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4
3
2
1
Grade 12
/ Annals on differential equations
11 exercises (including 9 corrected)
a
Former annuals (before 2012)
(11 exercices)
1.
F
ormer
annuals
(b
efor
e
2012)
E.3134
Throughout
the
exercise,
–
denotes
a
real
n
um
b
er
in
the
in
terv
al
]0
;
1]
.
1
W
e
prop
ose
to
study
the
functions
deriv
able
on
−∞
;
1
2
v
erifying
the
differen
tial
equation
:
(
E
λ
)
:
y
=
y
2
+
–
·
y
and
the
condition
y
(0)
=
1
.
It
is
assumed
that
there
exists
a
y
0
solution
of
(
E
λ
)
strictly
p
ositiv
e
on
−∞
;
1
2
.
W
e
la
y
on
−∞
;
1
2
:
z
=
1
y
0
W
rite
a
simple
differen
tial
equation
satisfied
b
y
the
func-
tion
z
.
2
Course
question
:
PRE-REQUISITES
The
solutions
of
the
differen
tial
equation
y
=
−
–
·
y
are
the
functions
:
x
↦−
→
C
·
e
−
λx
where
C
is
a
real
constan
t.
a
Demonstrate
the
existence
and
uniqueness
of
the
solu-
tion
z
of
the
differen
tial
equation
:
(
E
λ
)
:
z
=
−
(
–
·
z
+
1)
suc
h
that
:
z
(0)
=
1
b
Giv
e
the
expression
of
this
function,
whic
h
will
b
e
de-
noted
z
0
.
W
e
no
w
w
an
t
to
sho
w
that
the
function
z
0
do
es
not
cancel
on
the
in
terv
al
−∞
;
1
2
.
3
a
Demonstrate
that
:
ln
(1+
–
)
>
–
–
+1
.
W
e
can
study
on
]0
;
1]
the
function
f
defined
b
y
:
f
(
x
)
=
ln
(1+
x
)
−
x
x
+
1
.
b
Deduce
that
:
1
–
·
ln
(1+
–
)
>
1
2
.
4
Deduce
that
the
function
z
0
do
es
not
cancel
on
the
in
ter-
v
al
−∞
;
1
2
.
Sho
w
then
that
(
E
λ
)
admits
a
strictly
p
ositiv
e
solution
on
−∞
;
1
2
to
b
e
sp
ecified.
E.3180
MCQ
:
for
eac
h
question,
only
one
of
the
prop
osed
answ
ers
is
correct.
No
justification
is
required.
Eac
h
correct
answ
er
earns
0.75
p
oin
t,
eac
h
error
deducts
0.25
p
oin
t,
no
answ
er
is
w
orth
0
p
oin
t.
If
the
total
p
oin
ts
of
the
exercise
are
negativ
e,
the
mark
is
reduced
to
0.
Answ
er
on
y
our
cop
y
,
indicating
the
question
n
um
b
er
and
the
letter
corresp
onding
to
y
our
answ
er
:
1
The
equation
e
2
x
−
3e
x
−
4
=
0
admits
in
R
:
a
0
solution
b
1
solution
c
2
solutions
d
more
than
2
solutions
2
The
expression
−
e
−
x
a
is
nev
er
negativ
e
b
is
alw
a
ys
negativ
e
c
is
negativ
e
only
if
x
is
p
ositiv
e
d
is
negativ
e
only
if
x
ets
négatif
3
lim
x
↦→
+
∞
2e
x
−
1
e
x
+
2
=
a
−
1
2
b
1
c
2
d
+
∞
4
The
differen
tial
equation
y
=
2
y
−
1
has
as
set
of
solu-
tions
:
a
x
↦→
k
e
2
x
−
1
a
v
ec
k
∈
R
b
x
↦→
k
e
1
2
x
+
1
a
v
ec
k
∈
R
c
x
↦→
k
e
1
2
x
−
1
a
v
ec
k
∈
R
d
x
↦→
k
e
2
x
+
1
2
a
v
ec
k
∈
R
h
ttps:/
/c
hingmath.fr
chapExoCorrec/3134
sacados/3134
France Septembre 2006 6 points
chapExoCorrec/3180
sacados/3180
Antilles-Guyane Juin 2006 3 points
E.3259
P
art
A
:
a
differen
tial
equa-
tion
Consider
the
differen
tial
equation
:
(
E
)
:
y
−
3
y
=
−
3
e
1
+
e
−
3
x
2
W
e
giv
e
a
function
’
deriv
able
on
R
and
the
function
f
de-
fined
on
R
b
y
:
f
(
x
)
=
e
−
3
x
’
(
x
)
1
Sho
w
that
f
is
deriv
able
on
R
and
for
an
y
real
x
,
express
’
(
x
)
−
3
’
(
x
)
as
a
function
of
f
(
x
)
.
