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General: 
*  TMV Solver - Unless otherwise indicated,  all calculations will be with the TMV Solver.  To access
   this, press APPS, ENTER, ENTER. 
*  Most of these instructions will be carried out using a problem as an example.  Note that some of the  
   problems could be solved, possibly even easier, without the Finance APP, but this sheet deals with
   that APP only. 
*  Minus Signs - Note that some answers will have a minus sign before them.  These are there because 
   the calculator  follows the cash-flow sign convention in which cash outflows (investments for example)
   are negative and inflows are positive.  For many problems, you can ignore this sign.  When it's
   important, that will be indicated.
*  Setting N, P/Y, and C/Y - As a general rule, when there are no periodic payments, such as in  
   interest calculations, "N" is set equal to the number of years and P/Y is set at 1.  C/Y will be set to
   the number of compounding periods a year.  Notice that for daily compounding, C/Y will be set at 360
   for some problems.  For loans, annuities, and other such things with periodic payments, P/Y will be
   set for the number of payments a year, "N"  will be the number of payments, and C/Y will be set for
   the number of compoundings per year.

I.  Simple and Compound Interest.

    1. Simple Interest:
        A student had $5000 which she did not need for 11 months.  If she invested it for 11 months at 8%
        annual interest, how much did she have at the end of the 11 months?
        a)  Enter values so that the following display is completed:  N=1; I%=8*11/12; PV = 5000; PMT=0;
             P/Y =1; C/Y=1; END.
        b)  Set the cursor on FV and press ALPHA; SOLVE.  Note that SOLVE is the third function of the
             ENTER key. 
        c)  Note that if you want the interest accumulated, then just subtract $5000 from the answer
             obtained in the above operation. 

    2. Compound Interest:
        Ex 1:  Suppose that you invest $5000 for 6.5 years at 5.25% interest compounded quarterly,  
        how much money will you have at the end of the period? 
         a)  Enter values so that the following display is completed:  N=6.5; I%=5.25; PV = -5000; PMT=0;
              P/Y =1; C/Y=4; END.
        b)  Set the cursor on FV and press ALPHA; SOLVE.  Note that SOLVE is the third function of the
             ENTER key. Your answer should be 7017.93.
        c)  Note that if you want the interest accumulated, then just subtract $5000 from the answer
             obtained in the above operation. 

       Ex 2:  Suppose that you have $1200 and you need $1800 in 7 years,  at what interest compounded
       quarterly,  will you need to invest the money to earn this amount?
        a)  Enter values so that the following display is completed:  N=7; I%=5.25; PV = -1200;  
             PMT=0; P/Y =1; C/Y=4; END.
        b)  Set the cursor on I%, and press ALPHA; SOLVE.  Note that SOLVE is the third function of the
             ENTER key. Your answer should be 5.834 rounded to 3 decimal places.
        c)  Note that if you want the interest accumulated, then just subtract $5000 from the answer
             obtained in the above operation. 

        EX 3:  Interest Compounded Continuously:
            Although the formula A=Pe^rt is just about as easy as using the Finance APP, some users have difficulty
            rearranging the formula to obtain time or rate.  So, I will include this example of continuous compounding.
             Let's take the information in Ex 2 above except that we have interest compounded continuously.
             a)  Enter the information exactly as in Ex 2 except that for C/Y, enter 1E9.  Do that by pressing 2, 2ND
                  EE (the comma key), 9, ENTER.  Now, continue with item b) as in Ex 2. 

  3. Effective Interest Rate:
     Suppose that a one bank tells you that it pays 3.9% compounded monthly and another tells you
     that it pays 4% compounded semi-annually.  Which one is the best investment?
     a)  Press APPS, ENTER, Cursor to C:EFF( and press ENTER.  (Alternatively, you may press
          ALPHA C.)  "EFF (" will be pasted to the screen.
     b)  Enter 3.9, 12) and press ENTER.  Your interest will be 3.97%.
     c)  Press 2nd, ENTRY (the second function of ENTER); then edit the entry so that you have
          EFF(4, 2); then press ENTER.  Your answer will be 4.04.  So, this is the best investment.

II. Annuities and Mortgages:

     1. Ordinary Annuities:
         For our purposes, an ordinary annuity will be one in which equal payments are made at equal
         periods of time, the compounding period is the same as the payment period, and the payments
         are made at the end of the period. Note Well:  Because there are payments in an annuity, "N" in
         the TMV Solver must set equal to the number of payment periods.
         Ex. 1:  Suppose that you pay $20,000 each year into an annuity for 7 years.  If the interest is 6%
         compounded annually, how much will you have at the end of the period?
         a)  Enter values so that the following display is completed:  N=7; I%=6; PV = 0;PMT=-20000;
              P/Y =1; C/Y=1; END.
         b)  Set the cursor on FV and press ALPHA, SOLVE.  Note that SOLVE is the third function of the
             ENTER key. Your answer should be 167876.75.

