Math 21B
UC Davis, Winter 2010
Dan Romik
Solutions to Practice Exam 1
1
Evaluate
20
∑
k
=1
(2
k

2).
Solution.
Using the properties of Σnotation sums and the formula for the sum of
the first
n
integers, we get:
20
X
k
=1
(2
k

2)
=
20
X
k
=1
2
k

20
X
k
=1
2 = 2
20
X
k
=1
k

20
X
k
=1
2
=
2
×
20(20 + 1)
2

2
×
20 = 420

40 = 380
.
2
If
f
(
x
) is continuous and
25
R
0
f
(
x
)
dx
= 8, what is
5
R
0
f
(
x
2
)
x dx
?
Solution.
We perform the substitution
u
=
x
2
.
Using the substitution rule for
definite integrals, this gives that
Z
5
0
f
(
x
2
)
x dx
=
1
2
Z
5
0
f
(
x
2
)(2
x
)
dx
=
1
2
Z
25
0
f
(
u
)
du
=
1
2
Z
25
0
f
(
x
)
dx
=
1
2
×
8 = 4
.
3
Find the area of the region bounded between the curves
y
= 3
x
and
y
=
x
2
.
Solution.
First, it is a good idea to sketch a diagram showing the graphs of the
two curves and the region bounded between them. Figure 1 shows the result. Now
we find the
x
ordinates of the points of intersection of the two curves. This leads to
the equation
x
2
= 3
x
, which has the two solutions
x
= 0 and
x
= 3. Finally, the area
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '08
 Vershynin
 Differential Equations, Addition, Equations, Integers, dx

Click to edit the document details