Solver

1
Solver

• Finding maximum, minimum, or value by changing
  other cells
• Can add constraints
• Dont need to guess and check

2
Solver
3
Solver(show excel file-solverradiusbook1)

• Ex. Find the minimum surface area of a
  cylindrical soup can if the volume is 500 cubic
  centimeters and the height is three times the
  radius.

4
Solver

• Solution is surface area is approximately 354.85
  square centimeters.

5
Solver

• Sensitive to initial value
• Use graphical approximation to help solve project
• Use to verify/solve Questions 1 - 3
• Use to solve Questions 6 - 8

6
Solver

• Project (Answering Question 6)
• 6. What profit if price is 154.49? 7.83
  million

7
Solver

• Project (Answering Question 7)
• 7. How much would the company pay for a demand
  increase of 10? At most 3.90 million

8
Solver

• Project (Answering Question 8)
• 8. How would the price be affected by a demand
  increase of 10? No change

9
Solver

• Project (Answering Question 9)
• 9. Would it be wise to put 4,000,000 into
  training and streamlining if variable costs were
  reduced by 7? Yes, since profit increases

10
Solver

• Project (What to do)
• - Use Solver to answer Questions 1 - 4
• - Use Solver to answer Questions 6 - 9
• - Use separate computational cells to answer 1 -
  3, 4, 6, 7 - 8, and 9

11
Goals

• 1. What price should Save-it-All! put on the
  drives, in order to achieve the maximum profit?
• 2. How many drives might they expect to sell at
  the optimal price?
• 3. What maximum profit can be expected from
  sales of the SXL?
• 4. How sensitive is profit to changes from the
  optimal quantity of drives, as found in Question
  2?
• 5. What is the consumer surplus if profit is
  maximized?

12
Goals-Contd.

• 6. What profit could Save-it-All! expect, if
  they price the drives at 154.49?
• 7. How much should Save-it-All! pay for an
  advertising campaign that would increase demand
  for the SXL drives by 10 at all price levels?
• 8. How would the 10 increase in demand effect
  the optimal price of the drives?
• 9. Would it be wise for Save-it-All! to put
  4,000,000 into training and streamlining which
  would reduce the variable production costs by 7
  for the coming year?

13
Integration

• Revenue as an area under Demand function

14
Integration

• Total possible revenue

15
Integration

• Consumer surplus revenue lost by charging less
• Producer surplus revenue lost by charging more

16
Integration

• Approximating area under graph
• - Counting rectangles (by hand)
• - Using Midpoint Sum (by hand)
• - Using Midpoint Sums.xls (using Excel)
• - Using Integrating.xls (using Excel)

17
Integration

• Approximating area (Counting Rectangles)
• Ex.
• Approx. 9 rectangles
• Each rectangle is 0.25 square units
• Total area is approx. 2.25 square
  units

18
Integration

• Approximating area (Midpoint Sums)
• - Notation
• - Meaning

19
Integration

• Approximating area (Midpoint Sums)
• - Process
• Find endpoints of each subinterval
• Find midpoint of each subinterval

20
Integration

• Approximating area (Midpoint Sums)
• - Process (continued)
• Find function value at each midpoint
• Multiply each by and add them all
• This sum is equal to

21
Integration

• Approximating area (Midpoint Sums)
• Ex. Determine where .

22
Integration

• Approximating area (Midpoint Sums)
• Ex. (Continued)

23
Integration

• Approximating area (Midpoint Sums)
• Ex. (Continued)

24
Integration

• Approximating area (Midpoint Sums.xls)

25
Integration -showdemo-ex1Midpoint Sums.xls

• Ex1.
• Approximating area (Midpoint Sums.xls)

26
Integration

• Approximating area (Midpoint Sums.xls)

27
Integration, Integrals
Integration. Integrals
2. INTEGRALS What would happen if we computed
midpoint sums for a function which might assume
negative values in the interval a, b?
?
?
I
C
T
(material continues)
28
Integration, Integrals
the integral of f over a, b is and it
represents the algebraic sum of the signed areas
of the regions between the horizontal axis and
the graph of f, over a, b.
29
Integration

• Approximating area (Integrating.xls)
• - File is similar to Midpoint Sums.xls
• - Notation or or

30
Integration

• Approximating area (Integrating.xls)

31
Integrationshow ex1-Integrating.xls

• Approximating area (Integrating.xls)
• Ex1. Use Integrating.xls to compute

32
Integration

• Approximating area (Integrating.xls)
• Ex1. (Continued)
• So . Note that is the p.d.f. of
  an exponential random variable with parameter .
  This area could be calculate using the c.d.f.
  function .

33
Integration

• Approximating area (Integrating.xls)
• Ex. (Continued)

34
Integration

• Approximating area
• - Values from Midpoint Sums.xls can be positive,
  negative, or zero
• - Values from Integrating.xls can be positive,
  negative, or zero

35
Integration

• Ex2. Suppose a demand function was found to be
• . Determine the consumer surplus
  at a quantity of 400 units produced and sold.