     2. Annuities Due:
         Annuities Due have the same setup as ordinary annuities, except that BEGIN is highlighted
         instead of END.
         Ex. 1:  Suppose that you pay $500 each year into an annuity due for 7 years.  If the interest is
         6% compounded annually, how much will you have at the end of the year?
         a)  Enter values so that the following display is completed:  N=7; I%=6; PV = 0;PMT=-500;
              P/Y =1; C/Y=1; BEGIN
        b)  Set the cursor on FV and press ALPHA, SOLVE.  Note that SOLVE is the third function of the
             ENTER key. Your answer should be 4196.92, rounded to 2 decimal places.

     3. Sinking Funds:
         Sinking funds have the same characteristics as annuities, but they are for purposes other than an
         annuity.  They may be to accumulate enough money to buy a car, pay off a loan, or any other
         purpose.  Follow the same procedure for these as for annuities.

     4.  Mortgages:
          Suppose a family buys a home for $200000 and makes a down payment of $20000.  They take  
          out a $180000 mortgage at 7.5% for 30 years.  What is the monthly payment required to
          amortize this loan?
          a)  Enter values so that the following display is completed:  N=360; I%=7.5; PV =
              180000; FV=0; PMT=0; P/Y =12; C/Y=12; END.
          b)  Set the cursor on PMT and press ALPHA, SOLVE.  Note that SOLVE is the third function of 
               the ENTER key. Your answer should be 1258.59, rounded to 2 decimal places.
          Addendum:  To find the total interest paid on this loan, use this formula:
          Total Interest = Monthly Payment*Number of Months - Original Amount of Loan.
                              =  1258.59*360 -180000
                              = $273092.4

     5.  Amortization Table for a Loan:
           General:  The manual procedure, which I will explain first, takes a lot of time if you have to
           calculate  several loans or several lines on a table.   
           Therefore, I have added a little program that I wrote to save you some work.  The program follows  
           this explanation. 

           Manual Method:
              Suppose you have an 10-year loan of $80,000.00 at 8.5 percent with payments each month. 
              Make an amortization table for the first three payments.  You might first want to make a table  
              such as the following to enter your data.  The calculated data has already been entered in
              this table.
                 

                 1)  Press APPS, ENTER, ENTER.
                 2)  Put the following information in the display that appears:  N=10*12; I% = 8.5; PV=80000;
                      FV=0; P/Y=12;C/Y = 12; END.
                 3)  Put the cursor at PMT, press ALPHA, ENTER, and the payment of 991.885 will be displayed
                       opposite PMT.
                 4)  Press 2nd, QUIT, APPS, ENTER, 9, to display bal(.  We will now calculate the balance after
                      each of the three payments.
                 5)  Enter values so that your display looks like this:  bal(1,1  .  Press ENTER and the value
                       indicated in the table will be displayed.
                 6)  Press 2nd, ENTER, and the entry will be displayed again.  Edit the entry to look like this
                      bal(2,2  . Press ENTER and the new balance will be displayed.
                 7)  Follow this same procedure to calculate the balance after the third payment.
                 8)  Now, we will calculate the Principal Payments.  Press APPS, ENTER, 0 (zero), and ∑Prn( will
                      appear on the screen.
                 9)  To calculate the principal for the first payment, add numbers so that the entry looks as follows: ∑Prn(
                      1,1  .  For the second payment, change the display to ∑Prn( 2,2 and for the third to ∑Prn( 3,3  .  Of
                      course, press ENTER for each calculation.
                10) Finally, calculate the interest payments. Press APPS, ENTER, ALPHA, A, and the following will be
                     displayed:  ∑Int( .  Calculate the three interest amounts the using the same procedure as with the
                     principal balance and interest payments. 
                      Of course, you could have calculated all three amounts for the first payment; then the second etc.,
                      but that would have been much more work.

                      Using the Program:  This is a simple program that should take only a few minutes to enter if you  
                      have some rudimentary knowledge of how to enter programs.  You can find information on entering
                      programs in your TI User Manual in the programs section on this  Website.  (Click on TI  
                      Programming Keystrokes near the bottom of the navigation panel to the left.) STCC students may
                      call me at the Gill ASC and make an appointment to have the program electronically transferred to
                      their calculators.   It takes less than five minutes, including setup.  NOTE:  The colons to the left on
                      the lines of code are automatically entered when you enter the program  by hand.

