MATH 145
SAMPLE QUIZ 1 — ANSWERS
1. Many answers are possible. Here’s one:
2. f0(x) = e−x
(1 + e−x)2. For every value of x,f0(x) is positive. So, f(x) is always
increasing.
3. f0(x) = 1 + ln xand f00(x) = 1
x. For every value of x > 0, f00(x) is positive. So, f(x)
is concave up on the interval (0,∞).
4. f0(x) = −1
5xe−x2/10 and f00(x) = 1
5e−x2/10(1
5x2−1). The second derivative is equal
to 0 at x=±√5. So, these are possible inflection points. To show that these are
inflection points, set up a sign chart for f00(x) and show that concavity changes from
up to down at −√5 and from down to up at √5.
5. Note that, since Nis a density, we will only consider non-negative values of N.
f0(N) = 3(100 −N2)
(100 + N2)2. This derivative is equal to 0 at N= 10. Set up a sign chart
for f0(N) to show that f(N) is increasing for densities between 0 and 10.