College Algebra for Calculus
Final Exam Key
Instructions
1. Do NOT write your answers on these sheets. Nothing written on the test papers will be graded.
2. Please begin each section of questions on a new sheet of paper.
3. Please do not write answers side by side.
4. Please do not staple your test papers together.
5. Limited credit will be given for incomplete or incorrect justification.
6. All answers must be exact. Numeric approximations will be marked incorrect.
Questions
1. Solving (4 each)
(a) x3 − 19x + 30 = 0.
1
2
3
5
x−2
30
30
15
10
6
1 − 19 + 30
f (1)
=
=
12.
f (2)
=
8 − 38 + 30
=
0.
x2
x3
x3
+2x
+0
−2x2
2x2
2x2
−15
−19x +30
−19x
−4x
−15x
−15x
+30
+30
0
x2 + 2x − 15 = (x + 5)(x − 3).
The solutions are x = 2, 3, −5.
1
Final Exam
2
(b) 25x−3 + 7 = 39.
25x−3 + 7
2
5x−3
=
39.
+7−7
=
39 − 7.
5x−3
=
32.
5x−3
=
25 .
2
2
5x − 3
=
5.
5x − 3 + 3
=
5 + 3.
5x
5x
5
=
x
(c)
57
1+e0.2t
8.
8
=
.
5
8
.
=
5
= 1.
57
= 1.
1 + e0.2t
57 = 1 + e0.2t .
−1 + 57
=
−1 + 1 + e0.2t .
56
=
e0.2t .
ln(56) = 0.2t.
ln(56)
= t.
0.2
5 ln(56) = t.
Final Exam
(d)
√
3
5x + 1 −
√
x − 2 = 3.
√
√
5x + 1 − x − 2
√
5x + 1
√
( 5x + 1)2
=
3.
=
3+
5x + 1
=
4x − 6
=
2x − 3
=
2
=
x − 2.
√
(3 + x − 2)2 .
√
9 + 6 x − 2 + (x − 2).
√
6 x − 2.
√
3 x − 2.
√
(3 x − 2)2 .
4x − 12x + 9
=
9(x − 2).
(2x − 3)
2
2
p
Both solutions work.
=
√
4x − 21x + 27
=
0.
(4x − 9)(x − 3)
=
0.
x
=
3, 9/4.
p
√
5(3) + 1 − 3 − 2
√
√
16 − 1
=
4−1
=
5(9/4) + 1 − 9/4 − 2
p
p
49/4 − 1/4
=
7/2 − 1/2
=
p
=
3.
=
3.
Final Exam
4
2. Systems
(a) (6) Find all
x −2y
2x −3y
x
+y
solutions to the following using matrix notation
+z=5.
+z=8.
−z=6.


−2
1 5
−3
1 8  ∼
1 −1 6


1 −2
1
5
 0
1 −1 −2  ∼
0
3 −2
1


1 −2
1
5
 0
1 −1 −2  ∼
0
0
1
7


1 0 −1
1
 0 1 −1 −2  ∼
0 0
1
7


1 0 0 8
 0 1 0 5 
0 0 1 7
1
 2
1
R2 ← −2R1 + R2
R3 ← −1R1 + R3
R3 ← −3R2 + R3
R1 ← 2R2 + R1
R1 ← 1R3 + R1
R2 ← 1R3 + R2
The solution is (8, 5, 7).
(b) (8) Find the min and max values of P = 2y + 5x over the region inside x ≥ 0, y ≥ 0, 3x + 2y ≥ 6,
3x + 2y ≤ 18, y − 3x ≤ 0.
10
8
6
4
2
1
2
3
4
3x + 2y
=
6.
y
=
3x.
3x + 2(3x)
=
6.
9x =
6.
x =
2/3.
y
=
2.
3x + 2y
=
18.
y
=
3x.
5
6
Final Exam
5
3x + 2(3x)
=
18.
9x =
18.
x =
2.
y
=
6.
3x + 2(0)
=
6.
x =
2.
3x + 2(0)
=
x =
P (2/3, 2)
P (2, 6)
P (2, 0)
P (6, 0)
18.
6.
=
2(2) + 5(2/3)
=
22/3.
=
2(6) + 5(2)
=
22.
=
2(0) + 5(2)
=
10.
=
2(0) + 5(6)
=
30.
The max is at (6, 0) at 30 and the min is at (2/3, 2) at 22/3.
(c) (6) Find all solutions to 8y = x2 − 16 and x2 + 4y 2 = 16.
x2 + 4y 2
=
16.
2
=
8y + 16.
4y + 8y + 16
=
16.
x
2
2
4y + 8y
=
0.
4y(y + 2)
=
0.
y
=
0, −2.
x − 16
2
=
8(0).
2
=
16.
x
= ±4.
x
x2 − 16
=
8(−2).
2
=
0.
x
=
0.
x
The three intersections are (4, 0), (−4, 0), and (0, −2).
Final Exam
6
3. Arithmetic

1
 4
A=
 2
1
9
1
0
6
0
9
1
5


2
5
 0
2 
, B = 
 0
6 
2
6
0
0
1
0
0
1
0
0


0
1
0 
, C =  2
0 
1
−1

−2
1
−3
1 
1 −1
(a) (4) Calculate A − 3B.

1
 4

 2
1

1
 4

 2
1
(b) (4) Calculate BA.

