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3
2
1
Grade 12 - Comp.
/ Annales intégrales
6 exercises (100% corrected)
a
Integrals
(3 exercices)
Integrals and the intermediate value theorem
(1 exercice)
Integrals and convexity
(1 exercice)
Integrals, intermediate value theorem and convexity
(1 exercice)
x
x
y
y
-0,2
0
0,2
0,4
0,6
0,8
1
1,2
-0,2
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
1,8
2
2,2
2,4
2,6
C
g
C
f
1.
Inte
gr
als
E.7733
A
compan
y
man
ufactures
wire-
less
sp
eak
ers.
The
pro
duction
cost
of
a
sp
eak
er
is
300
euros.
Let
x
b
e
the
selling
price
of
a
sp
eak
er
in
h
undreds
of
euros.
Mark
et
researc
h
allo
ws
us
to
mo
del
the
situation
:
for
an
y
real
n
um
b
er
x
in
the
in
terv
al
3
;
10
,
if
the
selling
price
of
a
sp
eak
er
is
x
h
undred
euros,
then
the
n
um
b
er
of
buy
ers
is
mo
deled
b
y
:
f
(
x
)
=
e
−
0.25
·
x
+5
Th
us,
f
(
x
)
is
an
appro
ximation
of
the
n
um
b
er
of
buy
ers
for
a
selling
price
of
x
h
undred
euros.
F
or
example,
if
the
selling
price
of
a
sp
eak
er
is
set
at
400
euros,
the
n
um
b
er
of
buy
ers
is
appro
ximated
b
y
f
(4)
.
1
Giv
e
an
appro
ximate
v
alue
for
the
n
um
b
er
of
buy
ers
for
a
selling
price
of
400
dollars.
The
difference
b
et
w
een
the
amoun
t
obtained
from
the
sale
of
the
sp
eak
ers
and
their
pro
duction
cost
is
called
the
gross
margin.
2
What
is
the
gross
margin
of
this
compan
y
for
a
selling
price
of
400
euros
p
er
sp
eak
er?
Let
g
(
x
)
b
e
the
gross
margin,
in
h
undreds
of
euros,
ac
hiev
ed
b
y
the
compan
y
for
a
selling
price
of
x
h
undreds
of
euros
p
er
sp
eak
er.
3
Sho
w
that
for
an
y
real
n
um
b
er
x
b
elonging
to
the
in
terv
al
3
;
10
:
g
(
x
)
=
x
−
3
·
e
−
0.25
·
x
+5
4
A
computer
algebra
system
giv
es
the
follo
wing
results
:
factorize(deriv
e[(x-3)*exp(-0.25*x+5)])
−
x
−
7
4
·
e
−
1
4
x
+5
a
Using
the
result
from
the
computer
algebra
soft
w
are,
study
the
v
ariations
of
the
function
g
on
the
in
terv
al
3
;
10
.
b
A
t
what
unit
selling
price
will
the
compan
y
ac
hiev
e
the
maxim
um
gross
margin?
Giv
e
the
v
alue
of
this
gross
margin
rounded
to
the
nearest
euro.
5
Let
G
b
e
the
function
suc
h
that
G
(
x
)
=
−
4
·
x
−
4
·
e
−
0.25
·
x
+5
for
all
real
n
um
b
ers
x
of
3
;
10
.
a
Sho
w
that
G
is
a
primitiv
e
of
the
function
g
.
b
Let
I
=
10
3
g
(
x
)
d
x
.
Determine
the
exact
v
alue
of
I
.
E.7429
A
compan
y
w
an
ts
to
use
a
deco-
rativ
e
pattern
for
its
comm
unications.
T
o
create
this
pattern,
its
shap
e
is
mo
deled
using
t
w
o
func-
tions
f
and
g
defined
for
an
y
real
n
um
b
er
x
of
0
;
1
b
y
:
f
(
x
)
=
1
−
x
·
e
3
x
;
g
(
x
)
=
x
2
−
2
·
x
+
1
Their
represen
tativ
e
curv
es
will
b
e
denoted
C
f
and
C
g
.