2
Determine
f
so
that
’
is
a
solution
of
(
E
)
at
R
and
v
erifies
’
(0)
=
e
2
.
P
art
B
:
study
of
a
function
Let
f
b
e
the
function
f
defined
on
R
b
y
:
f
(
x
)
=
e
1
−
3
x
1+e
−
3
x
W
e
denote
b
y
C
its
represen
tativ
e
curv
e
in
the
plane
pro
vided
with
an
orthonormal
reference
frame
of
graphic
unit
2
cm
.
1
Determine
the
limits
of
f
in
−∞
and
in
+
∞
,
then
study
the
v
ariations
of
f
.
2
T
race
C
.
3
F
or
¸
real
non-zero,
w
e
p
ose
:
I
α
=
α
0
f
(
x
)
dx
a
Giv
e
the
sign
and
a
graphical
in
terpretation
of
I
α
as
a
function
of
¸
.
b
Express
I
α
as
a
function
of
¸
.
c
Determine
the
limit
of
I
α
when
¸
tends
to
+
∞
.
P
art
C
:
etude
d’une
suite
W
e
define
on
N
∗
the
sequence
(
u
n
)
b
y
:
u
n
=
1
0
f
(
x
)
·
e
x
n
dx
où
f
is
the
function
defined
in
part
B.
No
attempt
will
b
e
made
to
calculate
u
n
.
1
a
Giv
e,
for
an
y
n
from
N
,
the
sign
of
u
n
.
b
Giv
e
the
direction
of
v
ariation
of
the
sequence
(
u
n
)
.
c
Is
the
sequence
(
u
n
)
con
v
ergen
t?
2
a
Sho
w
that
for
an
y
n
of
N
:
I
1
u
n
e
1
n
·
I
1
où
I
1
is
the
in
tegral
of
the
part
B
obtained
for
¸
equal
to
1
.
b
Deduce
the
limit
of
the
sequence
(
u
n
)
.
Giv
e
its
exact
v
alue.
E.3150
P
art
A
The
function
f
is
defined
on
the
in
terv
al
[0
;
+
∞
[
b
y
:
f
(
x
)
=
20
x
+
10
e
−
1
2
x
Note
C
the
represen
tativ
e
curv
e
of
the
function
f
in
an
orthonormal
reference
frame
O
;
−
→
i
;
−
→
j
(unité
gr
aphique
1
cm
)
.
1
Study
the
limit
of
the
function
f
in
+
∞
.
2
Study
the
v
ariations
of
the
function
f
and
dra
w
up
its
table
of
v
ariations.
3
Establish
that
the
equation
f
(
x
)
=
10
admits
a
single
strictly
p
ositiv
e
solution
¸
in
the
in
terv
al
0
;
+
∞
[
.
Giv
e
an
appro
ximate
decimal
v
alue
to
the
nearest
10
−
3
of
¸
.
4
Dra
w
the
curv
e
C
.
5
Calculate
the
in
tegral
:
I
=
3
0
f
(
x
)
d
x
.
P
art
B
Let
y
(
t
)
b
e
the
v
alue,
in
degrees
Celsius,
of
the
temp
erature
of
a
c
hemical
reaction
at
time
t
,
t
b
eing
expressed
in
hours.
The
initial
v
alue,
at
time
t
=
0
,
is
y
(0)
=
10
.
