Chapter 4

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ g(x)=\frac{1}{\sin (3 x)} $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=3 \sqrt{x+1} $$

Assume that \(f(x)\) and \(g(x)\) are differentiable at \(x\). Find an expression for the derivative of \(y .\) $$ y=[-2 f(x)-3 g(x)] g(x)+\frac{2 g(x)}{3} $$

Differentiate the functions with respect to the independent variable. $$ f(x)=\ln (\cos (1-x)) $$

In Problems , find the tangent line, in standard form, to \(y=\) \(f(x)\) at the indicated point. $$ y=-2 x^{3}-3 x+1, \text { at } x=1 $$

The volume \(V\) of a spherical cell of radius \(r\) is given by $$V(r)=\frac{4}{3} \pi r^{3}$$ If you can determine the radius to within an accuracy of \(3 \%\), how accurate is your calculation of the volume?

Assume that \(N(t)\) denotes the size of a population at time \(t\) and that \(N(t)\) satisfies the differential equation $$ \frac{d N}{d t}=3 N\left(1-\frac{N}{20}\right) $$ Let \(f(N)=3 N\left(1-\frac{N}{20}\right)\) for \(N \geq 0\). Graph \(f(N)\) as a function of \(N\) and identify all equilibria (i.e., all points where \(\frac{d N}{d t}=0\) ).

Find \(\frac{d y}{d x}\) by applying the chain rule repeatedly. \(y=\left(\frac{2 x+1}{3\left(x^{3}-1\right)^{3}-1}\right)^{3}\)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ (x)=\frac{1}{\sin \left(3 x^{2}-1\right)} $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=2 \sqrt{x^{2}-1} $$

Let \(B(t)\) denote the biomass at time \(t\) with specinc growth rate \(g(B) .\) Show that the specific growth rate at \(B=0\) is given by the slope of the tangent line on the graph of the growth rate at \(B=0\).

Differentiate the functions with respect to the independent variable. $$ f(x)=\ln \left(\tan x^{2}\right) $$

In Problems , find the tangent line, in standard form, to \(y=\) \(f(x)\) at the indicated point. $$ y=2 x^{4}-5 x, \text { at } x=1 $$

Assume that a species lives in a habitat that consists of many islands close to a mainland. The species occupies both the mainland and the islands, but, although it is present on the mainland at all times, it frequently goes extinct on the islands. Islands can be recolonized by migrants from the mainland. The following model keeps track of the fraction of islands occupied: Denote the fraction of islands occupied at time \(t\) by \(p(t)\). Assume that each island experiences a constant risk of extinction and that vacant islands (the fraction \(1-p\) ) are colonized from the mainland at a constant rate. Then $$ \frac{d p}{d t}=c(1-p)-e p $$ where \(c\) and \(e\) are positive constants. (a) The gain from colonization is \(f(p)=c(1-p)\) and the loss from extinction is \(g(p)=e p .\) Graph \(f(p)\) and \(g(p)\) for \(0 \leq p \leq 1\) in the same coordinate system. Explain why the two graphs intersect whenever \(e\) and \(c\) are both positive. Compute the point of intersection and interpret its biological meaning. (b) The parameter \(c\) measures how quickly a vacant island becomes colonized from the mainland. The closer the islands, the larger is the value of \(c\). Use your graph in (a) to explain what happens to the point of intersection of the two lines as \(c\) increases. Interpret your result in biological terms.

Find \(\frac{d y}{d x}\) by applying the chain rule repeatedly. \(y=\left(\frac{(2 x+1)^{2}-x}{\left(3 x^{3}+1\right)^{3}-x}\right)^{2}\)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ g(x)=\frac{1}{\csc ^{2}(5 x)} $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=4^{\sqrt{1-2 x^{3}}} $$

Let \(N(t)\) denote the size of a population at time \(t\). Differentiate $$ f(N)=r N\left(1-\frac{N}{K}\right) $$

Differentiate the functions with respect to the independent variable. $$ g(s)=\ln \left(\sin ^{2}(3 s)\right) $$

In Problems , find the tangent line, in standard form, to \(y=\) \(f(x)\) at the indicated point. $$ y=-x^{3}-2 x^{2}, \text { at } x=0 $$

Suppose that you are studying reproduction in moss. The scaling relation $$N \propto L^{2.11}$$ has been found (Niklas, 1994 ) between the number of moss spores \((N)\) and the capsule length \((L)\). This relation is not very accurate, but it turns out that it suffices for your purpose. To estimate the number of moss spores, you measure the capsule length. If you wish to estimate the number of moss spores within an error of \(5 \%\), how accurately must you measure the capsule length?

Consider the chemical reaction $$ \mathrm{A}+\mathrm{B} \longrightarrow \mathrm{AB} $$ If \(x(t)\) denotes the concentration of \(\mathrm{AB}\) at time \(t\), then $$ \frac{d x}{d t}=k(a-x)(b-x) $$ where \(k\) is a positive constant and \(a\) and \(b\) denote the concentrations of \(A\) and \(B\), respectively, at time \(0 .\) Assume that \(k=3, a=7\), and \(b=4\). For what values of \(x\) is \(d x / d t=0 ?\) Interpret the meaning of \(d x / d t=0\).