P
art
A
A
computer
algebra
system
giv
es
the
follo
wing
results
:
Deriv
e
(1-x)*exp(3x)
:-3x*exp(3*x)+2exp(3*x)
factorize
-3*xexp(3*x)+2*exp(3*x)
:exp(3x)*(-3x+2)
factorize(deriv
e(exp(3x)(-3x+2)))
:3*exp(3*x)(1-3x)
Reading
:
the
deriv
ativ
e
of
the
function
f
is
giv
en
b
y
:
f
(
x
)
=
−
3
x
·
e
3
x
+2
·
e
3
x
,
whic
h,
after
factorization,
giv
es
:
f
(
x
)
=
−
3
x
+2
·
e
3
x
1
Study
the
sign
of
the
deriv
ativ
e
function
f
on
0
;
1
,
then
giv
e
the
table
of
v
ariations
of
f
on
0
;
1
,
sp
ecify-
ing
the
useful
v
alues.
2
The
curv
e
C
f
has
an
inflection
p
oin
t.
Determine
its
co-
ordinates.
P
art
B
W
e
w
an
t
to
calculate
the
area
of
the
shaded
part
of
the
graph.
1
Chec
k
that
p
oin
ts
A
and
B
,
with
co
ordinates
(
1
;
0
)
and
(
0
;
1
)
resp
ectiv
ely
,
are
p
oin
ts
common
to
curv
es
C
f
and
C
g
.
2
W
e
assume
that
:
for
all
x
in
0
;
1
:
f
(
x
)
−
g
(
x
)
=
1
−
x
·
e
3
x
−
1
+
x
a
Justify
that
for
all
x
in
0
;
1
:
e
3
x
−
1
0
b
Deduce
that
for
all
x
in
0
;
1
:
e
3
x
−
1+
x
0
c
Study
the
sign
of
f
(
x
)
−
g
(
x
)
for
all
x
in
0
;
1
3
a
Calculate
1
0
g
(
dx
)
d
x
.
b
W
e
assume
that
:
1
0
f
(
x
)
d
x
=
e
3
−
4
9
Calculate
the
area
S
,
in
area
units,
of
the
shaded
part.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/7733
sacados/7733
chapExoCorrec/7429
sacados/7429
x
x
y
y
-0,2
0
0,2
0,4
0,6
0,8
1
1,2
-0,2
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
1,8
2
2,2
2,4
2,6
C
g
C
f
a
0
1
1
2
3
4
5
6
C
2
3
4
5
6
7
8
I
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
J
O
C
Round
the
result
to
the
nearest
ten
th.
E.6989
Consider
the
function
f
defined
on
the
in
terv
al
0
;
1
b
y
:
f
(
x
)
=
4+e
−
5
x
The
curv
e
C
represen
tativ
e
of
the
function
f
has
b
een
plotted
in
a
planar
co
ordinate
system.
The
hatc
hed
area
D
on
the
figure
is
the
area
b
ounded
b
y
the
curv
e
C
,
b
y
the
x-axis,
the
y-axis
and
the
straigh
t
line
of
equation
x
=
1
.
W
e
w
an
t
to
divide
the
hatc
hed
domain
in
to
t
w
o
domains
of
equal
area
b
y
a
straigh
t
line
of
equation
y
=
a
,
parallel
to
the
x-axis,
according
to
the
example
giv
en
b
elo
w.
1
Justify
that
the
v
alue
a
=
3
is
not
suitable.
2
Determine
to
the
nearest
0.1
a
suitable
v
alue
of
a
.
2.
Inte
gr
als
and
the
interme
diate
value
the
or
em
E.6974
The
two
p
arts
of
this
exer
cise
ar
e
indep
endent.
P
art
A
In
this
part,
answ
ers
will
b
e
giv
en
without
justification,
with
the
precision
allo
w
ed
b
y
the
graph
b
elo
w
:
This
graph
sho
ws,
in
a
co
ordinate
system
with
origin
O
,
the
represen
tativ
e
curv
e
C
of
a
function
f
defined
and
differen-
tiable
on
the
in
terv
al
0
;
7
.
1
Enclose
eac
h
of
the
solutions
to
the
equation
f
(
x
)
=
10
on
the
in
terv
al
0
;
7
b
et
w
een
t
w
o
consecutiv
e
in
tegers.
2
Giv
e
the
maxim
um
v
alue
of
the
function
f
on
the
in
terv
al
0
;
7
and
sp
ecify
the
v
alue
at
whic
h
it
is
reac
hed.
3
The
v
alue
of
the
in
tegral
3
1
f
(
x
)
d
x
b
elongs
to
only
one
of
the
follo
wing
in
terv
als.