                     :PROGRAM: LAONAMRT
                     :"FKIZER  V:050106"
                     : Disp "ENTR DATA IN APPS"
                     :Input "1ST PMT NO. ", B
                     :Input "LAST PMT NO. ", E
                     :1→X
                     :ClrList L₁, L₂, L₃, L₄
                     :For(P,B,E
                     :tmv_Pmt→L₁(X)
                     :∑Int( P,P→L₂(X)
                     :∑Prn( P,P→L₃(X)
                     :bal(P,P→L₄(X)
                     :X+1→X
                     :End
                     :Stop

                     Using the Program:  Here's how to use this program, assuming you already have it entered.
                      1)  Follow the first three steps in the manual method described above; then press 2nd, QUIT.
                      2)  Pres, PRGM; cursor to the program name and press ENTER.
                      3) The statement 1ST PMT NO. will appear.  Enter the number of the first payment you want to 
                          calculate data for and press ENTER. 
                      4)  LAST PMT NO. will then appear.  Enter the number for the last payment you want to calculate
                           and  press ENTER.  Obviously, if you want only one payment, that number will be entered for
                           both the first and last payment number.
                      5)  The calculator will store the amounts for Payment, Interest, Principal Payment, and Principal
                           Balance in that order in lists L_1, L₂, L₃, and L₄.
                      6)  To access the data tables, press STAT, ENTER.  
                      7)  You will notice that the data has only five entries (Numbers plus decimal and negative sign, if
                            any.).  You can read the actual number stores for entry by scrolling to that entry and reading the
                            entry at the bottom of the list.
                       
III.  Loans:
      Loans, car loans for example, have the same structure as ordinary annuities.  Let's do an example
      to demonstrate that.
      Ex 1:  Suppose that a car costs $26000, and you pay down $4000.  The balance will be paid off in
      36 monthly payments with a interest of 10% per year on the unpaid balance. Find the monthly
      payment.
      a)  Enter values so that the following display is completed:  N=36; I%=10; PV = 22000;PMT=0;
           FV=0; P/Y =12; C/Y=12; END.
      b)  Set the cursor on PMT and press ALPHA, SOLVE.  Note that SOLVE is the third function of
           the ENTER key. Your answer should be 709.88, rounded to 2 decimal places.

IV.  Investments:

      1. Bonds:
          Ex 1:  Suppose that a $1000, 10-year, 8% bond is issued when the market rate is 7.5%. 
          Interest is paid semiannually.  What can you expect to pay for the bond?
          a)  Enter values so that the following display is completed:  N=20; I%=7.5; PV =0;PMT=40;
           FV=1000; P/Y =2; C/Y=2; END.  It's important to realize that the cost is based on the interest
           to maturity.
          b)  Set the cursor on PV and press ALPHA, SOLVE.  Note that SOLVE is the third function of
           the ENTER key. Your answer should be -1034.74, rounded to 2 decimal places.

           Ex 2:  Suppose that you have to pay $1034.74 for a $1000, 10-year, 8% bond with interest paid
           twice a year.  What is the interest to maturity for the bond?
           a)  Enter values so that the following display is completed:  N=20; I%=0; PV =1034.74;PMT=40;
                FV=1000; P/Y =2; C/Y=2; END.
           b)  Set the cursor on I% and press ALPHA, SOLVE.  Note that SOLVE is the third function of
           the ENTER key. Your answer should be 7.5%.

       2.  Present value:
             The syntax for Net Present Value (NPV) is:  npv(interest rate, CFO, CFList[CFFreq]).  Now,
             let's define what these mean: 
             Interest Rate = the rate by which to discount the cash flows over one period.
             CFO = the initial cash flow at time zero.
             CFOList = A list of cash flow amounts AFTER the initilal cash flow, CFO.
             CFFreq = How many there are of each amount.  The default is 1.
             Ex. 1:  Suppose you are offered an investment that will pay the cash flows in the table below at
             the end of each year for the next 5 years.  How much would you be willing to pay for it if you
             wanted 10 percent interest per year?
             

