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Transcript of Test1 Year 2010
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8/20/2019 Test1 Year 2010
1/12
Code: L725
Department of Mathematics
LJniversity
of Toronto
MAT 135Y
Term-Test
ffL
Thursday, November 11, 2010
Time allowed: 90 minutes
PleasePRINT
in
INK or
BALL-POINT PEN:
(Please
PRINT full name
and UNDERLINE
surname):
NAME OF STUDENT:
STUDENT NO.:
SIGNATURE OF STUDENT
(in
INK or BALL-POINT
PEN):
TUTORIAL
CODE
(..e.,
M4A, R5D, etc.):
TUTORIAL TIME
(".S.,
T4,
R5,
F3, etc.):
NAME
OF
YOUR TA:
NOTE:
1. Before
you
start, check
hat this test
has 12
pages.
There are NO blank
pages.
2.
This test has two
parts:
PART
A
[50
marks]:
10
multiple
choice
questions
PART B
[50
marks]:
7 written
questions
Answers to both PART A and PART B are
to be
given
in this booklet.
No
computer
cards
will
be used.
3.
No
aids allowed.
No calculators
DO NOT TEAR
OUT
ANY
PAGES.
FOR MARKERS
ONLY
QUESTION
MARK
PAFUT
A
150
B1
l7
B2
lB
B3
l8
B4
/ c
B5
l. ?
/ t
B6
l7
B7
l8
TOTAL
ln0
Page1
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12
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Code: L725
PART A
[50
marks]
Please
ead carefullv:
PART A consists
of
10
multiple-choice
questions,
each
of
which
has exactlv
one correct arswer.
Indicate
your
arrswer o
each
question
by completely
filling in
the appropriate
circle with
a
dark
pencil.
MARKING
SCHEME:
5
ma,rks for
a correct answer,
0
for
no answer or a
wrong answer. You
ale not
required to justify your arurr/ers n PARjT A. Note that for PART A, only your final answers (as
indicated by the circles
you
darken)
countl
your
computations
and
answers indicated
elsewhere will
NOT count.
DO NOT TEAR
OUT
ANY PAGES.
1. Find
the value
of l mrz
-
3r
-
4
n -+4 f r2 -2 r -8
@
O
@
o
@
2. Findthevalueof
m
ffi
1
t
o
6
3
4
_t
3
undefined
@
@
e
o
@
t
I
5
a
D
_1
2
_3
2
undefined
Page2
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8/20/2019 Test1 Year 2010
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3.
Find
the value
@o
@
undefined
@*
o*
oi
Code: L725
of lim
(sin
3r)
(cos
2r)
(csc
6r)
;;o
(sin
4r)(cos
5r)(csc *)
curve
A
-
2r3
-
4r
*
5
at
the
point
(1,
3) will
intersect
.
The
line
tangent
to the
the
y-axis
at
the
point
@
@
@
o
@
(0,0)
(0,
3)
(0,
3)
(0,
1)
(0,
2)
Page
3 of12
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8/20/2019 Test1 Year 2010
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5.
Letf(*)- [crz-2r
1f
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8/20/2019 Test1 Year 2010
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Code: L725
7
If
f
(r)
-
(ln
)r"
",
then
'
(r)
-
-e
tre
,
e'
1
e
?
e
B. It
f
(r)
-
cos(2rr),
find
f(at)(#),
i.e., ind
the
value
of the
61"t
derivative
of
f
atr-h.
O
-26ontr
@
-26on6o
@
_267n6r
O
261n6t
@
26onGL
@
@
@
e
@
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Code:
L725
9.
Let
(*)
:
tre*2
nd
9
:
f
(f(")).
Then
g'(l
-
(2" '+
1)"" '
2(2e2
1)""'
3(2e2 I)e"2
2(2e2
1)
"L+e2
3(2e2 1)sL+"2
@
@
@
@
e
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Code:
L725
10.
Find
the
value
of
lrg
cos2
-
cosz(2r)
.
,/,
1
r
t/Z
3
1
,
undefined
e
o
@
Page
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[7]
Code:
L725
PARI B
[50
marks]
Please ead
carefullv:
Present
your
complete solutions to
the
following
questions
in
the spaces
provided,
in
a neat and
logical fashion,
showing all
your
computations
a.nd
ustifications.
Any answer in PART
B without
proper
ustification
may receive ittle
or
no
credit.
Use
he back
of each
page
or
rough work only. If
you
must continue
your
formal
solution on the back
of a
page,
you
should indicate
clearly, in LARGE
Ietters, SOLUTIONCONTINUEDON THE BACK OF PAGE
In this case,you may get
credit for
what
you
write on the
back of that
page,
but
you
may also
be
penalized
for mistakes
on
the back of that
page.
MARKS
FOR EACH
QUESTION
AR^E NDICATED
BY
II.
DO NOT TEAR OUT ANY
PAGES.
1. Let
f
(a)
:
x2+3. Find
/'(c)
from
first
principles
i.e.
by using
onlv the definition
of the
derivative)
Page
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Codez 7725
2.
Use
arry
suitable method to frnd
ftfor
eachof the following.
There is no
need o simplify
your
fi.nalanswers
or this
question.
arctan
\ a )
Y :
2+es " -
t/,1
( b ) y
- ( r '
+8 ) t o ( " -5 ) t .
t/rl
3.
For
this
question,
simplify
your
final
answers
as much
as
possible.
(a)
Find
f '( i l
i t
f
(r)
-
ln(sin'
).
Rr-,
2J3
-
r t r
- ' -
t 4 l
L
' J
(b)
Find
'(4)
it
f
(r)
:
2fr.
t+rn-
,
LL
:
t/,1
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[5]
Code: L725
4. Find
\il:n
J
r'z
-
2r
+
t
-
r) .
5.
Find
the horizontal
asymptotes
(if
any) of each
of the following
functions.
Remember
to
justify
yor[
aflswers.
3lu
{al t lt l
:
-----r----:-.
2x -8
t4l
t3l
(u)
(")
-
2e-"
I.
,\,
l.IVt-)
_
|
d
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Code:
L725
6.
Find the
tangent
ine
to the curve 2A*sin(2g)
-
r-
2
at the
point
(2,0).
[7]
(\,*.
Z=
(x
-L)
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7
' l { -
I
-
tsl
Code:
1225
,
7.
(NS g:
This is not zrn
easy
question.
Very
little or
no credit
will be
given
unless
your
solution is completelv
correct.)
A circle s
centered t the
poi:rt
(6,0)
andhas adius
5. Find
the slope
fthe
line
passing
hrough
the
point
1-f,
O1 na a.ngento the upper
half
of the
circle.
o
Page
12 of72