qwerty

1. 1
Derivatives
Difference quotients are used in many
business situations, other than
marginal analysis (as in the previous
section)

2. 2
Derivatives
• Difference quotients
•
• Called the derivative of f(x)
• Computing
Called differentiation
     
h
h
x
f
h
x
f
lim
x
f
m
h 2
0







 
x
f 

3. 3
Derivatives
• Ex. Evaluate
if
  6
2 
 x
x
f
     
5452
.
5
002
.
0
0110903561
.
0
002
.
0
99445674
.
13
0055471
.
14
001
.
0
2
6
2
6
2
3
999
.
2
001
.
3










f
 
3
f 

4. 4
Derivatives
• Numerical differentiation is used to avoid tedious
difference quotient calculations
• Differentiating.xls file (Numerical differentiation utility)
• Graphs both function and derivative
• Can evaluate function and derivative

5. 5
Derivatives
• Differentiating.xls
Increment
x f (x ) f ' (x ) a b h s
= = #VALUE! 0.000001 t
u
v
w
Constants
Definition Plot Interval
Formula for f (x )
Computation
FUNCTION
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
x
f (x )
DERIVATIVE
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
x
f ' (x )

6. 6
Derivatives
• Use Differentiating.xls to graph the derivative of
on the interval [-2, 8]. Then
evaluate .
  6
2 
 x
x
f
 
3
f  DERIVATIVE
0
50
100
150
200
-5 0 5 10
x
f ' (x )
  5452
.
5
3 

f

7. 7
Important
• If f '(x) is constant, the displayed plot will
be distorted.
• To correct this, format the y-axis to have
fixed minimum and maximum values.
• Eg: Lets try to plot g(x)=10x in [-2,8]

8. 8
Derivatives
• Properties
If then
If then
If then
If then
   
x
f
a
x
g 
    
x
f
a
x
g 



     
x
g
x
f
x
h 
      
x
g
x
f
x
h 




  b
mx
x
f 
   m
x
f 

  c
x
f    0

 x
f

9. 9
Derivatives
• Tangent line approximations
• Useful for easy approximations to complicated
functions
• Need a point and slope (derivative)
• Use y = mx +b

10. 10
Derivatives
• Ex. Determine the equation of the tangent line to
at x = 3.
• Recall and we have the point (3, 14)
• Tangent line is y = 5.5452x – 2.6356
  6
2 
 x
x
f
  5452
.
5
3 

f
 
3
f  The slope of the graph of f
at the point (3,14)

11. 11
Derivatives
• Project (Marginal Revenue)
- Typically
- In project,
-
   
q
R
q
MR 

   
q
R
q
MR 

1000
     
h
h
q
R
h
q
R
q
R
2





Why ?

12. 12
Recall:Revenue function-R(q)
• Revenue in million dollars R(q)
• Why do this conversion?
Marginal Revenue in dollars per drive
   
q
R
q
MR 

   
 
q
R
q
R
q
MR





1000
1000
1000000

13. 13
Derivatives
• Project (Marginal Cost)
- Typically
- In project,
-
   
q
C
q
MC 

   
q
C
q
MC 

1000
     
h
h
q
C
h
q
C
q
C
2






14. 14
Derivatives
• Project (Marginal Cost)
- Marginal Cost is given in original data
- Cost per unit at different production levels
- Use IF function in Excel

15. 15
Derivatives
• Project (Marginal Profit)
MP(q) = MR(q) – MC(q)
- If MP(q) > 0, profit is increasing
- If MR(q) > MC(q), profit is increasing
- If MP(q) < 0, profit is decreasing
- If MR(q) < MC(q), profit is decreasing

16. 16
Derivatives
• Project (Marginal Revenue)
- Calculate MR(q)
-    
   
       
       
h
h
q
D
h
q
h
q
D
h
q
h
h
h
q
R
h
q
R
q
R
q
MR
h
q
D
h
q
h
q
D
h
q
2
2
1000
2
1000
1000
1000
1000


























17. 17
Derivatives
• Project (Marginal Cost)
- Calculate MC(q)
- IF(q<=500,115,IF(q<=1100,100,90))

18. 18
Derivatives
• Project (Maximum Profit)
- Maximum profit occurs when MP(q) = 0
- Max profit occurs when MR(q) = MC(q)
- Estimate quantity from graph of Profit
- Estimate quantity from graph of Marginal Profit

19. 19
Derivatives
• Project (Maximum Profit)
- Create table for calculations
q D (q ) R (q ) C (q ) P (q ) MR (q ) MC (q ) MP (q )
575.644 167.70
$ 96.53
$ 86.66
$ 9.87
$ 100.000
$ 100.000
$ 0.000
$

20. 20
Derivatives
• Project (Answering Questions 1-3)
1. What price? $167.70
2. What quantity? 575,644 units
3. What profit? $9.87 million
q D (q ) R (q ) C (q ) P (q ) MR (q ) MC (q ) MP (q )
575.644 167.70
$ 96.53
$ 86.66
$ 9.87
$ 100.000
$ 100.000
$ 0.000
$

21. 21
Derivatives
• Project (Answering Question 4)
4. How sensitive? Somewhat sensitive
-0.2%
-4.7%
q D (q ) R (q ) C (q ) P (q ) MR (q ) MC (q ) MP (q )
575.644 167.70
$ 96.53
$ 86.66
$ 9.87
$ 100.000
$ 100.000
$ 0.000
$
q D (q ) R (q ) C (q ) P (q ) MR (q ) MC (q ) MP (q )
565.644 168.86
$ 95.52
$ 85.66
$ 9.85
$ 103.644
$ 100.000
$ 3.644
$
q D (q ) R (q ) C (q ) P (q ) MR (q ) MC (q ) MP (q )
585.644 166.51
$ 97.52
$ 87.66
$ 9.85
$ 96.287
$ 100.000
$ (3.713)
$
q D (q ) R (q ) C (q ) P (q ) MR (q ) MC (q ) MP (q )
525.644 173.29
$ 91.09
$ 81.66
$ 9.42
$ 117.527
$ 100.000
$ 17.527
$
q D (q ) R (q ) C (q ) P (q ) MR (q ) MC (q ) MP (q )
625.644 161.53
$ 101.06
$ 91.66
$ 9.40
$ 80.745
$ 100.000
$ (19.255)
$

22. 22
Derivatives
• Project (What to do)
- Create one graph showing MR and MC
- Create one graph showing MP
- Prepare computational cells answering your team’s
questions 1- 4