER, ENTER.
You should now have
XY_{1} displayed opposite Y_{2}.
d) Press WINDOW and set Xmin=0, Xmax =
25000, Ymin=0. Press ZOOM, 0 (ZoomFit), and the graph
will be
displayed.
e) Press 2ND, CALC, 4(maximum) and
the graph will be again displayed.
f) From the graph screen, move the
cursor somewhat to the left of the peak of the graph and press ENTER.
Move the
cursor to slightly to the right of the peak and press ENTER again.
Finally, move the cursor
to the
approximate peak of the graph and press ENTER again. X=12500 and Y=468750
will be displayed.
These are the
quantity at which maximum revenue occurs and the maximum revenue at that quantity.
Detailed Calculations:
Since many teachers use the
detailed calculations as an instructional method, I will now go through a
fairly detailed
calculation of the constants “a” and “b.” Assume that we have the x and
yvalues as listed in the table in the first part
of this problem.
L_{1 Quantity} (x) (x 10^{3 }) 
L_{2 Price} (y)
_{ } 
L_{3} x² 
L_{4} xy 
1000 
72 
1E6 
72000 
4000 
63 
1.6E7 
252000 
9000 
48 
8.1E7 
432000 
14000 
33 
1.96E8 
462000 
20000 
15 
4E8 
300000 
∑x_{i}
= 48000 
∑y_{i}=
231 
∑ x_{i}²
= 6.94E8 
∑x_{i}y_{i}=1.52E6

We
will now calculate the values for x² and xy which have already been entered in
the table on this page.
a) Press STAT, ENTER, ENTER to go to the list screen. Enter the values in the
table into the
first two columns of the lists as indicated in the table above.
b) Highlight the name, L_{3}, and the top of the lists and press 2^{nd},
L_{1}, x², ENTER. The squared
numbers will be stored in list L_{3 }.
c) Highlight the name for list L_{4} and press 2^{nd}, L_{1},
2^{nd}, L_{2, }ENTER.
Now, solve the system of equations indicated as follows:
∑x
_{i})b +(∑x_{i}²)m
= ∑x_{i}y_{i
}nb +(∑x _{i})m = ∑y_{i
} In these equations, “m” is the slope and “b” is the yintercept. Those
may be designated by such constants
as "a" and "b" in some texts.
In terms of our lists in the table above, these equations can be written as a
matrix as follows:
∑L_{1} +(∑L_{3}
= ∑L_{4
}n +(∑L_{2} = ∑L_{3}
Enter the Coefficients into the Matrix:
a) Press 2nd, MATRIX, go to EDIT and press
ENTER.
b) Create a 2x3 matrix by pressing 2,
ENTER, 3, ENTER. Then move the cursor to the first element to enter the
data.
We are now
going to sum the lists and enter the sums in the matrix that we just
edited.
c) Press 2nd, LIST, move the cursor to MATH, and
press 5 to paste sum( opposite 1,1 at the bottom of the screen.
d) Enter L1, ), so that you
have sum(L_{1}). Now press enter to get the sum in the first
element of the matrix.
e) Repeat this procedure for L_{3}
in element (1,2), L_{4} in element (1,3); then enter the number
for "n" in element (2,1).
f) In a similar manner, repeat the
sums for L_{2} in element (2,2) and L_{3} in element (2,3).
When you finish
solve the matrix for "a" and "b" as follows:
f) Press 2nd, QUIT to get out of the matrix.
Then press 2nd, MATRIX, move the cursor over to MATH, and then select rref(
and press.
e) Press 2nd, MATRIX and then ENTER if you
have entered your numbers in [A], otherwise select the proper matrix
and press
ENTER. You should now have rref([A] on the screen.
f ) Press ENTER to get the answer.
The first row will be "b" and the second will be "a." You'll get almost
the same answer
as with all of the
arithmetic.
COMMENT: The matrix method may seem like a
formidable number of steps, but is saves many, many individual arithmetic
steps.
II. Miscellaneous:
1. Harmonic mean:
Suppose that an
investor has $1000 to invest each month and he pays $8.00, $8.50, $9.00, $9.50,
$10.00, and $10.50 for the
for the first six months.