Whic
h
one?
a
9
;
17
b
18
;
26
c
27
;
35
P
art
B
The
curv
e
giv
en
in
part
A.
is
the
represen
tation
of
the
func-
tion
f
defined
and
differen
tiable
on
the
in
terv
al
0
;
with
expression
:
f
(
x
)
=
2
·
x
·
e
−
x
+3
Recall
that
f
denotes
the
deriv
ativ
e
of
the
function
f
.
1
Sho
w
that
for
an
y
real
n
um
b
er
x
in
the
in
terv
al
0
;
7
:
f
(
x
)
=
−
2
x
+
2
·
e
−
x
+3
2
a
Study
the
sign
of
f
(
x
)
on
the
in
terv
al
0
;
7
and
then
deduce
the
table
of
v
ariations
of
the
function
on
this
same
in
terv
al.
b
Calculate
the
maxim
um
v
alue
of
the
function
f
on
the
in
terv
al
0
;
7
3
a
Justify
that
the
equation
f
(
x
)
=
10
has
t
w
o
solutions
on
the
in
terv
al
0
;
7
,
whic
h
w
e
will
denote
b
y
¸
and
˛
with
¸
<
˛
.
b
W
e
assume
that
¸
≈
0.36
to
10
−
2
.
Giv
e
the
v
alue
of
˛
,
rounded
to
10
−
2
.
4
Consider
the
function
F
defined
on
the
in
terv
al
0
;
7
b
y
:
F
(
x
)
=
−
2
x
−
2
·
e
−
x
+3
a
Justify
that
F
is
a
primitiv
e
of
f
on
the
in
terv
al
0
;
7
.
b
Calculate
the
exact
v
alue
of
the
area,
in
area
units,
of
the
plane
domain
b
ounded
b
y
the
lines
with
equations
x
=
1
,
x
=
3
,
the
x-axis,
and
the
curv
e
C
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6989
sacados/6989
Antilles Juin 2017
a
0
1
1
2
3
4
5
6
C
chapExoCorrec/6974
sacados/6974
2
3
4
5
6
7
8
I
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
J
O
C
2
3
4
5
6
7
I
2
3
4
5
6
7
8
9
J
O
A
B
5
The
function
f
studied
mo
dels
a
compan
y’s
profit,
in
thousands
of
euros,
from
the
sale
of
x
h
undreds
of
items
(
x
b
etwe
en
0
and
7
)
.
a
Calculate
the
a
v
erage
profit,
to
the
nearest
euro,
when
the
compan
y
sells
b
et
w
een
100
and
300
items.
b
The
compan
y
w
an
ts
its
profit
to
b
e
greater
than
10
000
euros.
Determine
the
n
um
b
er
of
items
the
compan
y
will
need
to
sell
to
ac
hiev
e
its
goal.
3.
Inte
gr
als
and
c
onvexity
E.6978
Let
f
b
e
a
function
defined
on
the
in
terv
al
0.7
;
6
;
w
e
assume
that
f
is
differen
tiable.
P
art
A
:
graphical
analysis
The
function
f
is
represen
ted
in
the
graph
b
elo
w.
1
The
tangen
t
at
the
p
oin
t
with
abscissa
3
to
the
curv
e
represen
ting
f
passes
through
the
p
oin
ts
A
(
3
;
4
)
and
B
(
4
;
0
)
.
Determine
f
(3)
.
2
Based
on
the
graph
ab
o
v
e,
giv
e
the
table
of
signs
for
f
on
the
in
terv
al
0.7
;
6
.
P
art
B
:
theoretical
study
W
e
assume
that
the
function
f
is
defined
b
y
:
f
(
x
)
=
x
2
−
2
·
x
+
1
·
e
−
2
x
+6
1
Sho
w
that
:
f
(
x
)
=
−
2
x
2
+
6
x
−
4
·
e
−
2
x
+6
where
f
denotes
the
deriv
ativ
e
of
the
function
f
.
2
Study
the
direction
of
v
ariation
of
the
function
f
on
the
in
terv
al
0.7
;
6
and
dra
w
up
a
table
of
v
ariations
of
the
function
f
on
the
in
terv
al
0.7
;
6
.
Y
ou
are
not
required
to
calculate
the
ordinates.
3
Using
computer
algebra
soft
w
are,
w
e
obtain
the
results
b
elo
w,
whic
h
can
b
e
used
without
b
eing
pro
v
en.
