Consumer resource models often have the following general form $R^{\prime}=f(R)-g(R, C) \quad C^{\prime}=\varepsilon g(R, C)-h(C)$ where R is the number of individuals of the resource and C is the number of consumers. The function $f(R)$ gives the rate of replenishment of the resource, g(R, C) describes the rate of consumption of the resource, and h(C) is the rate of loss of the consumer. The constant $\varepsilon,$ where $0<\varepsilon<1,$ is the conversion efficiency of resources into consumers. Find all equilibria of the following examples and determine their stability properties. A chemostat is an experimental consumer-resource system. If the resource is not self-reproducing, then it can be modeled by choosing $f(R)=\theta, g(R, C)=b R C,$ and $h(C)=\mu C.$ Suppose $\theta=2, b=1, \varepsilon=1, \text { and } \mu=1.$

Evaluate the limits. $\lim _{x \rightarrow 1} \frac{x^{1 / 3}-1}{x^{2 / 3}-1}$

Evaluate the integrals by using the properties of the definite integral and interpreting integrals as areas. $\int_{-3}^{3}(2+t) \sqrt{9-t^{2}} d t$

What is a doubling time? Suppose a population has a doubling time of 25 years. By what factor will it grow in 25 years? in 50 years? in 100 years?

Algebraically determine the general solution to the equation $4 \cos ^{2} x-1=0$. Give your answer in radians.

Two resistors (A and B) are connected in series to a 12-V battery. Resistor A ends up with 9 V across it. Which resistor has the least resistance: (a) A, (b) B, (c) both have the same, or (d) you can't tell from the data given?

Gregor, the cockroach, is on the edge of a Ferris wheel that is rotating at a rate of 2 rev/min (counterclockwise as you observe him). Gregor is crawling along a spoke toward the center of the wheel at a rate of 3 in/min. (a) Using polar coordinates with the center of the wheel as origin, assume that Gregor starts (at time t = 0) at the point r = 20 ft, $\theta=0$. Give parametric equations for Gregor’s polar coordinates r and $\theta$ at time t (in minutes). (b) Give parametric equations for Gregor’s Cartesian coordinates at time t. (c) Determine the distance Gregor has traveled once he reaches the center of the wheel. Express your answer as an integral and evaluate it numerically.

(a) Use the Intermediate Value Theorem to show that the function $f(x)=e^{x}+x-2$ has a zero in the interval [0, 2]. (b) Approximate the zero rounded to three decimal places.

Determine the intervals of constant concavity of the given function, and locate any inflection points. $f(x)=\left(2+2 x-x^{2}\right)^{2}$

(a) Investigate $\lim _{x \rightarrow 0} \cos \frac{\pi}{x^{2}}$ by using a table and evaluating the function $f(x)=\cos \frac{\pi}{x^{2}}$ at x=-0.1,-0.01,-0.001, -0.0001,..., 0.0001,0.001,0.01,0.1. (b) Investigate $\lim _{x \rightarrow 0} \cos \frac{\pi}{x^{2}}$ by using a table and evaluating the function $f(x)=\cos \frac{\pi}{x^{2}}$ at $x=-\frac{2}{3},-\frac{2}{5},-\frac{2}{7},-\frac{2}{9}, \ldots, \frac{2}{9}, \frac{2}{7}, \frac{2}{5}, \frac{2}{3}$. (c) Compare the results from (a) and (b). What do you conclude about the limit? Why do you think this happens? What is your view about using a table to draw a conclusion about limits? (d) Use technology to graph f. Begin with the x -window $[-2 \pi, 2 \pi]$ and the y -window [-1,1]. If you were finding $\lim _{x \rightarrow 0} f(x)$ using a graph, what would you conclude? Zoom in on the graph. Describe what you see.

Find each one-sided limit using properties of limits. $\lim _{x \rightarrow 3^{+}} \frac{x^{2}-9}{x-3}$

Does either the four-firm concentration ratio or the HHI directly measure the amount of competition in an industry? Why or why not?

What are the two key characteristics of public goods?

What is predatory pricing?