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).