What is his average cost per share?
a) Press STAT,
ENTER to go to the lists and enter the data in list L_{1}.
b) Place the cursor on
the name for L_{2} and enter 1, ÷, 2nd, L_{1},
so that you have 1/L_{1}.. Press ENTER and the reciprocals will
be displayed
c) Press 2nd, QUIT to leave the list editor and press 1,
÷, (. Press 2nd, LIST, move the cursor to MATH,
and press 5 for sum(.
d) Press 2nd, L_{2}, ),
÷, 6, ). You should now have 1/(sum(L_{2})/6)
e) Press ENTER and
$9.1706....will be displayed as opposed to the arithmetic mean of $9.25.
2. Geometric
mean:
Suppose that during
the past five years you has 75%, 10%, 30%, 20%, and 60% returns on your
investment. What would be
your average return for
the fiveyear period? You can enter ((1.75*1.1*1.3*1.2*.4)^(1/5))1, but
if you have a more complex
problem or you don't like
entering number, you can do it as follows:
a) Press STAT,
ENTER and enter the percentages in list L_{1}.
b) Move the
cursor to highlight the name for L_{2} and press 2nd, L_{1},
÷, 100, +, 1. So that you have L_{1}/100
+1 and press ENTER.
c) Move the cursor to the
first blank space in L_{2} and press (,; l then press 2nd, LIST.
Move the cursor to MATH and press 6 to
paste prod( to the bottom of the list screen.
d) Press 2nd, L_{2}, ),
^, (1, ÷, 2nd, 5), 1. You should now have (prod(L_{2}))^(1/5) 1
at the bottom of the list screen.
e) Press ENTER and you
will get .03734 or 3.734% rather than 15% for the arithmetic mean.
3) Median of Grouped data:
Consider the following table.
Age 
514 
1524 
2534 
3544 
4554 
Midpoint 
9.5 
19.5 
29.5 
39.5 
49.5 
Freq (People) 
750 
2005 
1950 
195 
100 
a) To
find the median class, enter the midpoints in L_{1} and the frequencies
in L_{2 }.
b) Press STAT, move the cursor to highlight CALC and press ENTER. 1Var Stats
will be displayed on the home screen.
c) If the data are in lists L_{1} and L^{2} just press
ENTER. If they are in other lists, you must enter the lists. For
example, ^{ }
press 2^{ND}, L_{2}, comma, 2^{ND}, L_{3}
and then press ENTER
d) Press ENTER and scroll down to Med=19.5. That is the median of the class
15
24. So 1524 is the median class.
e) Enter the appropriate data into the following formula:
Median = L + I *(N/2  F)/f
Where
L = lower boundary of the interval containing the median.
I = width of the interval containing the median.
N = total number of respondents.
F = cumulative frequency of those below the median class.
f = number of cases in the median class.
f) When you are finished entering, you should have this:
14.5+10(5000/2750)/2005
g) Press ENTER and you should get 23.228… Notice that the answer is different
form
the value of 19.5 given by the calculator. That value of 19.5 was chosen
by merely finding the midpoint
of the median class.
4. How Long to Double Interest:
Although this could be found using the Finance
Application, here's a quick way to do it.
An investment has a 7% interest compounded annually.
How long will it take for the value
to double?
Preliminaries: From the equation
A=P(1+r/n)^{nt} , we have 2P=P(1+0.07)^{t} . Cancel "P"
and we have
2=1.07^{t} , or 2=1.07^{x} for the
calculator.
a) Press Y= and enter 2 opposite Y_{1};
then enter 1.07^x opposite Y_{2}.
b) Press WINDOW and set Xmin=0, Xmax=15, Ymin=0
and Ymax=3. Press GRAPH.
c) Press 2ND, CALC, 5 (for Intersect).
d) From the graph screen that appears, press
ENTER, ENTER; then move the cursor near the intersection
of the two lines
and press ENTER again. The answer of 10.24 years will be displayed.
Note that you can find other factors such
as triple or quadruple by merely changing Y_{1} to 3 or 4.
5. Leonfief InputOutput Problem:
Suppose that an economy consist of the sectors of
food, clothing, and shelter. In that economy, producing one unit of food
requires 0.4 units of food, 0.2 unit of clothing, and
0.2 units of shelter.
