XemP 08H CalB-222 Ch03 Derivative E.pdfsx

Calculus in your nuts

1. HCMUT –DEPARTMEND OF MATH. APPLIED
---------------------------------------------------------------------------------------------------------------------
CALCULUS FOR BUSINESS –221 Semester
CHAPTER 3: DERIVATIVE
•PhD. NGUYỄN QUỐC LÂN (November 2022)

2. RATE OF CHANGE: CONSTANT & VARIABLE
--------------------------------------------------------------------------------------------------------------------------------------------
Linearfunctiony=ax+bchangesatconstantrate→Slope
Example:Sincebeginningof
theyear,thepriceofabottle
ofsodahasbeenrisingata
constantrateof2
cents/month Price
functiony=2x+b(x:
numberofmonth)…
Generally,therateofchangeisnotconstant.Howtodescribe?
Answer:Usederivativef’(x)(whichistheslopeoftangentline).
-
F
General
+(x)
=>
Rate
=
f(x)
=
S
-
-------

3. PRACTICAL EXAMPLE
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----------------------------------------------------------------

4. DERIVATIVE
-----------------------------------------------------------------------------------------------------------------------------------
Derivativeoff(x),x=a:
Reading:f’(a)is“fprimeofa”,anddf/dxis“deefbydeex”.
=
(a)
-
! I

5. TECHNIQUE (HOFFMANN, CHAPTER 2, SECTION 2 →4).
-----------------------------------------------------------------------------------------------------------------------------------
() () () ()()
/
//1
/
2
/
3
/
2
,,,...3,2:Highschool 





===
−
v
u
uvvuxxxxxx


Tangentequationof(C):y=f(x)atx=a:
Rateatwhichy=f(x)changeswithrespecttoxatx=a:
()()()axafafy −=−
/
()af
/
()
()
()
12
2
/
0
2
1
2
0
BKEL Calculated questions
21
a/ Given ,evaluate approximately
1
b/ Let be the tangent equation of (C): 2 3 at 1.
Evaluate approximately .
x
y y x dx
x
ygx y x xx
g x dx
+
=

+
==+=



6. 12x
+
3x)'
=
4x
+3(or)
=
vv
+
uv
(cost)
=
-
sinx
a)y
=
x
+
1
↑
j
=
/
y
=I
5
b)y
=
2
x
3+3xat
x=
1
y(1)
=
5
y
=
6x2
+
3y'(1)
=
9
y
-
5
=
9(x
-
1)
=>
y
=
gx
-
4
=
1!
S
19x
-
4) 2dx
=
=
=
x)
b)y
=
==
y'(2)
=
-
3
c)FindWate
(x
-
1)2 y
=
xXchanges
at
x
=
1
,
5
=
2
a)
y
=
f()
=
2
+
3x
=
f()
=
5
Bam may
:)k
=
2
*
Eme
(x
=
15
=
9,45
=>
f
' (1)
=
7
=>
Tangent
:
y
-
5
=
7(x
-
1)
=
y
=
7x
-
2
+(g(x)dx
·
may

7. RATE OF CHANGE
-----------------------------------------------------------------------------------------------------------------------------------
21
Example: Find the rate at which changes with respect to at 2
1
x
yxx
x
+
==
−
Rateatwhichy=f(x)changeswithrespecttoxatx=a:()af
/

8. Ans
:
P'(x)
=
-
800x
+
6800
=
4'/9)
=
-
400
dollars/thounds
unit
Conclusion
:
Profitdecreasing
atrateof400
dollars/thousands
units
p'(g)
lim
Dollars
ds
-
>
unitoffunction
~
variable

9. CHAIN RULE
-----------------------------------------------------------------------------------------------------------------------------------

