Math 1A, Spring 2008, Wilkening
Sample Final Exam 2
You are allowed one 8
.
5
×
11 sheet of notes with writing on both sides. This sheet
must be turned in with your exam.
Calculators are not allowed.
0. (1 point) write your name, section number, and GSI’s name on your exam.
1. (3 points) give precise deﬁnitions of the following statements or expressions:
(a)
f
(
x
) is neither even nor odd
(b)
R
f
(
x
)
dx
(c)
R
b
a
f
(
x
)
dx
Solution:
(a) There exist numbers
x
1
and
x
2
such that
f
(

x
1
)
6
=
f
(
x
1
) and
f
(

x
2
)
6
=

f
(
x
2
).
(b)
R
f
(
x
)
dx
is any antiderivative of
f
(
x
), i.e. a function
F
(
x
) such that
F
0
(
x
) =
f
(
x
).
(c) the deﬁnite integral is deﬁned as
Z
b
a
f
(
x
)
dx
=
lim
max Δ
x
i
→
0
n
X
i
=1
f
(
x
*
i
)Δ
x
i
,
where the limit is over all partitions
a
=
x
0
< x
1
<
···
< x
n

1
< x
n
=
b
of the
interval [
a, b
] into subintervals of length Δ
i
=
x
i

x
i

1
, and
x
*
i
is a sample point in
the
i
th interval [
x
i

1
, x
i
].
2. (4 points) Show that the tangent lines to the curves
y
=
x
3
and
x
2
+ 3
y
2
= 1 are
perpendicular where the curves intersect.
Solution:
The slope of the tangent line of the ﬁrst curve is
m
1
=
y
0
= 3
x
2
.
For the second, diﬀerentiate implicitly:
2
x
+ 6
yy
0
= 0
⇒
y
0
=

x
3
y
When the curves intersect, we have
y
=
x
3
, so
m
2
=
y
0
=

1
/
(3
x
2
). Since
m
2
=

1
/m
1
, these tangent lines are perpendicular.
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View Full Document3. (3 points) Evaluate
Z
1
0
tan

1
x
1 +
x
2
dx
.
Solution:
Let
u
= tan

1
x
. Then
du
=
dx
1 +
x
2
and the limits of integration become
x
= 0
→
u
= 0
,
x
= 1
→
u
=
π
4
.
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 Spring '08
 WILKENING
 Math, Calculus, Trigraph, iL, dx, ln x2, tan1 x dx

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