Producing one unit of clothing requires 0.1 units of
food, 0.2 units of clothing, and 0.3 units of shelter.
Producing one unit of shelter requires 0.3 units of food, 0.1 unit of clothing,
and 0.1 unit of shelter. The economic demands are
$100million of food, $30million of clothing and $250million of shelter.
Find the level of production needed to meet these
demands.
We can write the following inputoutput matrix:

F 
S 
C 
F 
0.4 
0.2 
0.2 
C 
0.1 
0.2 
0.3 
S 
0.3 
0.1 
0.1 
And the following for the demand matrix:
From the Leontief inputoutput model we have this:
XAX =D
X=(1A)^{1} D
a) Press 2ND, MATRIX, move the cursor to EDIT and enter the
inputoutput matrix in matrix [A]. Press 2ND, QUIT to
leave the matrix editor.
b) Press 2ND, MATRIX, move the cursor to EDIT and enter the demand
matrix in matrix [B]. Press 2ND, QUIT to
leave the matrix editor.
Now, we want to enter the appropriate expression
for (1A)^{1} D.
c) Press 2ND, MATRIX, move the cursor to MATH and press 5(identity), 3, ),
 (minus sign).
d) Press 2ND, MATRIX, ENTER, ). You should now have
(identity(3)[A]) on the home screen.
e) Press x^{1} , 2ND, MATRIX , 2(for matrix [B]), You should now
have (identity(3)[A])^{1} [B] on the home screen.
f) Press ENTER to get 396.34 for food, 252.22 for clothing, and 437.80 for
shelter. All are rounded to two decimal
places.
III. Linear
Programming:
1. Graphical Method:
Ex 1: Suppose
that we want to maximize the objective function with function and constraints
defined
as follows:
z=2x+5y
3x+2y≤6
x+2y≤
4
x≥0,
y≥0
a) First rewrite the
inequalities in slopeintercept form:
y≤(3/2)x
+3
y≤x/2+2
b) Press Y= and enter the
expressions in Y1 and Y2. If you wish, you can do the shading by moving
the
cursor
over to the left and pressing ENTER until you get the thirdquadrant triangle
for <, but I prefer to
just graph
the equations.
c) Press WINDOW and set
the following: Xmin=0, Xmax=5, Ymin=0, Ymax=4. Press GRAPH and the
graphs will be displayed. You can read the points (0,2) and (2,0) from the
graph, but we want to find
the point of intersection.
d) To find the
intersection point press 2ND, CALC, 5(Intersect), move the cursor back to about
x=.7 and
press ENTER. Press ENTER, ENTER after the cursor has moved to the other
curve and x=.5, y=2.25
will be displayed.
f) We now have the
points (0,2), (.5, 2.25), and (2,0) that we want to plug into the objective
function. You
could do this manually fairly easily, but there may be occasions when it is not
done so easily by hand.
Do the calculations with the calculator as follows:
1) Press 0, STO, X (use the [x,T,0,n] button for X), ALPHA, : (the decimal
button), 2, STO, ALPHA, Y,
ALPHA, :, 2, X, +, 5, ALPHA, Y. You should now have 0→X:2→Y:2X+5Y.
2) Press ENTER and 10 will be diplayed.
3) Now, press 2ND, ENTER and change the stored values of X and Y, so that
you have the following:
.5→X:2.5→Y:2X+5Y.
Press ENTER and 13.5 will be displayed.
3) Now, press 2ND, ENTER and change the stored values of X and Y, so that
you have the following:
2→X:0→Y:2X+5Y.
Press ENTER and 4 will be displayed.
So, the point (.5, 2.25) maximizes the objective function.
Using the Inequalz APP:
2) Graphical Linear Programming with the
Inequalities Application:
Do the same problem as above, that is, find the maximum of the objective function z=2x +5y,
subject to the following constraints:
3x+2y≤6
x+2y≤
4
x≥0,
y≥0
a) First rewrite the inequalities
in slopeintercept form:
y≤(3/2)x
+3
y≤x/2+2
b) Enter the right side of those equations opposite Y1 and Y2 respectively and enter 0
opposite Y3.
c) Set the WINDOW at Xmin = 1, Ymin=1, Ymax= the largest value of "b"
plus a few units, say 5 in
this problem.