10. -0
=
y
=
f(g(x))
-
+
=
4
dc()
+
d
=
y
=
+
(g(x))'
-
g(x)
=
+
=
8
dx4
25Khainrutel
=>
c'(x)
+
v()
,
((x)
=
1x
+
4x
+
53
=>
(y
*
+
4x
+
53)
·
10
,
2
+
+0
,
037)
x(+)
=
0
,
2+
2+
0
,
037
3
=10
,
13
c(x(
+
)
=
1(0
,
2
+
+0
,
03+
12
+
410
,
2+
2+0
,
037)
+
53-
Simply
-
>
C'(xk
+
1)
c'(
+
)
=
0
,
4+
+0
,
03
x
+
xy
+
y
-
-
19
=
0
10
,
4+
+0
,
03
Cau
y
=
-
SL y'(x0)x=
2
+
y
=
3N
=oy
=
XXX
2x+y
+
2y y
+
xy
=
0
=>
-
2x
-
y
=
=
x
,
75
+
-
1
,
53??
-
2y
+x
-

11. MARGINAL ANALYSIS
--------------------------------------------------------------------------------------------------------------------------------------------
Ineconomics,theuseofthederivativetoapproximatethechange
inaquantitythatresultsfromone(1)–unitincreasein
productioniscalledmarginalanalysis
Whichargumentcouldweusetotrytobuyoneairticketfor
HochiminhCity–HaNoiin(forexample!)100kVND?

12. Remember
:
*
fin
f(x)
=
=>
When
x=x
,
f(x)L
En
x+x0 x0
fk)
=
1
2
*
f(a)=
lim
I
If
+x
=
1(00x
=
x
-
a
=
1
=x
=a
+
1)
in
gritson
=>
f(a+h)
-
f(a)
=
+'(a)(
*
)
=
When
x=
0
:
+(a)
zotef(a)
.
*
Cost function
:
Cale()
:
units"Thechangeofproduct
((9)
(((9)
102
unit
nd
-
(19)
:
Thecostof10unit

13. MARGINAL ANALYSIS
--------------------------------------------------------------------------------------------------------------------------------------------
CostC(x)MarginalCostMC(a)=C’(a)C(a+1)–C(a)=
Costofthe(a+1)
th
product

14. -
changeofcost
when
x
=
a
1
unit
((x)
=
292
+
49
,
970
2)
Use
marginal
costtoestimatethecostof
15th
un
it
1)
Marginal
costat
4?
((is)
-
((14)
=
C'(14)
C'(q)
=
4q
+4
=
C(4)
=
20
=
cost
increase
by20$
T
I
-
15 14 14
When
a
increase
from4
by
T
unit Nhof(a
+
1)
-
f(a)
=
f'(a)

15. a)Mc(x)
=
+x
+
3
b)
m
((9)
-
...
((8)
((9)
-
C(8)=C'(8)
=
59
c)
Actualcost
=
((g)
-
(28)
=
(
+
3
.
9
+
98)
-
(
+3
.
8
+
98)
=
=
5
,
12
-x
=
10
-
??
=
1)

16. EXAMPLE
--------------------------------------------------------------------------------------------------------------------------------------------
Amanufacturerofdigitalcamerasestimatesthatwhenxhundred
camerasareproduced,thetotalprofitwillbeP(x)=–0.0035x
3
+
0.07x
2
+25x–200thousanddollars.
(a)Findthemarginalprofitfunction
(b)Whatisthemarginalprofitwhenthelevelofproductionisx=
10,x=50andx=80?
(c)Interprettheseresults.
RevenueR(x)Mar.Reve.MR(a)=R’(a)R(a+1)–R(a)
ProfitP(x)MarginalProfitMP(a)=P’(a)P(a+1)–P(a)

17. #
M
.
prote
aP(x)
=
-
0
,
0035x3
+0
.
07x
+
25x
-
200/thousanddollars)
=>
MP
=
P
=
-
3
.
0
,
0035x
+0
,
14x+
25
b)P'(10)
=
25
,
35
P(55
-c)
x
=
10
=
)q
=
10x100
=
1000
=
44about25
,
35x1000
=
25350
dollarswhen
qtby
1000
cameras
x
=
50
=q
=
5000
=
PTabout
5
,
75x1000
=
5750
dollars
when
↑by
5000
cameras
x
=
80
=q
=
8000
=
4tabout31000dollars
When
qt
by
8000
dollars

