Math 101 Quiz Week 6 Submit your answers directly to the facilitator by Day 3; please
show all necessary work. For questions 1-2, convert to logarithmic equations. For
example, the logarithmic form of "23 = 8" is "log2 8 = 3".
(2 points) 16 3/2 = 64
log16 64 = 3/2
(2 points) ex = 5
loge 5 = x (also ln 5 = x)
For questions 3-4, write the logarithmic equation in exponential form. For example, the
exponential form of "log5 25 = 2" is "52 = 25".
(2 points) log 3 27 = 3
33 = 27
(2 points) log 125 25 = 2/3
1252/3 = 25
For questions 5 and 6, recall that, when interest is compounded continuously, the balance
in an account after t years is given by A = Pert, where P is the initial investment and r is
the interest rate.
(5 points) Maya has deposited $600 in an account that pays 5.64% interest, compounded
continuously. How long will it take for her money to double?
1200 = 600e0.0564t
2 = e0.0564t
ln (2) = ln (e0.0564t)
0.0564t= ln (2)
t = ln (2) / 0.0564
t = 12.29 years
(5 points) Suppose that $2000 is invested at a rate of 6% per year compounded
continuously.
What is the balance after 1 yr?
Balance = 2000e0.06(1) = 2000(1.061837) = $2,123.67
After 2 yrs?
Balance = 2000e0.06(2) = 2000(1.127497) = $2,254.99
(5 points) A computer is infected with the Sasser virus. Assume that it infects 20 other
computers within 5 minutes; and that these PCs and servers each infect 20 more machines
within another five minutes, etc. How long until 100 million computers are infected?
Initially, at time t = 0, the number of infected computers is 1
At t = 5, the number of infections is 1 + 20 = 21
At t = 10, the number of infections is 21 + 21(20) = 21(1+20) = 21(21) = 212
At t = 15, the number of infections is 212 + (20)212 = 212(1 + 20) = 212(21) = 213
At t = 20, the number of infections is 213 + (20)213 = 213(1 + 20) = 213(21) = 214
The total number of infected computers after t minutes is 21t/5
Then: 100 x 106 = 21t/5
log21 (100 x 106) = log21 21t/5
t/5 = log21(100 x 106)
t = 5 log21(100 x 106)
t = 5 log (100 x 106) / log (21)
t = 30.25 minutes.
Rounding up to a multiple of 5 minutes gives t = 35 minutes
(40 points) Evaluate each of the functions below at x = 1, 2, 4, 8, and 16. Plot the graph
of each function. Classify each as linear, quadratic, polynomial, exponential, or
logarithmic, and explain the reasons for your classifications. Compare how quickly each
function increases, based on the evaluations and graphs, and rank the functions from
fastest to slowest growing.
f(x) = x3 - 3x2 - 2x + 1
x
1
2
4
8
16
y = f(x)
-3
-7
9
305
3297
(x, y)
(1, -3)
(2, -7)
(4, 9)
(8, 305)
(16, 3297)
The graph is polynomial because it includes numeric powers of x.
The graph looks like this:
f(x) = ex
x
1
2
4
8
16
y = f(x)
2.72
7.39
54.60
2980.96
8,886,110.52
(x, y)
(1, 2.72)
(2, 7.39)
(4, 54.60)
(8, 2980.96)
(16, 8886110.52)
The graph is exponential because the variable x is an exponent
The graph looks like this:
f(x) = 3x – 2
x
1
2
4
8
16
y = f(x)
1
4
10
22
46
(x, y)
(1, 1)
(2, 4)
(4, 10)
(8, 22)
(16, 48)
This graph is linear because the highest order of the variable is one.
The graph looks like this:
f(x) = log x
x
1
2
4
8
16
y = f(x)
0
0.3010
0.6021
0.9031
1.2041
(x, y)
(1, 0)
(2, 0.3010)
(4, 0.6021)
(8, 0.9031)
(16, 1.2041)
This graph is logarithmic because it is plotting the logarithm of the variable x.
The graph looks like this:
f(x) = x2 - 5x + 6
x
1
2
4
8
16
y = f(x)
2
0
2
30
182
(x, y)
(1, 2)
(2, 0)
(4, 2)
(8, 30)
(16, 182)
This graph is quadratic because the highest order of exponent is 2.
The graph looks like this:
Fastest to slowest growing
f(x) = ex
f(x) = x3 - 3x2 - 2x + 1
f(x) = x2 - 5x + 6
f(x) = 3x – 2
f(x) = log x
exponential
polynomial
quadratic
linear
logarithmic