1093 MAT Test 3 Review
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Match the point in polar coordinates with either A, B, C, or D on the graph.
1) 3,
3
5
4
B
A
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5r
-1
-2
C
D
-3
-4
-5
A) A
2) - 3,
B) B
C) C
D) D
C) C
D) D
4
3
5
4
B
A
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5r
-1
C
-2
D
-3
-4
-5
A) A
B) B
1
3) 3, -
5
3
5
4
B
A
3
2
1
-5
-4
-3
-2
-1
1
2
3
5r
4
-1
C
-2
D
-3
-4
-5
A) A
B) B
C) C
D) D
The polar coordinates of a point are given. Find the rectangular coordinates of the point.
3
4) 5,
4
A) -
52
52
,2
2
5) (- 7, 120)
773
A) ,
2
2
B) -
B)
5252
,
2
2
C)
7
73
,2
2
5252
,
2
2
C) -
773
,
2
2
The rectangular coordinates of a point are given. Find polar coordinates for the point.
6) (3 , - 3 )
3
A) 3 2 ,
B) - 3 2 ,
C) - 3 2, 4
4
4
D)
52
52
,2
2
D) -
7
73
,2
2
D) 3 2 , -
4
7) (- 3 , -1)
A) 2, -
5
6
B) 2,
5
6
C) 2,
6
D) 2, -
6
The letters x and y represent rectangular coordinates. Write the equation using polar coordinates (r, ).
8) x = - 7
A) r cos = - 7
B) r sin = -7
C) r sin = 7
D) r cos = 7
9) y = 3
A) r = 3
B) r sin = 3
C) r cos = 3
D) sin cos = 3
B) r sin2 = 4 cos
C) r = 4 cos
D) r cos2 = 4 sin
10) x2 + y2 = 4x
A) r = 4 sin
2
11) xy = 1
A) r2 sin 2 = 2
B) 2r2 sin cos = 1
C) r sin 2 = 2
D) 2r sin cos = 1
12) x2 + 4y2 = 4
A) r2 (4 cos2 + sin2 ) = 4
B) r2(cos2 + 4 sin2 ) = 4
D) 4 cos2 + sin2 = 4r
C) cos2 + 4 sin2 = 4r
Transform the polar equation to an equation in rectangular coordinates. Then identify and graph the equation.
13) r = 3
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
123456
r
A)
B)
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
(x -
6
5
4
3
2
1
123456
r
-6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
32
9
) + y2 = ; circle,
2
4
radius
x2 + (y -
3
3
, center at ( , 0) in
2
2
radius
rectangular coordinates
123456
32 9
) = ; circle,
2
4
3
3
, center at (0, ) in
2
2
rectangular coordinates
3
r
C)
D)
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
6
5
4
3
2
1
r
123456
-6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
x = 3 ; vertical line 3 units to the right
of the pole
14) = -
123456
r
x2 + y2 = 9 ; circle, radius 3,
center at pole
3
5
-5
r
5
-5
A)
B)
5
5
-5
5
r
-5
5
r
-5
y=-
-5
;
3
y=-
3 x; line through the pole making
an angle of with the polar axis
3
horizontal line units below the pole
3
4
C)
D)
5
-5
5
5
r
-5
-5
5
r
-5
y=
3 x; line through the pole making
an angle of with the polar axis
3
x-
2
2
; circle, radius ,
+ y2 =
3
9
3
center at
, 0 in rectangular coordinates
3
15) r sec = -6
6
5
4
3
2
1
123456r
-6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
A)
B)
6
5
4
3
2
1
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
123456r
-6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
123456r
x = -6;
vertical line 6 units to the left of the pole
y = - 6;
horizontal line 6 units below the pole
5
C)
D)
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
6
5
4
3
2
1
123456r
-6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
(x + 3)2 + y2 = 9; circle, radius 3
center at (- 3, 0) in rectangular coordinates
123456r
x2 + (y + 3)2 = 9; circle, radius 3,
center at (0, -3) in rectangular coordinates
Match the graph to one of the polar equations.
16)
5
4
3
2
1
-5 -4 -3 -2 -1
-1
1
2
3
4
5
r
-2
-3
-4
-5
A) r = -2
B) r = - 4 cos
C) r = - 4 sin
D) r sin = - 2
Write the complex number in polar form. Express the argument in degrees, rounded to the nearest tenth, if necessary.
17) 3 - 3 i
A) 9 (cos 300 + i sin 300 )
B) 9 (cos 315 + i sin 315 )
C) 3 2 (cos 300 + i sin 300)
D) 3 2 (cos 315 + i sin 315)
18) - 15 - 20i
A) 25(cos 233.1 + i sin 233.1)
C) 25(cos 126.9 + i sin 126.9)
B) 25(cos 53.1 + i sin 53.1)
D) 25(cos 306.9 + i sin 306.9)
Write the complex number in rectangular form.
