1The population $p_t$ of honey bees after $t$ weeks in a hive fluctuates according to the linear discrete-time dynamic system $p_{t+1} = -0.25p_t + 1500$. Which one of the following is a general solution to this system when $p_0 = 600$?
2For which value(s) of $a$ is the function $f(x) = \begin{cases} e^{2x} + 3, & x < -2 \ |ax-1|, & x \ge -2 \end{cases}$ continuous for all real numbers $x$?
3Which one of the following is the derivative of the function $h(t) = \ln \left( \frac{3}{\sqrt{t^2+1}} \right)$?
4The second derivative of the function $g(x) = \sin(x^2)$ is:
5Let $f(x)$ be a differentiable function defined on the closed interval $[2, 3]$. Assume that $f(2) = 7$ and $f(3) = 3$. Which two of the following statements must be true? (Choose two!)
6Find the equation of the tangent line to the graph of the equation $xe^y + y = x^2$ at the point $(1, 0)$.
7The cost (in dollars) to produce 100 hammers with steel of purity index $p$ is $C(p) = 2p^3 + 9p^2 - 60p + 300$, for $0 \le p \le 5$. What is the maximum possible cost and the minimum possible production cost of producing 100 hammers?
8Consider the equation $x^3 - x - 1 = 0$. Use Newton's method to find an approximation of a root of this equation starting with initial approximation $x_0 = 2$. Perform two iterations and round the final result to five decimal places.
9Calculate the exact value of the definite integral $I = \int_{1}^{e^3} \left( x^2 - 4 + \frac{1}{x} \right) dx$.
10A farmer's hay field production is modeled by $b_{t+1} = \frac{1}{2} b_t (10 - b_t)$. If a harvesting rate $h>0$ is introduced, the new relation is $b_{t+1} = \frac{1}{2} b_t (10 - b_t) - h b_t$. For which range of $h$ is the positive equilibrium stable?
