Math 132: Calculus for Physical and Life Sciences 2 Extra Credit Problem Set 10 Due Wednesday, April 30, no later than 5:00pm

Selected Answers

1. Center of mass at

\overline{x} = \frac{M_{y}}{M} = \frac{\frac{9π^{2}}{2} − 1}{6π + 1} ≐ 2.187.

3. (a) c = \frac{ln(.9)}{−6} ≐ .01756.

(c) \overline{t} ≐ 56.95 months.

4. (a) About 42% of the students will take between 1.5 and 2.5 hours.

(c) Median time = 2.523 hours hours.

5. (a)

f(x) = \frac{1}{\sqrt{450π}} e^{−(x−100)^{2}/450}.

(c) About 0.3% of the population.

6. The constant \frac{1}{\sqrt{2πσ^{2}}} in the formula for the normal density is irrelevant for the location of the critical number(s) and the inflection points, so we will ignore it and consider the following function g(x) and its derivatives:

g(x) = e^{−(x−μ)^{2}/(2σ^{2})}
⇒ g′(x) = −e^{−(x−μ)^{2}/(2σ^{2})} · (x − μ)/σ^{2} (chain rule)
⇒ g′′(x) = −e^{−(x−μ)^{2}/(2σ^{2})} (\frac{1}{σ^{2}} − \frac{(x − μ)^{2}}{σ^{4}} ) (product rule).