W
e
admit
that
the
function
whic
h,
to
an
y
real
t
b
elonging
to
the
in
terv
al
[0
;
+
∞
[
asso
ciates
y
(
t
)
,
is
solution
of
the
differ-
en
tial
equation
:
(
E
)
:
y
+
1
2
·
y
=
20
·
e
−
1
2
t
1
V
erify
that
the
function
f
studied
in
part
A
is
a
solution
of
the
differen
tial
equation
(
E
)
on
the
in
terv
al
0
;
+
∞
.
2
W
e
prop
ose
to
sho
w
that
this
function
f
is
the
unique
solution
of
the
differen
tial
equation
(
E
)
,
defined
on
the
in
terv
al
0
;
+
∞
,
whic
h
tak
es
the
v
alue
10
at
time
0.
a
Let
g
b
e
an
y
solution
of
the
differen
tial
equation
(
E
)
,
defined
on
[0
;
+
∞
[
v
erifying
g
(0)
=
10
.
Sho
w
that
the
function
g
−
f
is
a
solution,
on
the
in
terv
al
0
;
+
∞
,
of
the
differen
tial
equation
:
(
E
)
:
y
+
1
2
·
y
=
0
b
Solv
e
the
differen
tial
equation
(
E
)
.
c
Conclude.
3
After
ho
w
long
do
es
the
temp
erature
of
this
c
hemical
re-
action
drop
bac
k
to
its
initial
v
alue?
The
result
should
b
e
rounded
to
the
nearest
min
ute.
4
The
v
alue
„
in
degrees
Celsius
of
the
a
v
erage
temp
era-
ture
at
this
c
hemical
reaction
during
the
first
first
three
hours
is
the
mean
v
alue
of
the
function
f
o
v
er
the
in
ter-
v
al
0
;
3
.
Calculate
the
exact
v
alue
of
„
,
then
giv
e
the
decimal
ap-
pro
ximate
v
alue
of
„
rounded
to
the
degree.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/3259
sacados/3259
France Septembre 2003 10 points
chapExoCorrec/3150
sacados/3150
France Septembre 2005 7 points
E.3173
P
arts
A
and
B
are
indep
enden
t.
A
researc
h
lab
oratory
studies
the
ev
olution
of
an
animal
p
op-
ulation
that
app
ears
to
b
e
on
the
v
erge
of
extinction.
P
art
A
In
2000,
a
study
is
carried
out
on
a
sample
of
this
p
opulation
with
an
initial
size
of
one
thousand.
This
sample
ev
olv
es
and
its
size,
expressed
in
thousands
of
in-
dividuals,
is
appro
ximated
b
y
a
function
f
of
time
t
(expr
esse
d
in
ye
ars
fr
om
the
origin
2000)
.
A
ccording
to
the
ev
olution
mo
del
c
hosen,
the
function
f
is
deriv
able,
strictly
p
ositiv
e
on
[0
;
+
∞
[
,
and
satisfies
the
differ-
en
tial
equation
:
(
E
)
:
y
=
−
1
20
·
y
·
3
−
ln
y
1
Demonstrate
the
follo
wing
equiv
alence
:
a
function
f
,
deriv
able,
strictly
p
ositiv
e
on
[0
;
+
∞
[
,
v
eri-
fies
f
(
t
)
=
−
1
20
f
(
t
)
3
−
ln
f
(
t
)
,
for
an
y
t
from
[0
;
+
∞
[
if,
and
only
if,
the
function
g
=
ln
(
f
)
v
erifies,
for
an
y
t
of
0
;
+
∞
,
g
(
t
)
=
1
20
g
(
t
)
−
3
20
2
Giv
e
the
general
solution
of
the
differen
tial
equation
:
(
H
)
:
z
=
1
20
z
−
3
20
.
3
Deduce
that
there
exists
a
real
C
suc
h
that,
for
an
y
t
of
[0
;
+
∞
[
:
f
(
t
)
=
exp
3
+
C
·
exp
t
20
(the
notation
exp
denotes
the
natur
al
exp
onential
func-
tion
x
↦−
→
e
x
)
4
The
initial
condition
therefore
leads
us
to
consider
the
function
f
defined
b
y
:
f
(
t
)
=
exp
3
−
3
exp
t
20
a
Determine
the
limit
of
the
function
f
in
+
∞
.
b
Determine
the
direction
of
v
ariation
of
f
on
[0
;
+
∞
[
.
c
Solv
e
in
[0
;
+
∞
[
the
inequation
f
(
t
)
<
0.02
.