Xmax is a bit more difficult to anticipate. If there is an equality with a
negative slope, I usually
make Xmax
= 4/3*b/m, round it off to the next largest whole number and add a few units.
You can
enter the arithmetic opposite Xmax. You might want to press GRAPH
and see if all of the
corner points of the bounded region are on the screen.
Now we
will enter the inequality signs.
a) Move the cursor to the sign
(either equal or inequality) after Y1. If the inequality symbols do not
appear at the bottom of the screen, you will need to start the Inequality App.
Do that by pressing
APPS, move the cursor
down to Inequal, or Inequalz for the
international version, and press
ENTER, ENTER. The Y= editor screen should be displayed.
b)
Place the cursor on the equal sign opposite Y1 and press ALPHA, F3 (ZOOM).
The equal sign
should have been replaced by the inequality ≤.
c) Do the same for Y2; then opposite Y3, press ALPHA, F5 (GRAPH).
The symbol ≥ should have
replaced the equal sign before the 0.
d) Now, move the cursor up to the "X" in the upper left corner and press
ENTER. With the cursor on the
equal sign opposite X1, press ALPHA, F5 (GRAPH) to enter
≥; then enter a zero after that
symbol.
e) Press GRAPH to draw and shade the graphs.
f) Press ALPHA,
F1, 1 to define the feasible region.
If you only want to graph, you may stop here.
At this time we will find the x and yvalues of the corner points. We
will use a method to have the
calculator determine the corresponding values of the objective function.
If you prefer to determine
those values manually, just record them without pressing STO so that you can
substitute them into
the objective function.
a)
Press ALPHA, F3 (ZOOM) and x=.5, y=2.25 will be displayed. Record
these values if you choose to evaluate the objective
function by hand. Otherwise, press STO, ENTER.
b) Press the down arrow to move to the point x=2, y=0 and press STO, ENTER.
Note that different sequences of the cursor
keys may be needed to read the points. It takes a little experimenting.
c) Press the right arrow; then then down arrow to read the last point,
x=0, y=2.
Having the calculator evaluate the objective function.
a) Press STAT, ENTER, to bring up the lists with the stored data.
b) First we will name the list after list INEQY. We will name it
OBJ. Move the cursor to the name
block at the top of that list. Press 2ND, ALPHA, O, B, J, ENTER.
Now, we will define the list OBJ.
c) With the curosor on the list name, press ALPHA, " (the + key),
2, x(multiply), 2ND, LIST
and scroll down to
INEQX and press ENTER.
d) Press +,
5, x (multiply), 2ND, LIST, scroll to INEQY and press ENTER.
Press ALPHA, ". You should now have
"2*└INEQX+5*└INEQY" at the bottom left of the screen after "OBJ=". Press
ENTER and the values
of the objective function for each set of coordinates will be displayed in the OBJ list.
Admittedly, this is a rather formidable list of steps, but after a little
practice, it becomes quite easy.
2.
Application Using the Graphical Method:
Suppose a
company manufactures two types of products Good and Better and it has two
machines 1 and 2. The
company will
realize a profit of $5 on each Good item and $4 on each Better item.
To manufacture a Good
item, 6
minutes are required on machine 1 and 5 minutes on machine 2. To
manufacture a Better item,
9 minutes are
required on machine 1 and 4 minutes on machine 2. If 5 hours are available
on machine 1 and
3 hours are
available on machine 2, how many of each should be manufactured to make the most
profit?
To facilitate
graphing, let x = the number of good items and y the number of best items.
Then we want to
maximize the
following profit function:
P=5x+4y
We can make a
table for our constraints as follows:
Machine 
G 
B 
Totals 
1 
6 
9 
300 
2 
5 
4 
180 
So, we can write the following constraints:
6x+9y≤300
5x+4y≤180
x ≥ 0, y ≥0
a) First rewrite the inequalities in slopeintercept form:
y≤2/3x
+300/9
y≤5/4x+45
b) Press Y= and enter the
expressions in Y1 and Y2. If you wish, you can do the shading by moving
the
cursor
over to the left and pressing ENTER until you get the thirdquadrant triangle
for <, but I prefer to
just graph
the equations.
c) Press WINDOW and set
the following: Xmin=0, Xmax=50, Ymin=0, Ymax=45. Press GRAPH and the
graphs will be displayed.
d) To find the
intersection point press 2ND, CALC, 5(Intersect), ENTER, ENTER.