36
Integrationshow ex2-Integrating.xls

• Ex. (Continued)
• Total Revenue at 400
  units produced and sold

37
Integration

• Ex. (Continued)

38
Integration

• Ex. (Continued)
• Calculate Revenue at 400 units

39
Integration

• Ex. (Continued)
• Take total revenue possible and subtract revenue
  at 400 units.
• 107,508.80 83,569.60 23,939.20
• So, the consumer surplus is 23,939.20

40
Integration

• Formula for consumer surplus
• Income stream
• - revenue enters as a stream
• - take integral of income stream to get total
  revenue

41
Class Project Q5

1. (continued)

Marketing Project
42
Integration

• Project (What to do)
• - Calculate consumer surplus to answer question
  5
• - Use Integrating.xls
• 108.70 96.53 12.17 million

43
(No Transcript)
44
Recommendations (continued)

1. Consumer surplus Additional amount that
   customers who bought the drive would have paid.

Marketing Project
45
Integration Applications

• Fundamental Theorem of Calculus
• -
• Example applies to p.d.f.s and c.d.f.s
• Recall from Math 115a

46
Integration, Applications
Example 4. The Plastic-Is-Us Toy Company
incoming revenue -as an income stream(rather than
a collection of discrete payments) At a time t
years from the start of its fiscal year on July 1
the company expects to receive revenue at the
rate of A(t) million dollars per year Records
from past years indicate that Plastic-Is-Us can
model its revenue rate A(t) ?110?t5 330?t4
? 330?t3 110?t2 3.174 million dollars per
year.
47
Integration, Applications
The chief financial officer wants to compute
the total amount of revenue that Plastic-Is-Us
will receive in one year. The income stream,
A(t), is a rate of change in money, given in
dollars per year. the units along the t-axis
are years the area of a region under the graph
of A(t) is given in (millions of
dollars/year)?(years) millions of dollars.
48

• Since gives the area between the
  t-axis and the graph of
• A(t), over the interval 0, T, it can be shown
  that the integral gives the total amount of
  money, in millions of dollars, that will be
  received from the income stream in the first T
  years.

49
Integration, Applications
Use Integrating.xls to compute the total
income received by Plastic-Is-Us during the
period from 0 to 1 year. (Remember that we must
use x, not t, as the variable of integration in
Integrating.xls.)

50
Integration, Applications
Integration. Applications page 12
In addition to the total revenue, a company
would often like to know the present value of its
income stream during the next T years (0 ? t ?
T), assuming that money earns interest at some
annual rate r, compounded continuously.
51
Integration, Applications
Example 5. We return to the Plastic-Is-Us Toy
Company that we considered in Example 4. Recall
that they have an income stream of A(t) ?110?t5
330?t4 ? 330?t3 110?t2 3.174 million dollars
per year. The management of Plastic-Is-Us would
like to know the present value of its income
stream during the next year (0 ? t ? 1), assuming
that money earns interest at an annual rate of
5.5, compounded continuously.
Applying the integral formula for present value
to Plastic-Is-Us, we use Integrating.xls to find
that the present value of their income stream for
one year, starting on July 1, is million
dollars.
52
Integration, Calculus
the inverse connection between integration and
differentiation is called the Fundamental Theorem
of Calculus.
Example 7. Let f(u) 2 for all values of u.
If x ? 1, then integral of f from 1 to x is the
area of the region over the interval 1, x,
between the u-axis and the graph of f.
53
Integration, Calculus
The region whose area is represented by the
integral is rectangular, with height 2 and width
x ? 1. Hence, its area is 2?(x ? 1) 2?x ? 2,
and
54
Integration, Calculus
Example 8. Recall the income stream of A(t)
?110?t5 330?t4 ? 330?t3 110?t2 3.174 million
dollars per year that was expected by the
Plastic-Is-Us toy company in Example 4 of
Applications. Let G(T) be the total income that
is expected during the first T years, for 0 ? T ?
1. Picking a time T 0.5 years, we will check
that the instantaneous rate of change of G(T),
with respect to T, is the same as A(T). Note
that We now wish to
compute G?(0.5). Recall that G?(T) is
approximated by the difference quotient for small
values of h. We will let h 0.0001, and use
Integrating.xls to evaluate G(0.5 0.0001) and
G(0.5 ? 0.0001). Integrating.xls rounds the
numerical values of integrals to four decimal
places. For the present calculation, we gain
extra precision by copying the values from Cell
N20 and keeping all of their decimal
places. G(0.5 0.0001) G(0.5001)
2.79078611562868 G(0.5 ? 0.0001) G(0.4999)
2.78946381564699
Show files ex-feb16-Integrating.xls
ex2-feb16-Integrating.xls
55
Integration, Calculus