Midterm 1 Practice Exam Solutions - Math 32, University Of California, Berkeley, Fall 2012
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4MATH 32 FALL 2012
MIDTERM 1 - PRACTICE EXAM SOLUTIONS
(1) Find all values of
satisfying the inequality
1
5
1
1
1
Solution: Case 1:
0. This happens when
0. Then
=
.
1
5
1
5
1
5
1
1
1
Case 2:
0. This happens when
0. Then
=
.
1
5
1
5
1
5
1
1
Putting these cases together, we see that the solutions are
0
0
.
5
5
Note that
= 0 is not included in either case - no division by 0!
(2) Consider the polynomials
2
( ) = 2
+ 1
3
( ) =
+ 1
(a) Write the product ( )( ) in expanded form (i.e. as a sum of terms, each of which is
a constant times a power of )
Solution:
2
3
( )( ) = (2
+ 1)(
+ 1)
5
3
2
3
= (2
2
+ 2
) + (
+ 1)
5
3
2
= 2
+ 2
+ 1
(b) Write the composition (
)( ) in expanded form.
Solution:
3
2
(
)( ) = 2(
+ 1)
+ 1
6
4
3
4
2
3
= 2(
+
+
+
+ 1) + 1
6
4
3
2
= 2(
2
+ 2
+
2 + 1) + 1
6
4
3
2
= 2
4
+ 4
+ 2
4 + 3
(3) Let
be the line containing the points (1 1) and (5 13).
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