After
ho
w
man
y
y
ears,
according
to
this
mo
del,
will
the
sample
size
b
e
less
than
t
w
en
t
y
individuals?
P
art
B
In
2005,
this
researc
h
lab
oratory
dev
elop
ed
a
test
to
detect
the
disease
resp
onsible
for
this
disapp
earance
and
pro
vided
the
follo
wing
information
:
ˇ
The
p
opulation
tested
includes
50
%
diseased
animals.
If
an
animal
is
ill,
the
test
is
p
ositiv
e
in
99
%
of
cases
;
if
an
animal
is
not
ill,
the
test
is
p
ositiv
e
in
0.1
%
of
cases
ı.
Note
M
the
ev
en
t
ˇ
the
animal
is
sic
k
ı,
M
the
opp
osite
ev
en
t
and
T
the
ev
en
t
ˇ
the
test
is
p
ositiv
e
ı.
1
Determine
P
(
M
)
,
P
M
(
T
)
,
P
M
(
T
)
.
2
Deduct
P
(
T
)
.
3
The
lab
oratory
considers
a
test
to
b
e
reliable,
if
its
pre-
dictiv
e
v
alue,
i.e.
the
probabilit
y
that
an
animal
will
b
e
ill
kno
wing
that
the
test
is
p
ositiv
e,
is
greater
than
0.999
.
Ho
w
reliable
is
this
test?
E.3244
P
art
A
Let
f
b
e
the
function
defined
on
R
b
y
:
f
(
x
)
=
3e
x
4
2+e
x
4
1
Sho
w
that
:
f
(
x
)
=
3
1+2e
−
x
4
.
2
Study
the
limits
of
the
function
f
in
+
∞
and
in
−∞
.
3
Study
the
v
ariations
of
the
function
f
.
P
art
B
1
The
ev
olution
of
a
p
opulation
of
small
ro
den
ts
w
as
stud-
ied
in
the
lab
oratory
.
The
p
opulation
size,
at
time
t
,
is
denoted
g
(
t
)
.
W
e
th
us
define
a
function
g
of
the
in
ter-
v
al
0
;
+
∞
in
R
.
The
real
v
ariable
t
denotes
time,
ex-
pressed
in
y
ears.
The
unit
c
hosen
for
g
(
t
)
is
the
h
undred
of
individuals.
The
mo
del
used
to
describ
e
this
ev
olu-
tion
consists
in
taking
for
g
a
solution,
on
the
in
terv
al
0
;
+
∞
,
of
the
differen
tial
equation
:
(
E
1
)
:
y
=
y
4
a
Solv
e
the
differen
tial
equation
(
E
1
)
.
b
Determine
the
expression
of
g
(
t
)
when,
at
date
t
=
0
,
the
p
opulation
comprises
100
ro
den
ts,
i.e.
g
(0)
=
1
.
c
After
ho
w
man
y
y
ears
will
the
p
opulation
exceed
300
ro
den
ts
for
the
first
time?
2
In
realit
y
,
in
an
observ
ed
area
of
a
giv
en
region,
a
preda-
tor
prev
en
ts
suc
h
gro
wth
b
y
killing
a
certain
quan
tit
y
of
ro
den
ts.
W
e
denote
u
(
t
)
the
n
um
b
er
of
ro
den
ts
aliv
e
at
time
t
(expr
esse
d
in
ye
ars)
in
this
area,
and
w
e
admit
that
the
function
u
,
th
us
defined,
satisfies
the
conditions
:
(
E
2
)
:
u
(
t
)
=
u
(
t
)
4
−
u
(
t
)
2
12
u
(0)
=
1
for
an
y
real
n
um
b
er
t
p
ositiv
e
or
zero
and
où
u
denotes
the
function
deriv
ed
from
the
function
u
.
a
It
is
assumed
that,
for
an
y
p
ositiv
e
real
t
,
w
e
ha
v
e
u
(
t
)
>
0
.