Move the cursor to the
approximate intersection and press ENTER again. The values X=20, Y=20
will be displayed.
Now, we want to find the
point where the feasible region intersects the x and yaxes.
e) Press 2ND,
CALC, ENTER (for Value). The values x=0, y= 33.33.... will be displayed.
f) Press 2ND,
CALC, 2 (for zero) and make sure the equation 5/4x +45 is displayed at the top
of the screen.
If it is not, press the up arrow of the cursor control.
f) Move the
cursor down until you are at about x=3 or so and press ENTER. Move the
cursor to some small
negative number and press ENTER again. Finally, move the cursor to about zero
and press ENTER again.
The values X=36, Y=0 will be displayed
f) We now have the
points (0, 33.3 ), (20, 20), and (36,0) that we want to plug into the objective
function. You
could do this manually fairly easily, but there may be occasions when it is not
done so easily by hand.
Do the calculations with the calculator as follows:
1) Press 0, STO, X (use the [x,T,0,n] button for X), ALPHA, : (the decimal
button), 33, STO, ALPHA, Y,
ALPHA, :, 5, X, +, 4, ALPHA, Y. You should now have 0→X:33→Y:5X+4Y.
2) Press ENTER and 132 will be diplayed.
3) Now, press 2ND, ENTER and change the stored values of X and Y, so that
you have the following:
20→X:20→Y:5X+4Y. Press ENTER and 180 will be displayed.
3) Now, press 2ND, ENTER and change the stored values of X and Y, so that
you have the following:
36→X:0→Y:5X+4Y.
Press ENTER and 144 will be displayed.
So, the point (20, 20) maximizes the objective function.
3. Simplex Method:
By far
the simplest method for solving simplex problems is with a calculator program.
Later,
I'll give you
a location on my Website where you can find programs to download.
However,
many
students may not solve more than a halfdozen simplex problems in their college
career,
and some
professors will require doing simpler simplex problems by hand. So I will
include a semiautomated method
to reduce the
drudgery of arithmetic and to reduce errors.
Ex 1:
Suppose
that we want to maximize the objective function with the function and constraints
defined
as follows:
z=3x_{1}+2x_{2}+x_{3}
2x_{1}+
x_{2} +x_{3} ≤150
2x_{1}+2x_{2}+8x_{3} ≤
200
2x_{1}+3x_{2}+x_{3}
≤ 320
x_{1}≥0,
x_{2}≥0, x_{3
}
We introduce slack variables and have the following simplex tableau:
z=3x_{1}+2x_{2}+x_{3}
x_{1}+
x_{2} +x_{3}
s1 s2 s3  .
2 1 1
1 0 0 150
2 2 8
0 1 0 
200
2
3 1 0 0
1  320
3 2 1
0 0 0 0
Now, we want to solve this simplex problem. Using the Matrix operations,
do that as follows:
a) Press 2ND, MATRIX, move the cursor to EDIT and press ENTER.
b) Change the dimensions to 4 x 7 and enter the values into the matrix.
Just enter the number and
press ENTER to do that.
c) After you have completed entering the information, press 2ND, QUIT to
go the home screen.
d) Just as a precaution, let's store this in matrix [B] just so we won't
have to reenter everything
in case we make a mistake. So, press 2ND, MATRIX, ENTER, 2ND, MATRIX,
2(B). You should now
have [A]→[B] displayed on the screen. Press ENTER and the storing
will take place and the matrix
will be displayed.
e) Now, we want to decide where to pivot. Since 3 is the most negative
number, select that column.
Now, divide each number in that column into the numbers at the end. The
first number gives
150/2=75, which is smaller that the other quotients. So, we will pivot on
that point.
Row Operations:
Now, we want to use matrix row operations to leave a 1 in the pivot location and
zeros in all other
positions in that column. The syntax of the Operation that we will use is
as follows:
cc→M:c→A:c→R:*row+(M, [A], A,R)→[A], where c is a character (number, minus
sign, etc.) , M is the
row multiplier, A is the number of the row being multiplied, and R is the row
being replaced.
