Faculty of Arts and Science
University of Toronto
MAT 137Y1Y
Calculus!
April/May Examinations; April 17, 2000
Time Alloted: 3 hours
Instructors: G. Baumgartner, O. Calin, T. Haines, V. Jurdjevic, S. Lillywhite, R. Martinez
No aids allowed.
(9%)
1.
Given the sketch of the function
f
below, indicate on a chart whether
f
,
f
, and
f
are
positive, negative, or zero at the points
x
a
,
x
b
,
x
c
, and
x
d
.
d
a
b
c
(8%)
2.
Applying the
ε
,
δ
definition of limit, prove
lim
x
3
x
2
4
5
(7%)
3.
Let
f x
2sin
x
x
π
2
A
sin
x
B
π
2
x
π
2
cos
x
x
π
2
Find the values of
A
and
B
such that
f
is continuous, or show that the values do not exist.
1
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4.
Let
f x
sin
x
x
x
0
1
x
0
(7%)
(a)
Use the definition of derivative to compute
f
0 .
(6%)
(b)
Show there exists
c
0
π
such that
f
c
1
π
.
(7%)
5.
Let
f
be defined by the following graph.
a
2a
f
If
F x
x
0
f t dt
, draw the graph for
F
. (
f x
0 for all
x
a
2
a
.)
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 Calculus, Derivative, Taylor Series, Faculty of Arts and Science University of Toronto MAT, MAT 137Y1Y Calculus

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