Consider,
on
the
in
terv
al
0
;
+
∞
,
the
func-
tion
h
defined
b
y
h
=
1
u
.
Sho
w
that
the
function
u
sat-
isfies
the
conditions
(
E
2
)
if,
and
only
if,
the
function
h
satisfies
the
conditions.
(
E
3
)
:
h
(
t
)
=
−
1
4
h
(
t
)
+
1
12
h
(0)
=
1
for
an
y
real
n
um
b
er
t
p
ositiv
e
or
zero.
b
Giv
e
the
solutions
of
the
differen
tial
equation
:
y
=
−
1
4
y
+
1
12
and
deduce
the
expression
of
the
function
h
,
then
that
of
the
function
u
.
c
In
this
mo
del,
ho
w
do
es
the
size
of
the
study
p
opula-
tion
b
eha
v
e
when
t
tends
to
+
∞
?
h
ttps:/
/c
hingmath.fr
chapExoCorrec/3173
sacados/3173
chapExoCorrec/3244
sacados/3244
France Juin 2005 6 points
E.3247
W
e
prop
ose
to
sho
w
that
there
exists
a
single
function
f
deriv
able
on
R
v
erifying
the
condi-
tion
:
(
C
)
:
f
(
−
x
)
·
f
(
x
)
=
1
for
an
y
real
n
um
b
er
x
f
(0)
=
−
4
(où
f
denotes
the
function
derive
d
fr
om
the
function
f
)
and
find
this
function.
1
It
is
assumed
that
there
exists
a
function
f
satisfying
the
condition
(
C
)
and
w
e
then
consider
the
function
g
defined
on
R
b
y
:
g
(
x
)
=
f
(
−
x
)
f
(
x
)
a
Sho
w
that
the
function
f
do
es
not
cancel
at
R
.
b
Calculate
the
deriv
ativ
e
function
of
the
function
g
.
c
Deduce
that
the
function
g
is
constan
t
and
determine
its
v
alue.
d
Consider
the
differen
tial
equation
(
E
)
:
y
=
1
16
y
.
Sho
w
that
the
function
f
is
a
solution
of
this
equation
and
v
erifies
f
(0)
=
−
4
.
2
Course
question
a
It
is
kno
wn
that
the
function
x
↦−
→
e
x
16
is
a
solution
of
the
differen
tial
equation
(
E
)
.
Sho
w
then
that
the
set
of
solutions
of
the
equation
(
E
)
is
the
set
of
functions,
defined
on
R
,
of
the
form
x
↦−
→
K
e
x
16
,
où
K
is
an
y
real
n
um
b
er.
b
Sho
w
that
there
is
a
unique
solution
of
the
differen
tial
equation
(
E
)
taking
the
v
alue
−
4
and
0
.
3
Deduce
from
the
previous
questions
that
there
is
a
single
function
deriv
able
on
R
satisfying
the
condition
(
C
)
and
sp
ecify
that
it
is
this
function.
E.3674
The
t
w
o
parts
of
this
exercise
are
indep
enden
t
P
art
A
:
Consider
the
differen
tial
equation
:
(
E
)
:
y
+
y
=
e
−
x
1
Sho
w
that
the
function
u
defined
on
the
set
of
real
n
um-
b
ers
R
b
y
u
(
x
)
=
x
·
e
−
x
is
a
solution
of
the
differen
tial
equation
(
E
)
.
2
Consider
the
differen
tial
equation
:
(
E
)
:
y
+
y
=
0
Solv
e
the
differen
tial
equation
(
E
)
.
3
Let
v
b
e
a
function
defined
and
deriv
able
on
R
.
Sho
w
that
the
function
v
is
a
solution
of
the
differen
tial
equa-
tion
(
E
)
if
and
only
if
the
function
v
−
u
is
a
solution
of
the
differen
tial
equation
(
E
)
.
4
Determine
the
unique
solution
g
of
the
differen
tial
equa-
tion
(
E
)
suc
h
that
g
(0)
=
2
.