19) 8 cos + i sin
6
6
A)
1
3
i
+
4
4
20) 4(cos 300 + i sin 300)
A) - 2 3 - 2i
31
+i
4
4
C) 4 + 4 3 i
D) 4 3 + 4i
B) 2 - 2 3i
C) 2 3 - 2i
D) -2 + 2 3i
B)
6
Solve the problem. Leave your answer in polar form.
21) z = 10(cos 45 + i sin 45)
w = 5(cos 15 + i sin 15)
Find zw.
A) 5(cos 60 + i sin 60)
C) 50(cos 30 + i sin 30)
22) z = 8 cos
w = 3 cos
B) 5(cos 30 + i sin 30)
D) 50(cos 60 + i sin 60)
+ i sin
6
6
+ i sin
2
2
Find zw.
A) 12 cos
2
2
+ i sin
3
3
B) 24 cos
2
2
+ i sin
3
3
C) 12 cos
+ i sin
3
3
D) 24 cos
+ i sin
3
3
23) z = 5(cos 200 + i sin 200)
w = 4(cos 50 + i sin 50)
z
Find .
w
A)
5
(cos 40 + i sin 40)
4
B)
4
(cos 150 + i sin 150)
5
C)
5
(cos 150 + i sin 150)
4
D)
4
(cos 40 + i sin 40)
5
24) z =
w=
Find
A)
3 cos
6 cos
7
7
+ i sin
4
4
9
9
+ i sin
4
4
z
.
w
2
cos + i sin
2
2
2
C) 3 2 cos
B) 3 2 cos
3
3
+ i sin
2
2
D)
+ i sin
2
2
2
3
3
cos
+ i sin
2
2
2
Do indicated operation and write the expression in the standard form a + bi.
25) 2(cos 15 + i sin 15) 3
A) 4 + 4i
26)
2 cos
B) 4 2 + 4 2 i
C) 3 + 3i
D) 3 2 + 3 2i
B) 4i
C) - 4i
D) 4
B) 64i
C) 64 - 64 3 i
D) -64
3
3 4
+ i sin
4
4
A) - 4
27) (- 3 + i)6
A) - 64 3 + 64i
7
Determine whether the given function is exponential of the form f(x) = a x or not. If it is exponential, identify the
value of the base a.
28)
x
H(x)
9
-1
2
0
1
2
9
1
2
4
81
3
8
729
9
2
B) Exponential; a =
C) Exponential; a = 2
D) Not exponential
A) Exponential; a =
2
9
Solve the problem.
1
29) If 3 -x = , what does 9 x equal?
4
A) 16
B) 4
C) - 16
D)
1
16
Use transformations graph to the function. Determine the domain, range, and horizontal asymptote of the function.
30) f(x) = - 2 x+3 + 4
y
10
5
-10
-5
5
10
x
-5
-10
8
A)
B)
y
y
10
5
-10
10
5
-5
5
10
x
-10
-5
5
-5
x
-5
-10
10
-10
domain of f: (-, ); range of f: (-, - 4);
horizontal asymptote: y = - 4
domain of f: (-, ); range of f: (-, - 4);
horizontal asymptote: y = - 4
C)
D)
y
y
10
5
-10
10
5
-5
5
10
x
-10
-5
5
-5
x
-5
-10
10
-10
domain of f: (-, ); range of f: (- 4, );
horizontal asymptote: y = 4
domain of f: (-, ); range of f: (-, 4);
horizontal asymptote: y = 4
Solve the problem.
31) The function f(x) = 700(0.5)x/70 models the amount in pounds of a particular radioactive material stored in
a concrete vault, where x is the number of years since the material was put into the vault. Find the amount
of radioactive material in the vault after 50 years. Round to the nearest whole number.
A) 427 pounds
B) 490 pounds
C) 250 pounds
D) 265 pounds
B) - 3
C) 128
D) 3
B) {3}
C) { 35, -
Solve the equation.
5 - 3x
1
32) 4
=
256
A)
1
64
33) 2 x2 - 3= 64
A) {3, - 3}
9
35}
D) {6}
Change the exponential expression to an equivalent logarithm expression.
-3
1
34) 3 =
27
A) log
1
=3
- 3 27
1
B) log - 3 =
3
27
C) log
1
= -3
3 27
D) log
Change the logarithmic expression to an equivalent expression involving an exponent.
35) log
125 = - 3
1/5
1 -3
13
A) 125 1/5 = 3
B)
C)
= 125
= 125
5
5
Find the exact value of the logarithmic expression.
36) log
4
4
1
A)
B) 1
2
C) 4
1/27
3 = -3
D) (-3)1/5 = 125
D)
1
4
Use a calculator to evaluate the expression. Round your answer to three decimal places
log 9 + log 4
37)
ln 3 - ln 8
A) - 1.587
B) - 3.654
C) - 0.359
D) 0.490
B) (- 5 , )
C) (-, 5)(5, )
D) (0, )
B) (0, )
C) (- , 1 )
D) (1, )
B) (- 10, 10)
C) [- 10, 10]
D) (-100 , 100 )
Find the domain of the function.