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ g(x)=\frac{1}{\csc ^{3}\left(1-5 x^{2}\right)} $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ h(t)=5^{\sqrt{t}} $$

Let \(N(t)\) denote the size of a population at time \(t\). Differentiate $$ f(N)=r\left(a N-N^{2}\right)\left(1-\frac{N}{K}\right) $$

Differentiate the functions with respect to the independent variable. $$ f(x)=x \ln x $$

In Problems , find the tangent line, in standard form, to \(y=\) \(f(x)\) at the indicated point. $$ y=\frac{1}{\sqrt{2}} x^{2}-\sqrt{2}, \text { at } x=4 $$

Model Suppose that the rate of growth of a plant in a certain habitat depends on a single resource \(-\) for instance, nitrogen. Assume that the growth rate \(f(R)\) depends on the resource level \(R\) in accordance with the formula $$f(R)=a \frac{R}{k+R}$$ where \(a\) and \(k\) are constants. Express the percentage error of the growth rate, \(100 \frac{\Delta f}{f}\), as a function of the percentage error of the resource level, \(100 \frac{\Delta R}{R}\).

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ h(x)=\cot (3 x) \csc (3 x) $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ h(t)=6 \sqrt{6 t^{6}-6} $$

Consider the chemical reaction $$ \mathrm{A}+\mathrm{B} \longrightarrow \mathrm{AB} $$ If \(x\) denotes the concentration of \(\mathrm{AB}\) at time \(t\), then the reaction rate \(R(x)\) is given by $$ R(x)=k(a-x)(b-x) $$ where \(k, a\), and \(b\) are positive constants. Differentiate \(R(x)\).

Differentiate the functions with respect to the independent variable. $$ f(x)=x^{2} \ln x^{2} $$

In Problems , find the tangent line, in standard form, to \(y=\) \(f(x)\) at the indicated point. $$ y=3 \pi x^{5}-\frac{\pi}{2} x^{3}, \text { at } x=-1 $$

The reaction rate \(R(x)\) of the irreversible reaction $$\mathrm{A}+\mathrm{B} \rightarrow \mathrm{AB}$$ is a function of the concentration \(x\) of the product \(\mathrm{AB}\) and is given by $$R(x)=k(a-x)(b-x)$$ where \(k\) is a constant, \(a\) is the concentration of \(\mathrm{A}\) at the beginning of the reaction, and \(b\) is the concentration of \(\mathrm{B}\) at the beginning of the reaction. Express the percentage error of the reaction rate. \(100 \frac{\Delta R}{R}\), as a function of the percentage error of the concentration \(x, 100 \frac{\Delta x}{x}\)

Suppose that the rate of change of the size of a population is given by $$ \frac{d N}{d t}=r N\left(1-\frac{N}{K}\right) $$ where \(N=N(t)\) denotes the size of the population at time \(t\) and \(r\) and \(K\) are positive constants. Find the equilibrium size of the population-that is, the size at which the rate of change is equal to \(0 .\) Use your answer to explain why \(K\) is called the carrying capacity.

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ h(x)=\frac{3}{\tan (2 x)-x} $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ g(x)=2^{2 \cos x} $$

In Problems \(49-70\), differentiate with respect to the independent variable. $$ f(x)=\frac{3 x-1}{x+1} $$

Differentiate the functions with respect to the independent variable. $$ f(x)=\frac{\ln x}{x} $$

(Adapted from Valentine, \(1985 .\) ) Walker and Valentine (1984) suggested a model for species diversity which assumes that species extinction rates are independent of diversity but speciation rates are regulated by competition. Denoting the number of species at time \(t\) by \(N(t)\), the speciation rate by \(b\), and the extinction rate by \(a\), they used the model $$ \frac{d N}{d t}=N\left[b\left(1-\frac{N}{K}\right)-a\right] $$ where \(K\) denotes the number of "niches," or potential places for species in the ecosystem. (a) Find possible equilibria under the condition \(a

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ g(t)=\left(\frac{1}{\sin t^{2}}\right)^{3 / 2} $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ g(r)=2^{-3 \sin r} $$

Differentiate with respect to the independent variable. $$ f(x)=\frac{1-4 x^{3}}{1-x} $$

Differentiate the functions with respect to the independent variable. $$ h(t)=\frac{\ln t}{1+t^{2}} $$

In Problems , find the normal line, in standard form, to \(y=\) \(f(x)\) at the indicated point. $$ y=1-3 x^{2}, \text { at } x=-2 $$

Which of the following statements is true? (A) If \(f(x)\) is continuous, then \(f(x)\) is differentiable. (B) If \(f(x)\) is differentiable, then \(f(x)\) is continuous.

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ h(s)=\sin ^{3} s+\cos ^{3} s $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ g(r)=3^{r^{1 / 5}} $$

Differentiate with respect to the independent variable. $$ (x)=\frac{3 x^{2}-2 x+1}{2 x+1} $$

Differentiate the functions with respect to the independent variable. $$ h(t)=\sin (\ln (3 t)) $$