P
art
B
:
Consider
the
function
f
k
defined
on
the
set
R
of
real
n
um
b
ers
b
y
:
f
k
(
x
)
=
(
x
+
k
)
·
e
−
k
où
k
is
a
giv
en
real
n
um
b
er.
Note
C
k
the
represen
tativ
e
curv
e
of
the
function
f
k
in
an
orthogonal
reference
frame.
1
Sho
w
that
the
function
f
k
admits
a
maxim
um
in
:
x
=
1
−
k
.
2
Note
M
k
the
p
oin
t
on
the
curv
e
C
k
with
abscissa
1
−
k
.
Sho
w
that
the
p
oin
t
M
k
b
elongs
to
the
curv
e
Γ
of
equa-
tion
y
=
e
−
x
.
3
On
the
graph
giv
en
in
App
endix
1
(to
b
e
r
eturne
d
with
the
c
opy)
,
the
reference
frame
is
orthogonal
but
the
unit
on
the
x-axis
and
y-axis
and
the
n
um
b
ers
of
the
curv
es
do
not
app
ear.
T
w
o
curv
es
ha
v
e
b
een
dra
wn
on
this
graph
:
the
curv
e
Γ
of
equation
y
=
e
−
x
.
the
curv
e
C
k
of
equation
y
=
(
x
+
k
)
·
e
−
x
for
a
giv
en
real
n
um
b
er
k
.
a
Iden
tify
the
curv
es
and
name
them
on
App
endix
1
(r
e-
turn
with
c
opy)
.
b
Explaining
the
approac
h
used,
determine
the
v
alue
of
the
corresp
onding
real
n
um
b
er
k
and
the
graphical
unit
on
eac
h
axis.
4
Using
in
tegration
b
y
parts,
calculate
:
2
0
(
x
+
2)
·
e
−
x
dx
.
Giv
e
a
graphical
in
terpretation
of
this
in
tegral.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/3247
sacados/3247
sacados/3674
E.3675
1
In
this
question,
the
c
andidate
is
aske
d
to
demonstr
ate
know
le
dge.
The
follo
wing
result
is
assumed
to
b
e
kno
wn
:
The
function
x
↦−
→
e
x
is
the
only
function
’
deriv
able
on
R
suc
h
that
’
=
’
,
and
’
(0)
=
1
.
Let
a
b
e
a
giv
en
real.
a
Sho
w
that
the
function
f
defined
on
R
b
y
f
(
x
)
=
e
ax
is
a
solution
of
the
equation
y
=
a
·
y
.
b
Let
g
b
e
a
solution
of
the
equation
y
=
a
·
y
.
Let
h
b
e
the
function
defined
on
R
b
y
h
(
x
)
=
g
(
x
)
·
e
−
a
·
x
.
Sho
w
that
h
is
a
constan
t
function.
c
Deduce
the
set
of
solutions
to
the
equation
:
y
=
a
·
y
.
2
Consider
the
differen
tial
equation
:
(
E
)
:
y
=
2
y
+
cos
x
.
a
Determine
t
w
o
real
n
um
b
ers
a
and
b
suc
h
that
the
func-
tion
f
0
defined
on
R
b
y
:
f
0
(
x
)
=
a
·
cos
x
+
b
·
sin
x
or
a
solution
f
0
of
(
E
)
.
b
Solv
e
the
differen
tial
equation
:
(
E
0
)
:
y
=
2
y
.
c
Demonstrate
that
f
is
a
solution
of
(
E
)
if
and
only
if
f
−
f
0
is
a
solution
of
(
E
0
)
.
d
Deduce
the
solutions
of
(
E
)
.
e
Determine
the
solution
k
of
(
E
)
v
erifying
k
ı
2
=
0
.
E.3839
1
Solv
e
the
differen
tial
equation
:
2
·
y
+
y
=
0
(
E
)
whose
unkno
wn
is
a
function
defined
and
deriv
able
on
R
.
2
Consider
the
differen
tial
equation
:
2
·
y
+
y
=
e
−
x
2
·
(
x
+
1)
(
E
)
a
Determine
t
w
o
real
m
and
p
suc
h
that
the
function
f
defined
on
R
b
y
:
f
(
x
)
=
e
−
x
2
·
m
·
x
2
+
p
·
x
soit
solution
de
(
E
)
b
Let
g
b
e
a
function
defined
and
deriv
able
on
R
.