L1
f’(x)
:=(-2x^2+6x-4)*e^(-2x+6)
↦→
f
(
x
)
=
(
−
2
x
2
+
6
x
−
4)e
−
2
x
+6
L2
g(x)
:=Dérivée[f’(x)]
↦→
g
(
x
)
=
−
16
x
e
−
2
x
+6
+
4
x
2
e
−
2
x
+6
+
14e
−
2
x
+6
L3
Factoriser[g(x)]
↦→
2e
−
2
x
+6
2
x
2
−
8
x
+
7
L4
Résoudre[g(x)=0]
↦→
x
=
−
√
2
+4
2
;
x
=
√
2
+4
2
L5
F(x)
:=Primitive[f(x)]
↦→
F
(
x
)
=
1
4
−
2
x
2
+
2
x
−
1
e
−
2
x
+6
a
Determine
the
largest
in
terv
al
o
v
er
whic
h
the
function
f
is
conca
v
e
b
Do
es
the
graph
of
the
function
f
ha
v
e
an
y
inflection
p
oin
ts?
If
so,
giv
e
their
x-co
ordinates.
c
W
e
set
I
=
5
3
f
(
x
)
d
x
.
Calculate
the
exact
v
alue
of
I
,
then
the
v
alue
rounded
to
10
−
1
.
4.
Inte
gr
als,
interme
diate
value
the
or
em
and
c
onvexity
E.6988
The
curv
e
C
b
elo
w
is
the
repre-
sen
tativ
e
curv
e
in
the
plane
equipp
ed
with
an
orthonormal
co
ordinate
system
of
a
function
f
defined
and
t
wice
differen-
tiable
on
the
in
terv
al
−
4
;
10
.
W
e
denote
f
as
the
deriv
ativ
e
of
f
,
and
f
as
its
second
deriv
ativ
e.
The
tangen
t
to
the
curv
e
C
at
p
oin
t
A
with
abscissa
−
2
is
parallel
to
the
abscissa
axis.
The
shaded
area
S
in
the
figure
is
the
area
b
et
w
een
the
curv
e
C
,
the
x-axis,
the
line
with
equation
x
=
2
,
and
the
line
with
equation
x
=
4
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6978
sacados/6978
2
3
4
5
6
7
I
2
3
4
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7
8
9
J
O
A
B
chapExoCorrec/6988
sacados/6988
Antilles Juin 2017
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C
P
art
A
1
Determine
the
v
alue
of
f
(
−
2)
and
justify
y
our
answ
er.
2
Based
on
the
graph,
what
do
es
the
sign
of
f
(4)
app
ear
to
b
e?
3
Using
the
graph,
determine
a
range
of
t
w
o
consecutiv
e
in
tegers
for
the
area
of
the
shaded
region
S
in
the
figure.
P
art
B
The
previous
function
f
is
defined
on
the
in
terv
al
−
4
;
10
b
y
:
f
(
x
)
=
x
+
4
·
e
−
0.5
x
1
a
Sho
w
that
:
f
(
x
)
=
−
0.5
·
x
−
1
·
e
−
0.5
x
b
Study
the
v
ariations
of
the
function
f
on
the
in
terv
al
−
4
;
10
.
c
Sho
w
that
on
the
in
terv
al
1
;
6
,
the
equation
f
(
x
)
=
1.5
has
a
unique
solution.
Let
us
denote
this
unique
solution
b
y
¸
.
d
Donner
of
¸
,
rounded
to
10
−
2
of
¸
2
On
admit
that
the
second
deriv
ativ
e
of
f
is
defined
b
y
:
f
(
x
)
=
0.25
·
x
·
e
−
0.5
x
a
Étudier
the
con
v
exit
y
of
the
function
f
on
the
in
terv
al
−
4
;
10
.
b
Deduce
that
the
curv
e
C
admits
a
single
p
oin
t
of
in-
flection
I
whose
co
ordinates
will
b
e
calculated.
3
a
On
consider
the
function
F
defined
b
y
:
F
(
x
)
=
−
2
x
−
12
·
e
−
0.5
x
Ho
w
can
w
e
sho
w
that
F
is
a
primitiv
e
of
f
on
the
in
terv
al
−
4
;
10
?
This
c
hec
k
is
not
required.
b
Calculate
:
S
=
4
2
f
(
x
)
d
x
.
Giv
e
the
exact
v
alue
and
then
the
v
alue
rounded
to
the
h
undredth.
h
ttps:/
/c
hingmath.fr
-5
-4
-3
-2
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C
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