Vitousek and Farrington (1997) investigated nutrient limitations in soils of different ages. In the abstract of their paper, they say, Walker and Syers (1976) proposed a conceptual model that describes the pattern and regulation of soil nutrient pools and availability during long-term soil and ecosystem development. Their model implies that plant production generally should be limited by N [nitrogen] on young soils and by P [phosphorus] on old soils; N and P supply should be more or less equilibrate on intermediate aged soils. Vitousek and Farrington tested this hypothesis by conducting fertilizer experiments along a gradient of soil age, measuring the average increment in diameter (in mm/yr) of Metrosideros polymorpha trees. Denote by D(N, P, t) the diameter increment (in mm/yr) as a function of the amount of nitrogen (N) added, the amount of phosphorus (P) added, and the age (t) of the soil. Vitousek and Farrington’s experiments showed that $\frac{\partial D}{\partial t}(\mathrm{N}, 0, t)<0$ and $\frac{\partial D}{\partial t}(0, \mathrm{P}, t)>0$ for their choices of N > 0 and P > 0. Explain why their results support theWalker and Syers hypothesis.

The functional responses of some predators are sigmoidal; that is, the number of prey attacked per predator as a function of prey density has a sigmoidal shape. If we denote the prey density by N, the predator density by P, the time available for searching for prey by T, and the handling time of each prey item per predator by $T_{h},$ then the number of prey encounters per predator as a function of N, T, and $T_{h}$ can be expressed as $f\left(N, T, T_{h}\right)=\frac{b^{2} N^{2} T}{1+c N+b T_{h} N^{2}}$ where b and c are positive constants. (a) Investigate how an increase in the prey density N affects the function $f\left(N, T, T_{h}\right).$ (b) Investigate how an increase in the time T available for search affects the function $f\left(N, T, T_{h}\right).$ (c) Investigate how an increase in the handling time Th affects the function $f\left(N, T, T_{h}\right).$ (d) Graph $f\left(N, T, T_{h}\right)$ as a function of N when T = 2.4 hours, $T_{h}=0.2$ hours, b = 0.8, and c = 0.5.

Suppose that an object has temperature T and is brought into a room that is kept at a constant temperature $T_{a}.$ Newton’s law of cooling states that the rate of temperature change of the object is proportional to the difference between the temperature of the object and the surrounding medium. (a) Denote the temperature at time t by T(t), and explain why $\frac{d T}{d t}=k\left(T-T_{a}\right)$ is the differential equation that expresses Newton’s law of cooling. (b) Supposethat it takes the object 20 min to cool from $30^{\circ} \mathrm{C}$ to $28^{\circ} \mathrm{C}$ in a room whose temperature is $21^{\circ} \mathrm{C}.$ How long will it take the object to cool to $25^{\circ} \mathrm{C}$ if it is at $30^{\circ} \mathrm{C}$ when it is brought into the room?

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $f(x)=\ln \frac{x}{x+1}$

Evaluate the given indefinite integrals. $\int x \sqrt{3 x+1} d x$

Use l’Hospital’s rule to find the limits. $\lim _{x \rightarrow 0} \frac{\sin x}{x \cos x}$

Evaluate the given definite integrals. $\int_{0}^{\pi / 2} x \sin x d x$

Assume that f(x) is differentiable. Find an expression for the derivative of y at x = 1, assuming that f(1) = 2 and $f^{\prime}(1)=-1.$ $y=2 x f(x)$

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line with slope −1 and y-intercept (0,−3)

In Problems use (4.12) to find the derivative of the inverse at the indicated point. Let Denote the inverse of y = arcsin x, −1 ≤ x ≤ 1. Show that (REFERENCE 4.12) Derivative of an Inverse Function If f (x) is one to one and differentiable with inverse function f −1(x) and f ‘[ f −1(x)] ≠ 0, then f −1(x) is differentiable and

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $f(x)=\left(\ln \left(1-x^{2}\right)\right)^{3}$

Use a graphing calculator to sketch the graphs of the functions. $y=x^{3 / 2}, x \geq 0$

Use a table or a graph to investigate each limit.

$$\begin{aligned} &\lim s\left(s^{2}-4\right)\\ &s \rightarrow 2 \end{aligned}$$

Write each number in standard form a+bi. $\frac{-5+\sqrt{-50}}{10}$