Remember that the arrow indicates pressing the STO key, the colons are obtained
by pressing
ALPHA and the decimal key, and *row+( is obtained by pressing 2ND, MATRIX,
moving the
cursor to MATH, scroll down to *row+( and pressing ENTER. The matrix
symbol [A], must be
obtained by pressing 2ND, MATRIX, ENTER.
f) Now, let's do row 2. We will have the following on the home screen:
1→M:1→A:2→R:*row+(M, [A], A,R)→[A]. Press ENTER and a zero will appear in
the appropriate
cell with the other cells in row two modified accordingly.
g) We want a zero in place of the 2 in row 3. Press 2ND, ENTER and
edit the number for R, so that the display
now reads as follows: 1→M:1→A:3→R:*row+(M, [A], A,R)→[A].
This will set up the matrix operation
as follows: *row+(1, [A], 1, 3)→[A]. Press
ENTER and the changed matrix will be displayed.
h) We want a zero in place of the 3 in row 4. Press 2ND, ENTER and
edit the numbers for M and R,
so that you have this: 3/2→M:1→A:4→R:*row+(M, [A], A,R)→[A].
This will set up the matrix operation
as follows: *row+(3/2, [A], 1, 4)→[A]. Press
ENTER and the changed matrix will be displayed.
i) We want to reduce the first number, 2 to 1. Normally, we would
use the multiply row function to
do that, but to reduce key strokes, we're going to do it another way. So,
press 2ND, ENTER and
edit the display so that it reads as follows: 1/2→M:1→A:1→R:*row+(M, [A], A,R)→[A].
Press
ENTER and the following will be displayed.
[[ 1 .5 .5
.5 0 0 75]
[ 0 1 7
1 1 0
50]
[0 2 0
1 0 1 17]
[ 0 .5 .5
1.5 0 0 225]]
Now, the pivot point will be on the 1 in column 2.
j) We want a zero in place of the 2 in row 3 Column2.. Press 2ND, ENTER and
edit the numbers for M,
and R, so that you have this: 2→M:2→A:3→R:*row+(M, [A], A,R)→[A].
Press ENTER and the changed
matrix will be displayed
k)
We want a zero in place of the .5 in row 1, column2. Press 2ND, ENTER and
edit the numbers for M and R,
so that you have this: .5→M:2→A:1→R:*row+(M, [A], A,R)→[A]. Press
ENTER and the changed
matrix will be displayed.
l) We want a zero in place of the .5 in row 4, column 2. Press 2ND, ENTER and
edit the numbers for M and R,
so that you have this: .5→M:1→A:4→R:*row+(M, [A], A,R)→[A]. Press
ENTER and the changed
matrix will be displayed.
Since there are no more negative numbers in the last row, the solution is now
complete. The solution
you should have is as follows:
[[ 1 0
3 1 .5
0 50]
[ 0 1 7
1 1 0
50]
[0
0 14
1 2 1 70]
[ 0
0 4 1 0 0 250]]
3. Simplex Method with a Calculator Program:
There are a number of program available on the Internet. You can
either use one of my programs or do a Google
search to find another source. Here's the URL to go directly to my
programs.
http://www.angelfire.com/pro/fkizer/ti_programs/tiprograms.htm
Notice that there are different program depending on the type simplex problems
you want to solve. My programs
are in text for manual entry and in .8XP format for download and entry by
computer. The instructions are included
with the programs.
IV. Calculus Applications:
1. Cost, Average Cost, and
Marginal Cost:
Suppose that a
company estimates that the cost (in dollars) of producing x units of a certain
product is given by
C(x) =
200+0.05x+0.0001x².
Cost of producing
500 units.
a) Press Y=
and enter the equation 200+.05x +.0002x². Use the [X,T,0,n] key to enter x and
the x² key to
enter the exponent after entering x.
Press WINDOW and set Xmin at 450; then press ZOOM, 0 (ZoomFit) display the
graph.
b) Press 2ND,
CALC, ENTER.
c) From the
displayed graph screen enter 500 opposite s and press ENTER. The cost of
250 will be displayed.
