Chapter 4

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

Assume that \(f(x)\) and \(g(x)\) are differentiable at \(x .\) Find an expression for the derivative of \(y .\) \(y=3 f(x) g(x)\)

Differentiate $$ R(T)=\frac{2 \pi^{5}}{15} \frac{k^{4}}{c^{2} h^{3}} T^{4} $$ with respect to \(T\). Assume that \(k, c\), and \(h\) are positive constants.

In Problems \(40-46\), find \(\frac{d y}{d x}\) by applying the chain rule repeatedly. $$ y=\left(\sqrt{x^{3}-3 x}+3 x\right)^{4} $$

Assume that the measurement of \(x\) is \(a c-\) curate within \(2 \% .\) In each case, determine the error \(\Delta f\) in the calculation of \(f\) and find the percentage error \(100 \frac{\Delta f}{f} .\) The quantities \(f(x)\) and the true value of \(x\) are given. \(f(x)=x^{1 / 4}, x=10\)

Find the derivative with respect to the independent variable. $$ f(x)=\frac{1}{\sin ^{2} x+\cos ^{2} x} $$

Differentiate the functions with respect to the independent variable. \(h(t)=4^{2 t^{3}-t}\)

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

In Problems \(42-48\), find the tangent line, in standard form, to \(y=f(x)\) at the indicated point. $$ y=3 x^{2}-4 x+7, \text { at } x=2 $$

In Problems \(40-46\), find \(\frac{d y}{d x}\) by applying the chain rule repeatedly. $$ y=\left(1+2(x+3)^{4}\right)^{2} $$

Assume that the measurement of \(x\) is \(a c-\) curate within \(2 \% .\) In each case, determine the error \(\Delta f\) in the calculation of \(f\) and find the percentage error \(100 \frac{\Delta f}{f} .\) The quantities \(f(x)\) and the true value of \(x\) are given. \(f(x)=\ln x, x=20\)

Find the derivative with respect to the independent variable. $$ f(x)=\frac{1}{\tan ^{2} x-\sec ^{2} x} $$

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

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

Find the tangent line, in standard form, to \(y=f(x)\) at the indicated point. $$ y=7 x^{3}+2 x-1, \text { at } x=-3 $$

In Problems \(40-46\), find \(\frac{d y}{d x}\) by applying the chain rule repeatedly. $$ y=\left(1+\left(3 x^{2}-1\right)^{3}\right)^{2} $$

Assume that the measurement of \(x\) is \(a c-\) curate within \(2 \% .\) In each case, determine the error \(\Delta f\) in the calculation of \(f\) and find the percentage error \(100 \frac{\Delta f}{f} .\) The quantities \(f(x)\) and the true value of \(x\) are given. \(f(x)=\frac{1}{1+x}, x=4\)

Find the derivative with respect to the independent variable. $$ g(x)=\frac{1}{\sin (3 x)} $$

Differentiate the functions with respect to the independent variable. \(f(x)=3^{\sqrt{x+1}}\)

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

In Problems \(40-46\), find \(\frac{d y}{d x}\) by applying the chain rule repeatedly. $$ y=\left(\frac{x}{2\left(x^{2}-1\right)^{2}-1}\right)^{2} $$

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?

Find the derivative with respect to the independent variable. $$ g(x)=\frac{1}{\sin \left(3 x^{2}-1\right)} $$

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

Diversity Index The diversity of a population that contains two species can be measured by the Gini-Simpson diversity index. If the fraction of organisms from species 1 is \(p\), then the diversity is given by: $$ H(p)=2 p(1-p) $$ Use the product rule to calculate \(H^{\prime}(p) .\) For what value of \(p\) does \(H^{\prime}(p)=0 ?\)

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

In Problems \(40-46\), find \(\frac{d y}{d x}\) by applying the chain rule repeatedly. $$ y=\left(\frac{2 x+1}{3\left(x^{3}-1\right)^{3}-1}\right)^{3} $$

The speed \(v\) of blood flowing along the central axis of an artery of radius \(R\) is given by Poiseuille's law, $$v(R)=c R^{2}$$ where \(c\) is a constant. If you can determine the radius of the artery to within an accuracy of \(5 \%\), how accurate is your calculation of the speed?

Find the derivative with respect to the independent variable. $$ g(x)=\frac{1}{\csc ^{2}(5 x)} $$

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

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

In Problems \(40-46\), find \(\frac{d y}{d x}\) by applying the chain rule repeatedly. $$ y=\left(\frac{(2 x+1)^{2}-x}{\left(3 x^{3}+1\right)^{3}-x}\right)^{2} $$

Suppose that you are studying reproduction in moss. (Niklas, 1994) found a scaling relation $$N \propto L^{2.11}$$ between the number of moss spores \((N)\) and the capsule length (L). 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?

