Download MTH 112 Practice Test 3 summer 2012.tst

MTH 112 Practice Problems for Test 3 - Summer 2012
Identify the intervals where the function is increasing, decreasing, or constant.
1)
5
y
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 x
-1
-2
-3
-4
-5
Find and simplify the difference quotient f(x + h) - f(x)
, h≠ 0 for the given function.
h
2) f(x) = 6x + 7
3) f(x) = x2 + 9x - 8
Find the inverse of the one-to-one function.
4) f(x) = 8x + 4
5) f(x) = 2x - 5
7
3
6) f(x) = x + 8
7) f(x) = (x + 2)3
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
8) log (3x)
3
9) log
10) ln 5
125
x
e5
9
11) logn x8
1
12) log 13 -2
2
13) log
7
5
y
14) logb (yz 4 )
15) log
16) log
17) log
5
7 · 11
13
3
x + 2
x5
2
x
4
18)
3
log2 4 x y
4
Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose
coefficient is 1. Where possible, evaluate logarithmic expressions.
1
19) (log9 x + log9 y) - 4 log9 (x + 8)
5
20) 9ln (x - 2) - 5 ln x
21) loga 51 + loga 3
22) ln 24 - ln 3
23)
1
2
loga x + loga y
9
3
24) ln (x2 - 4) - ln (x + 2)
Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places
25) log 40.1
15
26) log
27) log
π
17
0.5
20
2
Solve the equation by expressing each side as a power of the same base and then equating exponents.
28) 3 (1 + 2x) = 243
29) 3 (6 - 3x) = 1
27
30) 5 (x - 1)/4 = 5
31) 16x + 9 = 64x - 5
Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the
solution.
32) 10x = 3.06
33) 9ex = 25
34) e2x = 4
35) 7 x = 6 x + 7
Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic
expressions. Give the exact answer.
36) log (x - 2) = 3
2
37) log (x + 1) + log (x - 5) = 4
2
2
38) log 9 + log x = 1
4
4
39) log (x + 4) = log (4x - 5)
40) log x2 = log (8x + 9)
2
2
41) log
21
(x + 84) = 3 - log
21
x
Write the standard form of the equation of the circle with the given center and radius.
42) (-7, -4); 12
43) (-8, 0); 2
44) (-3, -6); 5
Find the center and the radius of the circle.
45) (x + 7)2 + (y + 8)2 = 49
3
Graph the equation and state its domain and range. Use interval notation
46) x2 + y 2 = 100
10
y
5
-10
-5
5
10 x
5
10 x
-5
-10
Graph the equation.
47) (x - 5)2 + (y - 3)2 = 9
10
y
5
-10
-5
-5
-10
48) x2 + y 2 - 4x - 6y + 9 = 0
10
y
5
-10
-5
5
10 x
-5
-10
Solve.
49) The value of a particular investment follows a pattern of exponential growth. In the year 2000, you invested
money in a money market account. The value of your investment t years after 2000 is given by the exponential
growth model A = 3300e0.053t. How much did you initially invest in the account?
4
Solve the problem.
50) A sample of 550 grams of radioactive substance decays according to the function A(t) = 550e-0.028t, where t is
the time in years. How much of the substance will be left in the sample after 40 years? Round to the nearest
whole gram.
Solve.
51) The population of a small country increases according to the function B = 1,900,000e0.04t, where t is measured
in years. How many people will the country have after 8 years?
Solve the problem.
52) The number of acres in a landfill is given by the function B = 2200e-0.02t, where t is measured in years. How
many acres will the landfill have after 6 years? (Round to the nearest acre.)
53) The formula A = 106e0.032t models the population of a particular city, in thousands, t years after 1998. When
will the population of the city reach 120 thousand?
Solve the system by the substitution method.
54) x + y = 6
y = x2 - 12x + 36
55) x2 + y2 = 61
x + y = -11
Solve the system by the addition method.
56) x2 + y2 = 9
x2 - y2 = 9
57) x2 + y2 = 25
25x2 + 16y2 = 400
58) x2 + y2 - 8x - 8y + 31 = 0
x2 - y2 - 8x + 8y - 1 = 0
Graph the solution set of the system of inequalities or indicate that the system has no solution.