1
38) f(x) = ln
x-5
A) (5 , )
39) f(x) = ln x
A) (- , 0)
40) f(x) = log9 (100 - x2)
A) (-, - 10) ( 10, )
Write as the sum and/or difference of simple logs. Express powers as factors.
20 r
41) log
11 s
A) log
C) log
11
11
(20 r) - log
20
11
s
B) log
1
log m log s
11
11
2
Express as a single logarithm.
42) 3 log x + 5 log (x - 6)
6
6
A) 15 log x(x - 6)
6
D) log
11
11
20 +
1
log r - log s
11
11
2
s - log
11
C) log x3 (x - 6)5
6
B) log x(x - 6)
6
10
20 -
1
log r
11
2
D) log x(x - 6)15
6
Solve the problem.
43) Which of the following is equivalent to log 1 -
1
?
x2
A) log(x - 1) + log( x + 1) - 2 log x
C) log(x - 1) - log(x + 1) - 2 log x
B) log 1 - 2 log 1 - 2 log x
D) log 1 - log 1 + 2 log x
Use the Change-of-Base Formula and a calculator to evaluate the logarithm. Round your answer to three decimal
places.
44) log
173
8.7
A) 0.420
B) 19.885
C) 2.382
D) 2.238
Solve the equation.
45) log (4 + x) - log (x - 5) = log 2
A)
B) - 14
3
2
C) 14
D)
46) log (x + 1) + log (x - 5) = 3
3
3
A) {9 }
B) {8 , - 4 }
C) {- 4 }
D) {8}
47) 2 (3x - 1) = 14
A) 0.23
C) 0.94
D) 2.67
B) 1.60
Use a graphing calculator to solve the equation. Round your answer to two decimal places.
48) log4 (x + 2) - log5(x - 1) = 1
A) {- 0.69}
B) {2.05}
C) {2.00}
D) {1.75}
49) ex - ln x = 3
A) {2.17}
B) {1.27}
C) {1.14}
D) {0.57}
Solve the problem.
50) During 1991, 200,000 people visited an Amusement Park. During 1997, the number had grown to 834,000. If
the number of visitors to the park obeys the law of uninhibited growth, find the exponential growth
function that models this data.
A) f(t) = 634,000e0.248t
B) f(t) = 634,000e0.238t
C) f(t) = 200,000e0.238t
D) f(t) = 200,000e0.248t
51) Susan purchased a painting in the year 2000 for $7000. Assuming an exponential rate of inflation of 3.2%
per year, how much will the painting be worth 5 years later?
A) $9176.03
B) $ 8214.58
C) $4891.02
D) $7812.90
52) How long will it take a sample of radioactive substance to decay to half of its original amount, if it decays
according to the function A(t) = 450e- 0.157t, where t is the time in years? Round to the nearest hundredth
year.
A) 38.91 yr
B) 70.65 yr
C) 43.33 yr
11
D) 4.41 yr
53) Newton's Law of Cooling states that if a body with temperature T1 is placed in surroundings with
temperature T0 different from T1 , then the body will either cool or warm to temperature T(t) after time t, in
minutes, where T(t) = T0 + ( T1 - T0 )e- kt.
A cup of coffee with temperature 103F is placed in a freezer with temperature 0F. After 8 minutes, the
temperature of the coffee is 55.2F. What will its temperature be 19 minutes after it is placed in the freezer?
Round your answer to the nearest degree.
A) 23F
B) 20F
C) 22F
D) 18F
54) The logistic growth model P(t) =
1890
represents the population of a bacterium in a culture tube
1 + 62e- 0.329t
after t hours. What was the initial amount of bacteria in the population?
A) 31
B) 35
C) 29
55) The logistic growth model P(t) =
D) 30
260
represents the population of a species introduced into a
1 + 31.5e- 0.175t
new territory after t years. When will the population be 70?
A) 14.01 years
B) 12.22 years
C) 0.79 years
12
D) 2.58 years
Answer Key
Testname: 1093 TEST 3 REVIEW F11
1) A
2) A
3) A
4) B
5) B
6) D
7) A
8) A
9) B
10) C
11) A
12) B
13) D
14) B
15) C
16) D
17) D
18) A
19) D
20) B
21) D
22) B
23) C
24) D
25) B
26) A
27) D
28) B
29) A
30) D
31) A
32) D
33) A
34) C
35) B
36) A
37) A
38) A
39) B
40) B
41) B
42) C
43) A
44) C
45) C
46) D
47) B
48) C
49) C
50) C
13
Answer Key
Testname: 1093 TEST 3 REVIEW F11
51) B
52) D
53) A
54) D
55) A
14