Find the derivative with respect to the independent variable. $$ g(x)=\frac{1}{\sin ^{2}\left(1-5 x^{2}\right)} $$

Differentiate the functions with respect to the independent variable. \(h(t)=5^{\sqrt{t}}\)

Small populations of organisms will often find themselves outcompeted by other species. Populations do not start to grow until they exceed some critical size. This is known as the Allee effect. One model for population growth that incorporates the Allee effect is: $$ \frac{d N}{d t}=f(N) \text { where } f(N)=r N(N-a)(1-N / K) $$ where \(r, a\), and \(K\) are all positive constants and \(K>a\). (a) Show that if \(N0\). (a) and (b) together imply that populations smaller than \(a\) will shrink, and populations larger than \(a\) will grow. (c) Show that \(f^{\prime}(0)<0\) and \(f^{\prime}(K)<0\). (d) Show that \(f^{\prime}(a)>0\). We will show in Chapter 8 how to use (c) and (d) to predict the size of the population as \(t \rightarrow \infty\)

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

Chewing in Mammals Druzinsky (1993) showed that chewing frequency (i.e., the number of mass an animal chews in one minute) is proportional to its body mass raised to the power of \(-0.128\), that is, if \(M\) is the body mass and \(c\) the chewing frequency. then, $$ c=k M^{-0.128} $$ for some positive constant \(k\). (a) Assume that the body mass of a particular mammal is given by a formula \(M(t)=1+2 \sqrt{t} .\) Calculate \(d c / d t\). (b) Druzinsky also found that the jaw length of the animal, \(L\), is proportional to its body mass raised to the \(0.312\) power, i.e., \(L=r M^{0.312}\) where \(r\) is another positive constant.

Suppose that the rate of growth of a plant in a certain habitat depends on a single resource-for instance, nitrogen. The dependence of the growth rate \(f(R)\) on the resource level \(R\) is modeled using Monod's equation $$f(R)=a \frac{R}{k+R}$$ 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, \(100 \frac{\Delta R}{R}\).

Find the derivative with respect to the independent variable. $$ h(x)=\cot (3 x) \csc (3 x) $$

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

Consider the chemical reaction: $$ \mathrm{A}+\mathrm{B} \longrightarrow \mathrm{AB} $$ If \(x\) denotes the concentration of \(\mathrm{AB}\) at time \(t\) and \(a, b\) are the initial amounts of \(A\) and \(B\) present, 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 \(R(x)\).

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

Wing Beat Frequency in Hummingbirds Altshuler and Dudley (2003) found that hummingbirds' wing beat frequency, \(f\), decreases with body mass, \(m\), according to $$ f=40-\frac{8}{5} m $$ where \(f\) is measured in beats per second and \(m\) in grams. Assume that the amount of thrust that a flying hummingbird can generate depends on its mass and wing beat frequency as follows $$ T=c f^{2} m^{4 / 3} $$ for some positive constant \(c\). (This equation is derived from the thrust mechanics of a moving wing.) (a) Equation (4.5) should only be used if \(m<25\). Why? (The largest hummingbirds have a mass of \(22 \mathrm{~g}\).) (b) Calculate \(d T / d m .\) (c) Show by plotting \(d T / d m\) that for larger hummingbirds, \(T\) decreases with \(m\). That is, show that \(d T / d m<0\), once \(m\) exceeds a critical threshold. (d) For a hummingbird to fly, its thrust must exceed its weight \(W=m g\) (where \(g\) is the acceleration due to gravity). Explain, using your answer to (d), why if the hummingbird's mass increases, then \(T\) will eventually be smaller than \(W\) (Hint: \(d W / d m=g\) is constant and positive), no matter what the value of \(c\) is.

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 error of the reaction rate, \(\Delta R\), as a function of the error of the concentration \(\Delta x\).

Find the derivative with respect to the independent variable. $$ h(x)=\frac{3}{\tan (2 x)-x} $$

Differentiate the functions with respect to the independent variable. \(g(x)=2^{2 \cos x}\)

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

In Problems 49-54, find the normal line, in standard form, to \(y=f(x)\) at the indicated point. $$ y=2+x^{2}, \text { at } x=-1 $$

Find the derivative with respect to the independent variable. $$ g(t)=\left(\frac{1}{\sin t^{2}}\right) $$