8.4: Exercises

8.1. Practicing the chain rule. Use the chain rule to calculate the following derivatives

(a) \(y=f(x)=(x+5)^{5}\)

(b) \(y=f(x)=4\left(x^{2}+5 x-1\right)^{8}\)

(c) \(y=f(x)=(\sqrt{x}+2 x)^{3}\)

8.2. Growth curve. An example of a growth curve in population biology is called the Bertalanffy growth curve, after Canadian biologist Ludwig von Bertalanffy. This curve is defined by the equation

\[N=\left(a-b 2^{-k t}\right)^{3}, \nonumber \]

where the constants \(a, b\) and \(k\) are positive and \(a>b ; N\) denotes the size of the population and \(t\) denotes elapsed time. Find the growth rate \(d N / d t\) of the population.

Note: if \(f(x)=2^{a x}\), then \(f^{\prime}(x)=0.6931 \cdot a 2^{a x}\). Derivatives of such exponential functions are studied in Chapter 10.

8.3. Earth’s temperature. We expand and generalize the results of Example 8.10. As before, let \(G\) denote the level of greenhouse gases on Earth, and consider the relationship of temperature of the earth to the albedo \(a\) and the emissivity \(\varepsilon\) given by Equation (8.1).

(a) Suppose that \(a\) is constant, but \(\varepsilon\) depends on \(G\). Assume that \(d \varepsilon / d G\) is given. Determine the rate of change of temperature with respect to the level of greenhouse gasses in this case.

(b) Suppose that both \(a\) and \(\varepsilon\) depend on \(G\). Find \(d T / d G\) in this more general case (hint: the quotient rule as well as the chain rule are needed).

8.4. Shortest path from nest to food sources.

(a) Use the first derivative test to verify that the value \(x=\frac{d}{\sqrt{3}}\) is a local minimum of the function \(L(x)\) given by Eqn (8.2)

(b) Show that the shortest path is \(L=D+\sqrt{3} d\).

(c) In Section \(8.2\) we assumed that \(d<<D\), so that the food sources were close together relative to the distance from the nest. Now suppose that \(D=d / 2\). How would this change the solution?

8.5. Geometry of the shortest ants’ path. Use the results of Section \(8.2\) to show that in the shortest path, the angles between the branches of the Y-shaped path are all \(120^{\circ}\). Recall that \(\sin (30)=1 / 2, \sin (60)=\) \(\sqrt{3} / 2\).

8.6. More about the ant trail. Consider the lengths of the \(\mathrm{V}\) and \(\mathrm{T}\)-shaped paths in the ant trail example of Section 8.2. We refer to these as \(L_{V}\) and \(L_{T}\); each depend on the distances \(d\) and \(D\) in Figure 8.3.

(a) Write down the expressions for each of these functions.

(b) Suppose the distance \(D\) is fixed. How do the two lengths \(L_{V}, L_{T}\) depend on the distance \(d\) ? Use your sketching skills to draw a rough sketch of \(L_{v}(d), L_{T}(d)\).

(c) Use you sketch to determine whether there is a value of \(d\) for which the lengths \(L_{V}\) and \(L_{T}\) are the same.

8.7. Divided attention. A bird in its natural habitat feeds on two kinds of seeds whose nutritional values are

• 5 calories per seed of type 1 , and
• 3 calories per seed of type 2 .

Both kinds of seeds are hidden among litter on the forest floor and have to be found. If the bird splits its attention into \(x_{1}\) (a fraction of 1 - its whole attention) searching for seed type 1 and \(x_{2}\) (also a fraction of 1 ) searching for seed type 2 , then its probability of finding 100 seeds of the given type is

\[P_{1}\left(x_{1}\right)=\left(x_{1}\right)^{3}, \quad P_{2}\left(x_{2}\right)=\left(x_{2}\right)^{5} . \nonumber \]

Assume that the bird pays full attention to searching for seeds so that \(x_{1}+x_{2}=1\) where \(0 \leq x_{1} \leq 1\) and \(0 \leq x_{2} \leq 1\).

(a) Give an expression for the total nutritional value \(V\) gained by the bird when it splits its attention. Use the constraint on \(x_{1}, x_{2}\) to eliminate one of these two variables (for example, let \(x=x_{1}\) and write \(x_{2}\) in terms of \(x_{1}\).)

(b) Find critical points of \(V(x)\) and classify those points.

(c) Find absolute minima and maxima of \(V(x)\) and use your results to explain the bird’s optimal strategy for maximizing the nutritional value of the seeds it can find.