The slope of the secant line is . Find c in [0,1] such
that, f ′(c) = e − 2, or f ′(c) − (e − 2) = 0. Since f ′(x) = e x – 2x, c can be
calculated by solving 0 = ex − 2x − (e − 2). The answer is 0.351.
(B)
Use disks; then ΔV = πR^2 H = π(ln y)^2 Δy. Note that the limits of the
definite integral are 1 and 2. Evaluate the integral
Alternatively, use shells*; then ΔV = 2πRHT = 2πx (2 − ex) Δx. Here, the
upper limit of integration is the value of x for which ex = 2, namely, ln 2.
Now evaluate
(C) Note that the rate is people per minute, so the first interval width
from midnight to 6 A.M. is 360 minutes. The total number of people is
estimated as the sum of the areas of six trapezoids:
