Cours : Taux d'évolution

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- 1 -

D. PINEL, Site Mathemitec : http://mathemitec.free.fr/index.php

• !"# $ $

%

• & $ '"# !"#

("# %

• ) $ $ *# !*#

!*# *# %

• + (# !" %

• , - ("# * . %

(2)

- 2 -

# 100

t

/ *("" ("#

5200 20 1040

×100=

0 -

1 25% 25 1 0.25

100 4

= = = 2

3 -

2

4 y₁ y₂ 5 y₁→y₂6

0 ! y₁ y₂ ² ¹

1

y y

t y

= −

/ 1

2 1 100 100%

100

P P P

t P P

= − = = = = !""#

7 t<0 ou y₂<y₁

2 1

0

t> ou y >y " #

8 !"# 1

-

• 0 1 !"#

• 0 9 1

• 0

$ # %

:

- 5 6

y1 y2 = +

( )

1 t y1

! " #$ % &%' y à y₁ ₂

2 1

1

y y

t y

= − $ %( y2− = × ⇔y1 t y1 y2 =ty1+y1⇔y2 = +

( )

1 t y1

/

1 !"# 1 10 0.9

−100= & ' )

' " ( & *' +

, !;<# 1 19.6 1.196

−100 = &

' ), ' " ( & - ),

(3)

- 3 -

8 y₁ y₂

y2 y₁

& '()

5 - 6 y₁ y₂

= !>

0 ₁ ₂

c c

y ^× y

÷

. ' / % &"" " ' 0 "

Rappels de Première : Applications

( * !"# $ $

% Pi

• ?!"# = $"!

= !> = !? "!= ";

• 0

0.9

i f

P ^× P

÷

1 1.11

i 0.9 f f

P = P ≈ P

Pf P_i 0 c' 1.11≈ t'= − ≈c' 1 0.11 11%=

1& ' & ' " ' 2 "

" +

1 2

y à y t

* y₂ y₁ c' 1

=c & &

, & y à y₂ ₁ 1 ' 1

t 1 + = t

+

!

( , , !"# ("#

+ - %

Pi

• c₁= −1 10% 0.9=

2 1 20% 0.8

c = − =

• 0

0.9 0.8 2

i f

P P P

× ×

→ → P_f =0.9 0.8× P_i =0.72P_i

"@(

• A = !> = ? != "@(? != $"(B

1& 3 #" %" 2 ( % %

" - % ( & 4

(4)

- 4 -

" +

1 2

y à y t₁ y₂ y₃ t₂

* 5- 6 y₁ y₃ c c c= ×₁ ₂

, - y à y₁ ₃ 1+ = +t

(

1 t1

)(

1+t2

)

0

( ) ( )

( )

1 2

1 1

2 1

t t

i f

t

P P P

× + × +

× +

→ →

( - , 5 6 (# !"

+ - %

C

( ) ( ) ( )

( )

1 2% 1 2% 1 2%

2 10

1

... ...

i

t

P P P

× + × + × +

× +

→ → →

0 ¹^{+ = +}^t

(

^{1 2%}

)

¹⁰ ⁼^1.02¹⁰ ^≈^1.22 t≈1.22 1 0.22− = - ((#

$ -

• t₁ t₂

(

1+t1

)(

1+t2

)

• t₂ t₁

(

1+t2

)(

1+t1

)

0 D

0 -

1, , ... ,2 _n

t t t

* 5- 6 c c c= × × ×₁ ₂ ... c_n

, - 1+ = +t

(

1 t1

)(

1+t2

)

× × +... 1

(

t_n

)

0

( ) ( ) ( )

( )

1 2 1

1 1

1 2

1

...

tn

t t

n t

P P P

× + × + × +

× +

→ → →

(5)

- 5 -

Taux d’évolution moyen ou équivalent

C 9

. /

0 1

xn =a x a= ¹ⁿ 5 1

n6

8 D 3

1 1

1

n n n

n n n n

x = a =a ^× =a =a =a

1

3 5 53

x = ⇔ =x ( ^ x^y E

E

1

53 ≈1.71 5 !F'6

1, 2,..., _n

c c c

0 . - c=

(

c c1× × ×2 ... c_n

)

¹ⁿ

2 ! /

$ ( ( ' % " ( & ( '

" /

5 9 26 -

' ! / t_M

# 0

( )

1 2

1

... _n

t

P P P

× +

→ → → G # # t_M

( ) ( ) ( )

( )

1 1 1

1 2

1

...

M M M

t t t

n t

P P P

× + × + × +

× +

→ → →

3

(

¹+tM

)

ⁿ = + ⇔ +¹ t ¹ tM = +

( )

¹ t ¹ⁿ G -

C . t_M

( )

¹

1+t_M = +1 t ⁿ G -

( * , - ("# *

. %

tM . 1 . = ("# !> = !(

4

(1 ) (1 ) (1 )

1 2

1.2

...

M M M

t t t

P P Pf

× + × + × +

×

→ → →

(

¹+tM

)

⁵=^1.2 G

1

1+t_M =1.25 ≈1.037 H t_M = !"'@? != ""'@

! 5 " % 6 % 47

"

(6)

- 6 -

( , , - '*# !"#

3 .

• - t₁= +1 35% 1.35= t₂ =1 10% 0.9− =

C

( ) 1.35 0.9

2 1

i f

t

P P P

× ×

× +

→ → 1+ =t 1.35 0.9× ⇔ =t 0.215 21.5%= -

>(!*#

• t_M . (I

/

(1 ) (1 ) (1 )

1 2

1.215

...

M M M

t t t

P P Pf

× + × + × +

×

→ → →

(

¹+tM

)

²⁴=^1.215⇔tM =^1.215²⁴¹ − ≈^{1 0.81%} . "B!#

Approximations des petits taux

/ . 2

Indice en base 100

, 9

D '

-

0 J 5(""(6

0

• 0 (""( *33

• C !

& % ( & " " % ( & (

" % ' 8 ' %' 9

1 !

0 % ,33,

K >!'# 5!!'$!""6 (""' K $<"*# (""I

K 2

K >("B# (""<

,33, ,33- ,334 ,335 ,336 I('" I@B" ';@I I;;( *!!"

,33, ,33- ,334 ,335 ,336 '("" '@B" '(*" I""" I('"

,33, ,33- ,334 ,335 ,336 I('" I@B" ';@I I;;( *!!"

*33

100 4780 113

×4230≈ ;';* !!B !("B

(7)

- 7 -

( % 0 *331

1 (""'

(""*

1

118 113 . 2%

13 4 4

t= − ≈

, L 0

1 9 M E - -

9 M

,33, ,33- ,334 ,335 ,336 / !*33 !!' ;';* !!B !("B / (*33 !!B! !"!< !(* !'((