MTH 234 Groupwork 12 - 4.4, 4.5
(1) Differentiate the following:
(a) y = −6e2x
(b) y = ex
6
(c) y = e−2x
3
(d) y = −7x2 e−3x
(e) y = ln(5x + 3)
(f) y = ln(2 + x2 )
(g) y = y =
ln(3x)
x−3
xex
(h) y = y =
ln(x2 − 1)
1
2
(i) y =
−4e8x
5x − 2
e−8x + e8x
(j) y =
x
2
(k) y = (2z + e−z )5
√
(l) y = e
7x−1
(m) y = ln(2x4 + 3x)7/5
(n) y = ln(ln(x))
3
(2) The Richter scale provides a measure of the magnitude of an earthquake. In
fact, the largest Richter number M ever recorded for an earthquake was 8.9
from the 1933 earthquake in Japan. The following formula shows a relationship between the amount of energy released
and the
Richter number.
E
2
M = log
3
0.007
where E is measured in kilowatt-hours.
a) For the 1933 earthquake in Japan, what value of E gives a Richter number
M = 8.9?
b) If the average household uses 247 kWh per month, how many monthes
would the energy released by an earthquake of this magnitude power 10 million households?
c) What happens to dM/dE as E increases?
(3) A friend concludes that because y = ln(x) and y = ln(6x) have the same
dy
derivative, namely dx
= x1 , these two functions must be the same. Explain
why this is incorrect. In addition to your explaination, provide your friend
with another example of two functions that have the same derivative but are
not the same.
4
(4) The passage of the Social Security Amendments of 1965 resulted in the creation of the Medicare and Medicaid programs. Since then, the percentage of
persons 65 years and older with family income below the poverty level has
declined. The percent can be approximated by the following function:
P(t) = 30.60 - 5.79ln(t)
where t is the number of years since 1965.
(a) Find the percent of persons 65 years and older with family income below
the poverty level and the rate of change in the following years: 1970,
1990, 2010.
(b) How would you describe the poverty level over time? The rate of change
of over time?
(c) Comment on the long-term accuracy of this model (for example, consider
the projected poverty level in 2200 and what this says about the model).