Sho
w
that
g
is
a
solution
of
the
equation
(
E
)
if,
and
only
if,
g
−
f
is
a
solution
of
the
equation
(
E
)
.
Solv
e
the
equation
(
E
)
.
3
Study
the
v
ariations
of
the
function
h
defined
on
R
b
y
:
h
(
x
)
=
1
4
·
e
−
x
2
·
x
2
+
2
x
.
4
Determine
the
limits
in
−∞
and
in
+
∞
of
the
function
h
.
5
In
the
orthonormal
plane
O
;
−
→
i
;
−
→
j
,
note
C
the
rep-
resen
tativ
e
curv
e
of
h
and
Γ
that
of
the
function
:
x
↦−
→
e
−
x
2
a
Study
the
relativ
e
p
ositions
of
C
and
Γ
.
b
Plot
these
t
w
o
curv
es
on
the
same
graph.
E.4080
P
art
A
:
Organized
kno
wledge
transfer
W
e
will
use
the
follo
wing
result
:
the
solutions
of
the
differen-
tial
equation
y
=
a
·
y
où
a
∈
R
are
the
functions
g
defined
on
R
b
y
:
g
(
x
)
=
K
·
e
a
·
x
où
K
∈
R
.
The
aim
of
this
part
is
to
determine
the
solutions
of
the
dif-
feren
tial
equation
:
(
E
)
:
y
=
a
·
y
+
b
où
a
∈
R
∗
and
b
∈
R
.
1
Demonstrate
that
the
function
u
defined
on
R
b
y
:
u
(
x
)
=
−
b
a
is
a
solution
of
(
E
)
.
2
Let
f
b
e
a
function
defined
and
deriv
able
on
R
.
Pro
v
e
the
follo
wing
equiv
alence
:
f
is
a
solution
of
(
E
)
⇐
⇒
f
−
u
is
a
solution
of
the
differ-
en
tial
equation
y
=
a
·
y
.
3
Deduce
all
solutions
of
the
differen
tial
equation
(
E
)
.
P
art
B
A
cyclist
is
riding
on
a
v
ery
long,
straigh
t
do
wnhill
road.
W
e
denote
v
(
t
)
his
sp
eed
at
time
t
,
où
t
is
expressed
in
seconds
and
v
(
t
)
in
meters
p
er
second.
It
is
further
assumed
that
the
function
v
th
us
defined
is
deriv-
able
on
the
in
terv
al
0
;
+
∞
.
A
simple
mo
del
allo
ws
us
to
consider
that
the
function
v
is
a
solution
of
the
differen
tial
equation
:
10
·
v
(
t
)
+
v
(
t
)
=
30
Finally
,
it
is
assumed
that,
when
the
cyclist
sets
off,
his
initial
sp
eed
is
zero,
i.e.
v
(0)
=
0
.
1
Demonstrate
that
:
v
(
t
)
=
30
·
1
−
e
−
t
10
2
a
Determine
the
direction
of
v
ariation
of
the
function
v
on
the
in
terv
al
0
;
+
∞
.
b
Determine
the
limit
of
the
function
v
in
+
∞
.
3
In
this
situation,
the
cyclist’s
sp
eed
is
considered
stabi-
lized
when
his
acceleration
v
(
t
)
is
less
than
0.1
m
·
s
−
2
.
Determine,
to
the
nearest
second,
the
smallest
v
alue
of
t
at
whic
h
the
cyclist’s
sp
eed
is
stabilized.
4
The
distance
d
co
v
ered
b
y
this
cyclist
b
et
w
een
instan
ts
t
1
and
t
2
is
giv
en
b
y
:
d
=
t
2
t
1
v
(
t
)
d
t
Calculate
the
distance
tra
v
eled
b
y
this
cyclist
in
the
first
35
seconds.
h
ttps:/
/c
hingmath.fr
sacados/3675
chapExoCorrec/3839
sacados/3839
chapExoCorrec/4080
sacados/4080
Nouvelle-Caledonie Mars 2011 6 points
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