Average cost of
producing 500 units.
a) Press Y=
and change the entry so tha tit is as follows: (200+.05x +.0002x²)/x
Press WINDOW and set Xmin=450; then press ZOOM, 0(ZoomFit) to display the graph.
b) Press 2ND,
CALC, ENTER and enter 500 opposite x.
c) Press
ENTER and x=500, y=.5 will be displayed. The average cost is $0.50.
Marginal Cost for
500 Units:
a)
Change the equation back to the cost equation: 200 +.05x+ .0001x².
b) From
the graphing screen press, 2ND, CALC, 6(dy/dx).
c)
Enter 500 and press ENTER. The equation dy/dx =.15 will be displayed.
2. AverageCost
Minimization:
Find the number of units that will minimize the average cost and the
corresponding cost.
We want
to find the point where C(x) = C'(x), so we'll need the first derivative: c'(x)=.05+.0002x.
a)
Enter (200+.005x+.0001x²)/x opposite Y1 and .05+.0002x opposite Y2.
b)
Set xmin = 450 and xmax = 200. Press ZOON, 0(ZoomFit) to display the
graphs.
Now find the intersection point.
c)
Press 2ND, Calc, 5 (intersect).
d)
From the graph screen, press ENTER, ENTER; then move the cursor approximately to
the intersection
and press ENTER again.
e)
The numbers x=1414.21... and y=.332... will be displayed. So, the number
to minimize average cost
is approximately 1414 and the cost is approximately .332 dollars.
3. Cost Minimization;
The cost C(x) of manufacturing x units of a produce is approximated by the
following: C(x) =100+10/x+x²/100.
Find the number of units that should be manufactured to minimize cost.
a) Press Y= and enter the equation 00+ 10/x + x²/100. Use the [X,T,0,n]
key to enter x and the x² key to
enter the exponent after entering x. Press WINDOW and set Xmin =0, Xmax=50,
Ymin=90 and Ymax =140.
b) Press 2ND, CALC, 3 (minimum), and move the cursor to the left of the
minimum (about 5) and press ENTER.
Move the cursor to the right of the minimum (about 20) and press ENTER.
Move the cursor back to the left
to approximately the minimum and press ENTER. The value 10 will be
displayed for the number to minimize
cost.
4. Complementary
or Substitute
Product Determination:
Suppose that the demands of two products x sells for $2.50 and product y sells
for $3.00 with unit prices p and q
are given by
the following equations:
x=8,000  0.09p² + 0.08p²
y=15000 + 0.04p²  0.3q²
Are the products substitute complementary or neither.
The decisionmaking conditions are summarized in the following table:
Partial Derivatives 
Products x and y 
∂f/∂q
> 0 and ∂f/∂p > 0 
Substitute 
∂f/∂q < 0
and ∂f/∂p < 0 
Complementary 
∂f/∂q >
0 and ∂f/∂p < 0 
Neither 
∂f/∂q <
0 and ∂f/∂p > 0 
Neither 
a) Press Y= and enter the following equations:
8000  0.09*2.50² +0.08x²
opposite Y1 and
15000+ 0.04*x²
0.3*3².
b) Move the cursor to the equal sign opposite Y2 and press ENTER to
disable that equation; then press 0 for
ZoomFit to GRAPH the equation.
c) Press 2nd, CALC, 6.
d) ENTER 3 and press ENTER to get 0.48.
b) Move the cursor to the equal sign opposite Y1and press ENTER to disable
that equation; then move the cursor to Y2
and press ENTER to enable that equation. Press ZoomFit to graph Y2.
e) Press 2nd, CALC, 6.
d) ENTER 2.5 and press ENTER to get 0.2.
f) Since we know that with substitute goods, all other things remaining
constant, if the price of increases the demand for the other
increases. In mathematical terms we can express that as
∂f/∂q > 0 and ∂f/∂p > 0. So, products x and y are substitute.
5. Equality of Marginal Product of Labor and Capital.
The productivity of a manufacturing organization is given by f(x,y) = 15x^{0.4}
y^{0.6} . If the company is utilizing 400 units of
labor, what units of capital will make the marginal productivity of labor and
the marginal productivity of capital equal? What is
the marginal rate at that level?
