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Exam Final MAT1720 B
Calcul différentiel et intégral I (University of Ottawa)
Studocu is not sponsored or endorsed by any college or university
Exam Final MAT1720 B
Calcul différentiel et intégral I (University of Ottawa)
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MAT1320B
Final Exam
8 December 2017
Calculus I
Elizabeth Maltais
LAST NAME:
FIRST NAME:
STUDENT NUMBER:
•
This is a three hour exam.
•
No calculators are permitted. No notes, books, papers of any kind, or any other aids. Scrap
paper will be provided on request.
•
Print your name and student number on this page.
•
Verify that your copy has all 15 pages (including this one).
•
Writeyoursolutionsinthespaceprovided(usethebacksofthepagesifnecessary). Youmust
show all of your work.
•
Cellular phones, unauthorized electronic devices or course notes are not allowed during this
exam. Phonesanddevicesmustbeturnedo
ff
andputawayinyourbag. Donotkeepthemin
yourpossession,suchasinyourpockets. Ifcaughtwithsuchadeviceordocument,thefollow-
ing may occur: you will be asked to leave immediately the exam, academic fraud allegations
will be filed which may result in you obtaining a 0 (zero) for the exam. By signing below, you
acknowledge that you have ensured that you are complying with the above statement.
•
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SIGNATURE:
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Do not write below this line.
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MAT1320B Calculus I
Final Exam
8 December 2017
page 2 of 15
1.
Find the derivative directly from the definition for the function
f
(
x
) =
1
x
+2
. You must use
[3]
the definition, not some other method.
2.
Find the derivative of each function.
[6]
a)
f
(
x
)=sin(ln(
x
2
))
b)
g
(
t
)=
t
ln(
t
)
fyy
=
qkjmo
fKth)g#
=fmsoFt¥yk+Fx
=lfjmo
k¥+2
-
¥2
=
him
a-
=
:*
.tn#t*ynfyEIgITItt*=ghjmoyt2-x-hx+ht2)KtZ)(h1f'K1=cos(lnk4)(xtz)Kx
)
ln(gHD=ln(
that
)
⇒
lnlgttt
)=lnHlntH
⇒
gttjg
'H=
ttbnttltlntttt
⇒
gtttgtttfflnttttbntttf
]
⇒
gytktfnty.ph#y
Trouvez la derivee de f(x) utilisant la definition de la derivee:
(N'employez pas les regles de derivation).
Trouvez la derivee de chacune des fonctions suivantes:
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MAT1320B Calculus I
Final Exam
8 December 2017
page 3 of 15
3.
Estimate the value of
f
(0
.
1) using a linearization of
f
(
x
)=tan(
x
)+1. Choose the point
a
of
[3]
the linearization appropriately.
4.
Show that
d
d
x
Z
x
2
3
1
/
2
1+
t
d
t
+
Z
2
tan
-
1
x
tan(
t
)d
t
!
is zero.
[3]
4×1
=
Ha
)
+
f
'
(a)
(
×
-
a)
Let
a=O
.
fkktank
)
+1
Hot
tank
+1
=
0+1=1
f
'
k
)
=
seek
)
f
'H=sec44=
¥
,p=
fz
=L
:
.
LK
)
=
It
1
(
x
-
o
)
=
Hx
%
f
(
0
.
1)
~~L(O
.
1)
=
It
0.1
By
FTC
1
,
a¥(B¥tat
+
t.IN?.nafttdt
)
=Y÷×zk4
'
+
-
da
,
fstantxtanttldt
=tgGy÷
-
(
tankan
'
'
KD
.
Clan
'
'
KH
)
=*×
.
-
Handy
.tt
)
)
=X
=
X
-
I
Fxz
ltxz
=
0
Estimez une valeur de
utilisant une linearisation de
Faites un choix approprie' du point a de linearisation.
Montrez que
= 0
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MAT1320B Calculus I
Final Exam
8 December 2017
page 4 of 15
5.
Consider the curve defined by
y
+
xe
y
=
x
2
.
[6]
a) Give
d
y
d
x
in terms of
x
and
y
.
b) Findallvaluesof
x
suchthatthepoint(
x,
0)isonthecurvedefinedbytheaboveequation.
For each of these give the slope of the tangent line to the curve at that point.
ytxey
=×2
⇒
da¥t1eY+xeY.dd¥
=2x
⇒
day
+
xeY.ge#=2x-eY
⇒
aduxtdtxey
)
=2x
-
EY
⇒
day
=
2×-1
ltxet
y+×eY=×2
at
(
×
,gy=o
⇒
0t×e°=×2
⇒
×=×2
=)
O=×2
-
×
⇒
o=x(
x-D
¢
x
X=O
11=1
at
(
0,01
,
slope
=
data
=
210164=-1
ate
,o
)
,
s1°pe=ad¥=2yfpe÷=2÷=z
Considerez la courbe definie par:
Trouvez
en fonction de x et y.
Trouvez la valeur de x telle que le point
est sur la courbe d'equation donnee ci-dessus.
Pour chaque valeur, trouvez la pente de la tangente a' la courbe en ce point.
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MAT1320B Calculus I
Final Exam
8 December 2017
page 5 of 15
6.
Find each of the limits.
[6]
a) lim
x
!
3
+
1
-
e
x
-
3
(
x
-
3)
2
b) lim
x
!
0
(
x
2
+1)
1
/x
→
1-
e3
-3
→
1-
e°→0
→
(
3
.
}p→o
(
indeterminate
form
g)
eighteen
d-
exsj
'
×→3tFx3py
=
,fng+Ty¥
,
→
-
e⇒→
-1
→
→
2Gt
.
3)
→
o+
to
=
-
oo
txhjmoebnlkzt
't
"
'
)
txhjnoetxlnatxy
=
@
(
xhfnotxlnatxzy
or
this
is
indeterminate
Of
=ekm→o¥'
"
)
yandex.mn#tf)=elxmso*j
←
correction
7¥
=e°
=L
=e
Evaluez les limites:
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