59) 3x - y ≤ -3
x + 4y ≥ -4
y
10
8
6
4
2
-10 -8 -6 -4 -2
-2
2
4
6
8 10 x
-4
-6
-8
-10
5
60) -1 ≤ y < 4
y
10
8
6
4
2
-10 -8 -6 -4 -2-2
-4
2 4
6 8 10
x
4
6
x
10
x
-6
-8
-10
61) y > -1
x ≥ 3
y
6
4
2
-6
-4
-2
2
-2
-4
-6
62) y > x2
3x + 6y ≤ 18
y
10
5
-10
-5
5
-5
-10
Write the augmented matrix for the system of equations.
63) -2x + 5y + 9z = 52
3x + 9y + 8z = 97
8x + 6y + 8z = 88
6
64)
6x + 6z = 90
4y + 7z = 88
4x + 3y + 2z = 68
Solve the system of equations using matrices. Use Gaussian elimination with back -substitution.
65) x + y + z = -1
x - y + 5z = 5
5x + y + z = 19
66) -4x - y - 3z = -22
-4x + 6z = 16
9y + z = 22
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
67) 5x + 2y + z = -11
2x - 3y - z = 17
7x - y = 12
68)
x + y + z = 9
2x - 3y + 4z = 7
x - 4y + 3z = -2
7
Answer Key
Testname: MTH 112 PRACTICE TEST 3 SUMMER 2012
1) increasing (-2, -1) ∪ (3, ∞)
decreasing, (1, 3)
constant, (-1, 1)
Note: Do not use brackets [ ]on these.
2) f(x+h) = 6(x + h) + 7= 6x + 6h + 7
f(x+h) - f(x) =6x+6h+7-(6x+7)
= 6x+6h+7-6x-7
= 6h
f(x+h)-f(x) 6h
= = 6
h
h
30) 3
31) 33
32) 0.49
33) 1.02
34) 0.69
35) 81.36
36) {10}
37) {7}
4
38) { }
9
3) 2x + h + 9
x - 4
4) f-1 (x) = 8
39) 3
40) {9, -1}
41) {63}
42) (x + 7)2 + (y + 4)2 = 144
7x + 5
5) f-1 (x) = 2
43) (x + 8)2 + y 2 = 4
44) (x + 3)2 + (y + 6)2 = 5
6) f-1 (x) = x3 - 8
3
7) f-1 (x) = x - 2
8) 1 + log x
3
9) 3 - log x
5
10) 5 - ln 9
11) 8logn x
45) (-7, -8), r = 7
46)
10
5
12) -2 log 13
2
1
13) log y
5
7
-10
-5
15) log 7 + log 11 - log 13
5
5
5
16) log (x + 2) - 5 log x
3
3
1
17) log x - 2
2
2
10 x
-10
Domain = (-10, 10); Range = (-10, 10)
47)
10
1
1
3
18) log2 x + log2 y - 4
2
4
19) log9 5
-5
14) logb y + 4 logb z
5
y
y
5
xy
(x + 8)4
-10
(x - 2)9
20) ln x5
-5
5
10 x
-5
21) loga 153
-10
22) ln(8)
Domain = (2, 8), Range = (0, 6)
23) loga (x2/3 y1/9 )
24) ln (x - 2)
25) 1.3631
26) 2.4750
27) -4.3219
28) {2}
29) {3}
8
Answer Key
Testname: MTH 112 PRACTICE TEST 3 SUMMER 2012
48)
61)
y
10
y
6
5
4
2
-10
-5
10 x
5
-6
-4
-2
2
4
6
x
10
x
-2
-5
-4
-10
-6
49) $3300.00
50) 179 grams
51) 2,616,543
52) 1951
53) 2002
54) {(5, 1), (6, 0)}
55) {(-5, -6), (-6, -5)}
56) {(3, 0), (-3, 0)}
57) {(0, 5), (0, -5)}
58) {(5, 4), (3, 4)}
59)
62)
y
10
5
-10
-5
5
-5
y
10
-10
8
6
63)
4
-2 5 9 52
3 9 8 97
8 6 8 88
2
-10 -8 -6 -4 -2
-2
2
4
6
8 10 x
64)
-4
6 0 6 90
0 4 7 88
4 3 2 68
65) {(5, -5, -1)}
66) {(2, 2, 4)}
67) ∅
68) Infinitely many solutions.
-6
-8
-10
60)
12
y
10
8
6
4
2
-10 -8 -6 -4 -2-2
2 4
6 8 10
x
-4
-6
-8
-10
-12
9