Take the partial derivatieves: ∂f/∂x=6x^{0.6} y^{0.6}
and ∂f/∂y=9x^{0.4} y^{0.4}
a) Press Y= and enter 6*400x^0.6)x ^0.04 opposite Y_{1}.
Opposite Y_{2}, enter 9(400x^0.4)x ^0.4.
b) Press WINDOW and set Xmin =0, Xmax =700, Ymin =0 and Ymax=10.
Press ENTER to display the graph.
c) Press 2ND, CALC, 5 (Intersect).
d) When the graph appears, press ENTER, ENTER; move the cursor near the
intersection and press ENTER again.
X=600 and Y=7.652.., So, 600 units of capital and 400 units of labor will
give marginal rates of 7.652.
6. Consumers' Surplus:
Assume the pricedemand equation as follows of p=200.05x. Find the
consumers' surplus at $8.
Note that we could integrate the function CS=(200.05x8)dx, but I have chosen
to simulate an actual integration and evaluate it
at both limits. First, we need to find the upper limit of integration,
that is, where the line y=8 intersects the demand curve.
a) Press Y= and enter the expression 200.05x opposite Y1. Enter 8
opposite Y2.
b) Press WINDOW and set Xmin = 0, Xmax = 250, Ymin=6, Ymax=20. Press GRAPH
to graph the equations.
c) Press 2nd, CALC, 5 for Intersect.
d) On the graph that appears, press ENTER, ENTER; then move the cursor
approximately to the intersection and press ENTER.
The intersection, 240 will be the upper limit of integration.
Now we want to integrate the areas separately.
a) Press Y= and move the cursor to the equal sign opposite Y2 and press ENTER to
disable that equation. Press GRAPH.
b) Press 2nd, CALC, 7 for integration. On the graph screen enter 0 for the
lower limit and press ENTER. Then ENTER 240 for
the upper limit and press ENTER. The value of the area, 3360, will be displayed.
c) Press Y=, move the cursor to the equal sign opposite Y1 and press ENTER
to disable that equation. Press GRAPH.
d) Press 2nd, CALC, 7 for integration. On the graph screen enter 0 for the
lower limit and press ENTER. Then ENTER 240 for
the upper limit and press ENTER. The value of the area, 1920, will be displayed.
e) The consumers' surplus, CS=33601920 =1440
7. Producers' Surplus:
Assume the pricesupply equation as follows of p=2+0.0002x². Find the
producers' surplus at $20.
Note that we could integrate the function CS=(202 0.0002x²)dx, but I have
chosen to simulate an actual integration and evaluate it
at both limits. First, we need to find the upper limit of integration,
that is, where the line y=20 intersects the supply curve.
a) Press Y= and enter the expression 2+0.0002x² opposite Y1. Enter 20
opposite Y2.
b) Press WINDOW and set Xmin = 0, Xmax = 300, Ymin=0, Ymax=25. Press GRAPH
to graph the equations.
c) Press 2nd, CALC, 5 for Intersect.
d) On the graph that appears, press ENTER, ENTER; then move the cursor
approximately to the intersection and press ENTER.
The intersection, 268.33, will be the upper limit of integration.
Now we want to integrate the areas separately.
a) Press Y= and move the cursor to the equal sign opposite Y1 and press ENTER to
disable that equation. Press GRAPH.
b) Press 2nd, CALC, 7 for integration. On the graph screen enter 0 for the
lower limit and press ENTER. Then ENTER 268.33 for
the upper limit and press ENTER. The value of the area, 2146.66, will be
displayed.
c) Press Y=, move the cursor to the equal sign opposite Y1 and press ENTER
to disable that equation. Press GRAPH.
d) Press 2nd, CALC, 7 for integration. On the graph screen enter 0 for the
lower limit and press ENTER. Then ENTER 268.33 for
the upper limit and press ENTER. The value of the area, 5366.6, will be
displayed.
e) The suppliers' surplus, CS=5366.62146.66 =3219.94.
Making it Better: I
would be grateful if you would report any errors or suggestions for improvements
to me. Just click "Email Webmaster," site the item number, and tell me
your suggested change.
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