18. APPROXIMATION BY INCREMENTS
--------------------------------------------------------------------------------------------------------------------------------------------
Ifthetotalrevenuefunctionofagoodisgivenby100Q–Q
2
write
theexpressionforthemarginalrevenuefunction.Ifthecurrent
demandis60,estimatethechangeinthevalueoftotalrevenuedue
toa2unitincreaseinQ
()xxyyx
yy
xx
xx 





=
0
/
0
small is If .
in change The :
in change The :
:At
himy
=>
X 0
by
zy
-
0x
-
R(q)
=
100Q
-
Q
=
Mr
=
R'(q)
=
100
-
29
q
=
60
:
2unitYing
=
q
=
2
=
0 RvR'(60)
.
09
RemainO
,
Susitdecrease
inq
=
(
-
20)
.
2
=
-
40
=>
rq
=
- 0
.
8=R
=
R160)
.
q
=R
decreasebyabout
40
=
=
20
.
-0
.
8
=
16

19. IMPLICIT FUNCTION
-----------------------------------------------------------------------------------------------------------------------------------
Thefunctionswehavemetsofarcanbedescribedbyy=f(x)
Somefunctions,however,aredefinedimplicitlybyarelationbetween
x&ysuchasx^3+y^3=6xyy=y(x):Implicitfunction
AnequationF(x,y)=0defines
implicitlyone(ormore)
functiony=y(x):
F(x,y(x))=0
Thesefunctionsarecalled
implicitfunction
x
=
0
+
y3
=
0
y
can'
+
3
*
=
0
+y3
-
18y
+
27
y
=
3
Zy
=
=

20. IMPLICIT DIFFERENTIATION
-----------------------------------------------------------------------------------------------------------------------------------
MethodofImplicitDifferentiation:Differentiatingbothsidesofthe
equationF(x,y)=0withrespecttox,regardingalwaysy=y(x),and
thensolvingtheresultingequationfory’.
3) (3;at 6 :(C) curve theo tangent t theFindb/
6 if Finda/ :Example
33
33/
xyyx
xyyxy
=+
=+
(x2)"
=
2x
(02)'
=
2u
.
r
3y-
yirongigtons
=>
(x
+
yS)y
=
(Gxy)x
not
Cy
Look
at
Hoffman
Implicitdiff
xy
+
xy
a)3x2
+
3y
.
y
=
G[y
+
xy]
=
(3y2
-
(x)y
=
by
-
3x
=
y
=
(2
-
x
y
-
-
2xb)
A+(3
,
3)
:
y
=
=>
Tangent
:
%
0
↓To
y
-
3
=
-
1(x
-
3)
=y
=
-
x+
6

21. ECONOMIC EXAMPLE
---------------------------------------------------------------------------------------------------------------------------------------------------
SupposetheoutputatacertainfactoryisQ=2x
3
+x
2
y+y
3
units,
wherexisthenumberofhoursofskilledlaborusedandyisthe
numberofhoursofunskilledlabor.Thecurrentlaborforceconsists
of30hoursofskilledlaborand20hoursofunskilledlabor.Estimate
thechangeinunskilledlaborythatshouldbemadetooffseta1–
hourincreaseinkilledlaborxsothatoutputwillbemainteinedatits
currentlevel.

