110 Math Exam 3: Winter 2010
4 Mar 2010 - 10 Mar 2010
1. Find the domain of the f g where f ( x) =
(b) { x| x
3, x
(c) { x| x
0, x
2/3}
2. Find the inverse of f ( x) =
(c) f ( x) =
1
4
x
4
x+
x+2
4
3, x
(f) { x| x
0}
0, x
(e) { x| x
0}
(b) f 1 ( x) =
and g( x) = 2 .
x
(d) { x| x
(a) { x| x
(a) f 1 ( x) =
x
x+3
2
3}
3}
0, x
2/3}
4
x+2 .
2
(d) f 1 ( x) = 2 x
2
(e) f 1 ( x) =
3. Find the domain of h( x) = log3
2
x
4
(f) f ( x) = x
1
x
x1
.
(a) (0, )
(d) [0, 1)
(b) (1, )
(e) (, 0] (1, )
(c) (, 0)
(f) (, 0) (1, )
1
4. Suppose that ln 7 = a and ln 2 = b. Express ln( 14 ) in terms of a and b.
(a) a + b
(d) (a + b)
(b) 1 a b
(e) 1 b
(c) 1 a + b
(f) 1 + a b
5. Solve 43 x = 32 for x.
(a) x =
(b) x =
(c) x =
(d) x =
5
6
2
3
1
2
6
5
(e) x = 2
(f) x =
1
3
2
6. Which of the following is equivalent to log5 9:
(a) log5 36 log5 3
(d) log5 1 + log5 18
(b) log5 45 1
(e) 3 log5 3
(c) log5 1 + log5 3
(f) 1 + log5 3
7. Use properties of logarithms to the nd the exact value of log6 9 + log6 4.
(a) 0
(d) 3
(b) 1
(e) 4
(c) 2
(f) none of the above
8. Express 21 log3
3
x + log3 (9 x2 ) log3 9 as a single logarithm.
(a) log3 (81 x9 )
(b) 21 log3 ( 3 x + 9 x2 9)
(c) log3 x
(d) log3
x 9 +9 x 2
9
(e) log3 (9 x)
9
(f) none of the above
9. Solve log x + log( x + 15) = 2 for x. Recall that log x = log10 x.
(a) x =
(d) x = 5
1500
99
(b) x = 20 and x = 5
(e) x = 5
(c) x = 20
(f) no solution exists
10. Solve log2 ( x + 1) log4 x = 1 for x.
(a) x = 1
(d) x = 1
(b) x = 0
(e) x = 2
(c) x =
(f) no solution exists
1
2
11. Solve 2 log5 x = 3 log5 4 for x.
(a) x = 16
(d) x = 8
(b) x = 16
(e) x = 8
(c) x = 16
(f) x = 8
2
12. Select the function that best ts the following graph.
(a) y = ln( x) 1
(d) y = ln( x) + 1
(b) y = ln( x)
(e) y = ln( x 1)
(c) y = ln( x 1)
(f) y = ln( x) + 2
2
3
13. ( Solve 2)2 x = 2 x for x
(a) x = 1, x =
(d) x =
2
3
(b) x = 1
(c) x = 2, x =
2
3
(e) x = 2
(f) x = 2, x =
1
3
2
3
14. The equation 32 x (4)3 x 2 = 5 has two solutions. Find the sum of the two
solutions.
(a) 0
(d) 3
(b) 1
(e) 4
(c) 2
(f) 5
15. How many years will it take for an initial investment of $10,000 to grow to
$30,000? Assume a rate of interest of 10% compounded continuously.
(a) ln 3
(d) 10 ln 3
(b) 100 ln 3
(e) 0.1 + ln 5
(c) ln 5
(f)
3
ln 3
10
16. What is the principal needed to get 1000 dollars after 2 years at 6% interest
compounded monthly?
(a) 1000 (1.005)24
(d) 1000 (0.995)12
(b) 1000 (1.005)12
(e) 1000 (0.995)24
(c) 1000 (1.005)24
(f) 1000 (1.005)12
17. A certain radioactive material decays according to the formula A(t) = A0 e0.0244t
where t is in years and A0 is the initial amount of the material. What is the half
life of this material?
(a)
ln 2
0.0244
(d)
ln 2
0.0244
A0 ln 2
0.0488
(b) ln 2
(e)
(c) 2(.0244)
(f) none of the above
18. Find the vertex, focus, and directrix of the parabola (y 2)2 = 8( x + 1).
(a) vertex (2,-1); focus (2,1); directrix x = 0
(b) vertex (2,-1); focus (1,2); directrix x = 3
(c) vertex (2,-1); focus (4,-1); directrix x = 0
(d) vertex (-1,2); focus (1,2); directrix x = 0
(e) vertex (-1,2); focus (1,2); directrix x = 3
(f) vertex (-1,2); focus (-1,4); directrix x = 4
19. Write the equation of a parabola with the focus at (-3,4) and directrix at the line
y = 2.
(a) ( x 3)2 = 4(y + 3)
(d) ( x 3)2 = 4(y + 3)
(b) ( x + 3)2 = 4(y 3)
(e) ( x + 3)2 = 4(y + 3)
(c) ( x + 3)2 = 4(y 3)
(f) ( x 3)2 = 4(y 3)
20. Write the equation of a parabola that has vertex (0,0) and focus (4,0).
(a) x2 = 16y
(d) x2 = 16y
(b) y2 = 8 x
(e) x2 = 8y
(c) y2 = 8 x
(f) y2 = 16 x
4
1. c
2. a
3. f
4. d
5. a
6. b
7. c
8. c
9. e
10. d
11. f
12. a
13. a
14. b
15. d
16. c
17. d
18. e
19. b
20. f
5