2 0
 0 0
BA = 
 0 1
2 0
0
9
1
5
9
1
0
6
0
9
1
5
5
2
6
6

1
0
 4
0 

0  2
1
−1
9
1
0
6
0
9
1
5
(2)(1) + (0)(4) + (0)(2) + (0)(1)
(0)(1) + (0)(4) + (1)(2) + (0)(1)
(0)(1) + (1)(4) + (0)(2) + (0)(1)
(2)(1) + (0)(4) + (0)(2) + (−1)(1)
=

=
0
1
0
0
9
1
0
6
2
 2

 4
1
18
0
0
1
1
9
12 −5


2 0
5
 0 0
2 
 − 3
 0 1
6 
2 0
6
 
−6
0
  0
0
+
  0 −3
−6
0

−5
9
 4
1

 2 −3
−5
6

0
0 
 =
0 
−1

0 0
−3 0 
 =
0 0 
0 3

0 5
6 2 
.
1 6 
5 9
0
1
0
0

5
2 

6 
6
(2)(9) + (0)(1) + (0)(0) + (0)(6)
(0)(9) + (0)(1) + (1)(0) + (0)(6)
(0)(9) + (1)(1) + (0)(0) + (0)(6)
(2)(9) + (0)(1) + (0)(0) + (−1)(6)
(2)(0) + (0)(9) + (0)(1) + (0)(5)
(0)(0) + (0)(9) + (1)(1) + (0)(5)
(0)(0) + (1)(9) + (0)(1) + (0)(5)
(2)(0) + (0)(9) + (0)(1) + (−1)(5)

10
6 

2 
4
(c) (3) Calculate det(B)

2
 0

 0
2

2
 0

 0
0

2
 0

 0
0
0
0
1
0
0
1
0
0
0
0
1
0
0
1
0
0
0
1
0
0
0
0
1
0

0
0 
 ∼
0 
−1

0
0 
 ∼
0 
−1

0
0 

0 
−1
det(B) = −(2)(1)(1)(−1) = 2.
(2)(5) + (0)(2) + (0)(6) + (0)(6)
(0)(5) + (0)(2) + (1)(6) + (0)(6)
(0)(5) + (1)(2) + (0)(6) + (0)(6)
(2)(5) + (0)(2) + (0)(6) + (−1)(6)
Final Exam
(d) (4) Calculate
7
√ 2
1+i 3
2
.
√ !2
1+i 3
=
2
√
√ 2
1 + 2i 3 + i2 3
=
4
√
−2 + 2i 3
=
4 √
−1 + i 3
.
2
Final Exam
8
4. Graphing
(a) (4) Graph −2f (x − 3) + 1. Note f (x) is in Figure 1.
Translate right 3, scale vertically by 2, reflect over the x-axis, shift up one
3
2
1
2.5
3.0
3.5
4.0
-1
(b) (8) Graph the conic 9x2 + 25y 2 − 54x + 200y + 256 = 0. Label center and vertices.
9x2 + 25y 2 − 54x + 200y + 256
2
=
0.
2
9(x − 6x) + 25(y + 8y) + 256 = 0.
9(x2 − 6x + 9 − 9) + 25(y 2 + 8y + 16 − 16) + 256 = 0.
9([x − 3]2 − 9) + 25([y + 4]2 − 16) + 256
2
2
9(x − 3) − 81 + 25(y + 4) − 400 + 256
2
2
-2
-1
-2
-3
-4
-5
-6
-7
2
=
0.
=
0.
9(x − 3) + 25(y + 4) − 225
=
0.
9(x − 3)2 + 25(y + 4)2
25(y + 4)2
9(x − 3)2
+
225
225
(x − 3)2
(y + 4)2
+
25
9
=
225.
=
1.
=
1.
4
6
8
Final Exam
9
(c) (8) Graph the conic 9x2 − 4y 2 + 36x + 8y = 4. Label center and vertices.
9x2 − 4y 2 + 36x + 8y
=
4.
9(x2 + 4x) − 4(y 2 − 2y)
=
4.
=
4.
=
4.
9[x + 2] − 36 − 4[y − 1] + 4
=
4.
9[x + 2]2 − 4[y − 1]2
9(x + 2)2
4(y − 1)2
−
36
36
(y − 1)2
(x + 2)2
−
4
9
=
36.
=
1.
=
1.
2
2
9(x + 4x + 4 − 4) − 4(y − 2y + 1 − 1)
2
2
9([x + 2] − 4) − 4([y − 1] − 1)
2
2
10
5
-15
-10
5
-5
10
-5
3
+4x
(d) (8) Graph including showing all asymptotes x2x−5x+6
.
Finding roots and vertical asymptotes (in that order).
x3 + 4x =
x(x2 + 4)
0.
x =
0.
x2 − 5x + 6
=
0.
(x − 3)(x − 2)
=
0.
x =
x
y
0.
=
−1 0
− 0
1
2 2.5
+ VA
−
2, 3.
3 4
VA +
Find the asymptotic action.
x2 − 5x + 6
x
x3
x3
+5
+0x2
−5x2
5x2
5x2
+4x
+6x
−2x
−25x
23x
+0
+0
+30
−30
Final Exam
10
50
-10
5
-5
10
-50
-100
-150
-200
1.0
0.5
-1.0
0.5
-0.5
-0.5
-1.0
Figure 1: Graph of function f (x)
1.0