Math 171 — Calculus I Spring 2016
Name:
Weekly Write-Up on Sections 5.3-5.6
Problem 1.
From the suggested homework:
[5.6 #20] To model the effects of a carbon tax on CO2 emissions, policymakers study
the marginal cost of abatement B(x), defined as the cost of increasing CO2 reduction
from x to x + 1 tons (in units of ten thousand tons - see figure). Which quantity is
represented by the area under the curve over [0,3] in the figure?
Problem 2.
From an old exam:
Compute the following limit:
x+h
Z
1
lim
f (s) ds
h→0 2h
x−h
1
2
Problem 3.
For in-class discussion:
The function ln x is sometimes defined for x > 0 by the integration:
Z
ln(x) =
1
x
1
dt.
t
Assume that a > 0 and x > 0. Derive the identity ln(ax) = ln(a) + ln(x) by carrying
out the following steps:
0
R ax
1. Show that 1 1t dt = x1 [ HINT: Use the Chain rule.]
2. Explain why the result in part (a) implies that ln(ax) = ln(x) + c.
3. By choosing an appropriate x, show that the value of the constant c in part (c) is
ln(a).