22. Output
a
=
f(x
,
y)
Q
=
2x
+
xy
+
y x
:
numbersofhoursof
skill
labors
y
:
numbersofhoursofunskill
labors
x
=
30
x
↓bythour
-
>
a
=
constant
=>
2x3
+
x
y
+
y
=
const
y
=
20
&
yt
by
how
much
&
um
y
=
y(x)
=
y
-y
=
y
-
0x
Xx
=
1
(2x3
+
y
+
y))
=
(const)
=
6x2
+
2xy
+
x
y3yy
=
0by
=
y
.x
I
=-
3
,
14
=>
y
=
eye-3 =>
Decreasey inabout
3
,
14hours

23. C
=
C(q)
unitF(c
,
q)
=
const
-
>
Implicit
C
=
Clq)
If
g
,
c
change/respect?[C
-
39
=
4275
Chain
a
9
,
9
=
g(t)
, d
I
give
n
=C=?
1500units
=>q
=
15=C
=27
=
120Cost
=
120000dollars
(c
2
-
39)
!
=
142751
++
2CC
-
99q
=
0
=>
2x120xc
-
9x15x
=>
c
=
1
,
6875
=>
Cost
increases
withrateof
1687
.
5
dollars
/week
.

24. RELATED RATES. EXAMPLE 2.6.8 (SECTION 2.6)
--------------------------------------------------------------------------------------------------------------------------------------------

25. RELATED RATES. EXAMPLE 2
-----------------------------------------------------------------------------------------------------------------------------------

26. DERIVATIVE & CONTINUOUS
--------------------------------------------------------------------------------------------------------------------------------------------
lim
#
im
+(x)
=
+(c)
-
hasderivative
Smooth
?
~
I
#
-
>
Derivative
=>
Continious
C#

27. EXAMPLE
--------------------------------------------------------------------------------------------------------------------------------------------
end
answer
x
+
0
,
=
8
continuous
ea
=
flo)lint
ing(x)
=
1
=
g(0)
=
+(
=
g()vxref
6
First
,
a
=
f(0)
=
limf(x)
x-0e
! 1x
-
>
0
=
I=L
When
a
=
1
+
d
=

28. INCREASING –DECREASING FUNCTION (HOFFMANN, CH. 3)
--------------------------------------------------------------------------------------------------------------------------------------------

29. RELATIVE (LOCAL) EXTREMA & MIN -MAX
--------------------------------------------------------------------------------------------------------------------------------------------
(C):y=f(x)hasarelative(local)maximumatx=ciff(c)f(x)x
nearc;relative(local)mininumatx=ciff(c)f(x)xnearc.
Relativemax,minarecalledrelativeextremaandareonlylocal.

30. EXTREMA & MIN MAX EXAMPLE
--------------------------------------------------------------------------------------------------------------------------------------------

31. DIMINISHING RETURNS
--------------------------------------------------------------------------------------------------------------------------------------------
TheoutputQ(t)ofafactoryworkerthoursaftercomingtowork.

32. INFLECTION POINTS
--------------------------------------------------------------------------------------------------------------------------------------------
Aninflectionpoint(orpointofinflection)isapoint(c,f(c))(C):y
=f(x)wheretheconcavity(thatmeansf’’)changes.Atsuchapoint,
eitherf’’(c)=0orf’’(c)doesnotexist.
() .752
poins Inflection :Example
46
ttttQ +−=

33. INFLECTION POINT OR DIMINISHING RETURN
--------------------------------------------------------------------------------------------------------------------------------------------
Anefficiencystudyofthemorningshiftbetween8:00A.M.and
12:00noonatafactoryindicatesthatanaverageworkerwillhave
producedQ(t)=–t
3
+9t
2
+12tthourslater.Atwhattimeduring
themorningistheworkerperformingmostandleastefficiently?

34. (PRICE ELASTICITY OF DEMAND)
--------------------------------------------------------------------------------------------------------------------------------------------

35. EOD EXAMPLE
--------------------------------------------------------------------------------------------------------------------------------------------

36. CLASSIFICATION
--------------------------------------------------------------------------------------------------------------------------------------------

37. CLASSIFICATION AND REVENUE
--------------------------------------------------------------------------------------------------------------------------------------------

38. EXAMPLE
--------------------------------------------------------------------------------------------------------------------------------------------