6. Sketch a graph that is consistent with the all the following information: • g(x) is defined at all x except for x = 0

• g(−2) = −4

• g′(x) < 0 for x < −2 and for x > 0

• g′(−2) = 0

• g′(x) > 0 for −2 < x < 0

• g′′(x) < 0 for x < −3

• g′′(x) > 0 for −3 < x < 0 and for x > 0

7. A particle moves so that r(t) = 3 cos tî + sin tĵ = (3 cos t, sin t) .

a. Compute the velocity vector v(t) and the speed. b. Give an equation in the variables x and y that describes the path of the

motion. Sketch the path for 0 ≤ t ≤ π, and indicate the direction of motion.

8. We find a Riemann sum approximation ∫_{0}^{π/6} sec^{3} x dx ≈ 0.60827 by using the right endpoints of 500 equal size subintervals. Does this Riemann sum over- or under-estimate the integral?

9. The local gas station/video rental/convenience store has three staff persons working from opening at 7 am to 2 pm, two working from 2 pm to 4 pm, four working from 4 pm to 10 pm, and one from 10 pm to closing at 1 am. One fine day everyone works as usual, except the two on duty from 2 pm to 4 pm lock up at 3 pm and sneak out to play street hockey on the roof for an hour. Graph the work done (in staff-hours) over the course of this day as a function of time. What is the average number of staff on duty that particular day?

10. a. Compute the Taylor polynomial of degree 2 for \sqrt{1 + x}, centered at x = 0 .

b. The electric potential V , at a distance r along the axis perpendicular to the center of a charged disc of radius a and charge density σ, is V = 2πσ(\sqrt{r^{2} + a^{2}} − r) . If r is large in comparison to a, then a/r is close to 0. Use part a. to give an approximation for \sqrt{r^{2} + a^{2}} − r as r times a polynomial in powers of a/r.

Unconstrained population growth

In many cases, the rate of change of a variable quantity is proportional to the quantity itself. Consider human population. If a city of 100,000 is increasing at a rate of 1,500 persons/year, we would expect a similar city of 200,000 to be increasing at a rate of 3,000 persons/year and a city of 600,000 to be increasing at a rate of what?

1. For the cities described above, if P represents the population at time t, and the net growth rate P′ is proportional to P , compute the value of the proportionality constant, which is customarily denoted by the letter k in chemistry and math, and by the letter r in biology. That is, if P′ = rP , determine the value of r. What are the units for r? This constant is generally called the per capita growth rate or intrinsic growth rate). Why is the