a) Press STAT, ENTER to go to the lists.  It there are numbers in the list you choose to use,
                   you can erase those numbers by highlighting the list name, for example L₁, pressing CLEAR;
                   then ENTER.  Do not use DEL.
               b) Enter the numbers starting with 100 in list L_1.   To enter a number, just enter it and press ENTER. 
               c)  Press 2nd, QUIT to leave the list.
               d)  Press APPS, ENTER, 7. "npv(" will be pasted to the home screen.
               e)  Make entries so that you have the following: npv(10, 0, L₁.  To enter L₁, press 2nd L₁.  (L₁
                    is the second function of the number 1 key.)
               f)  Press ENTER.  Your answer should be 1065.26 rounded to two decimal places.
               NOTE 1:  Instead of using the lists, you could enter the following:
               npv(10, 0, {100, 200, 300, 400, 500}). Then press ENTER.  I frankly prefer to use lists because
               of the increased flexibility.
               NOTE:  If you have several CONSECUTIVE cash flows, you can create a frequency table in
              another list, L_2, for example.  You will need to enter the frequency for each of the CFO values,
              even if it is 1.  Your entry then would be npv(10, 0 L₁, L₂ .
              Ex. 2: Suppose that we wanted to find the future value.  Rather than using the TMV solver for
              each cash flow and adding them up, just multiply the answer from Ex. 1 by (1+.10)^5.  To do
              that, press 2nd, Ans, x (multiply), (1+.10)^5.  Your answer should be 1715.61.
              Ex. 3: Suppose that you were offered the above investment for $800.  What is the NPV?
              CFO is now -800.  The cash outflow is negative.  So, we would enter, npv(10, -800, L₁.  Your
              answer should be 265.26 rounded to 2 decimal places.  
     
       3.  Internal Rate of Return (Irr):  
            Suppose you wanted to find the Irr for the npv example above.
           a)  Press APPS, ENTER, 8.  The term "irr(" will be displayed on the home screen.
           b)  Make entries so that you have the following:  irr(-800, L₁.Your answer should be 19.538.  This
                assumes that the numbers in the table of cash flows above have been entered in list L₁.

         4.  Modified Internal Rate of Return (MIrr):  
           Step 1:  First we'll find the Future Value:
           a) Press STAT, ENTER to go to the lists.  It there are numbers in the list you choose to use,
                   you can erase those numbers by highlighting the list name, for example L₁, pressing CLEAR;
                   then ENTER.  Do not use DEL.
           b) Enter the numbers starting with 100 in list L_1.   To enter a number, just enter it and press ENTER. 
           c)  Press 2nd, QUIT to leave the list.
           d) Press APPS, ENTER, ENTER to display the TMV Solver.
                e)  Enter values in the display as follows::  N=5; I%=0; PV =-800; PMT=0;
                 FV=1715.61; P/Y =1; C/Y=; END.
            Now, we want to enter a calculated value into FV. To do that, place the cursor opposite FV, press
            CLEAR to clear the value there; the do the following:
           f)  Press APPS, ENTER, 7. "npv(" will be pasted to the home screen.
           g)  Make entries so that you have the following: npv(10, 0, L₁).  To enter L₁, press 2nd L₁.  (L₁ is the
                second function of the number 1 key.)
           h) Now, we want to multiply this by (1.1)^5.  To do that enter 1.1^5.  You should now have
              the this expression:  npv(10, 0, L₁)1.1^5.  When you move the cursor away from FV you
              should have 1715.61
           i) Set the cursor on I% and press ALPHA, SOLVE.  Note that SOLVE is the third function of
                 the ENTER key. Your answer should be 16.48 rounded to two decimal places. 

V.  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 slope-intercept 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 third-quadrant 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. 

        1. 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 half-dozen simplex problems in their college career,
            so I will include a semi-automated method.
            Ex 1:  Suppose that we want to maximize the objective function with function and constraints defined
            as follows:
            z=3x₁+2x₂+x₃
            2x₁+ x₂ +x₃ ≤150
                2x₁+2x₂+8x₃ ≤ 200
            2x₁+3x₂+x₃ ≤ 320
            x₁≥0, x₂≥0, x₃
               We introduce slack variables and have the following simplex tableau:
               z=3x₁+2x₂+x₃
            x₁+ x₂ +x₃   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 re-enter 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 thenumber 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 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 you
                       now have this:  the display so that it
                       reads as follows:   -1→M:1→A:3→R:*row+(M, [A], A,R)→[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].  Press ENTER and the changed
                       matrix will be displayed.
                       reads as follows: *row+(3/2, [A], 1, 4)→[A]. 
                 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]]

VI.  Calculus Applications:
       1.  Cost, Average Cost, and Marginal:
           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. Average-Cost 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 x-min = 450 and x-max = 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.