Chapter 4

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

16\. Population Growth Assume that \(N(t)\) denotes the size of a population at time \(t\), and that in some conditions \(N(t)\) satisfies the differential equation $$ \frac{d N}{d t}=r N $$ where \(r\) is a constant. (a) Find the per capita growth rate. (b) Assume that \(r<0\) and that \(N(0)=20\). Is the population size at time 1 greater than 20 or less than \(20 ?\) Explain your answer.

Binding by Hemoglobin The fraction of hemoglobin, \(f\), that is bound to oxygen depends on the concentration of oxygen \(P\). This relationship is often modeled by Hill's equation, $$f(P)=\frac{P^{m}}{k^{m}+P^{m}}$$ where \(k\) and \(m\) are positive constants that vary from species to species and on whether the animal lives at sea level or at a higher altitude. (a) Show that \(f^{\prime \prime}(0)>0\) if \(m=2\) (b) Show that \(f^{\prime \prime}(0)<0\) if \(m=1\) In Chapter 5 we will show how these observations can be related to the different shapes that the graph of \(f(P)\) against \(P\) has for different values of \(m\).

Assume that \(x\) and \(y\) are differentiable functions of \(t\). Find \(\frac{d y}{d t}\) when \(y^{2}+(x+1)^{2}=1, \frac{d x}{d t}=1\) for \(x=-\frac{1}{2}\), and \(y>0\).

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ g(t)=\sqrt{t+\sqrt{t+1}} $$

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=\ln (1+x)\) at \(a=0\)

Compute \(f(c+h)-f(c)\) at the indicated point. Your answers will contain \(h\) as an unknown variable. \(f(x)=-2 x+1 ; c=2\)

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

Differentiate the functions with respect to the independent variable. \(f(x)=\exp [\sin (3 x)]\)

Apply the product rule to find the tangent line, in slope-intercept form, of \(y=f(x)\) at the specified point. \(f(x)=(1-2 x)(1+2 x)\), at \(x=2\)

Differentiate the functions given with respect to the independent variable. $$ f(s)=s^{3} e^{3}+3 e $$

\(N(t)\) is the size of the population at time \(t\). The population is modeled using a differential equation $$ \frac{d N}{d t}=r N $$ where \(r\) is a constant. (a) According to the differential equation what is the growth rate of the population? (b) Assume that \(r>0\). Explain why the population size at time \(t=1\) will be larger than the population size at time \(t=0\). (c) If \(r>0\), will the growth rate at time \(t=1\) be larger or smaller than the growth rate at time \(t=0\) ? (d) Answer (c) again but comparing the per capita growth rates at times \(t=0\) and \(t=1\).

Two atoms are modeled as interacting via a Lennard-Jones \(6-12\) potential. That is, the energy of interaction, \(V\), depends on their spacing, \(r\), according to a formula. $$V(r)=\frac{a}{r^{12}}-\frac{b}{r^{6}}, r>0$$ where \(a\) and \(b\) are positive constants. (a) The force between the atoms can be shown to be given by \(F(r)=-\frac{d V}{d r}\). When the atoms are in equilibrium, \(F(r)=0\). Find the equilibrium spacing of the pair of atoms, \(r\). (b) Generally the equilibrium spacing of a pair of atoms interacting with energy is stable if \(V^{\prime \prime}(r)>0\), and unstable if \(V^{\prime \prime}(r)<0\). For the Lennard-Jones \(6-12\) potential, show that the equilibrium spacing that you calculated in (a) is stable.

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ g(t)=\left(\frac{t}{t-3}\right)^{3} $$

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=\ln (1+2 x)\) at \(a=0\)

Compute \(f(c+h)-f(c)\) at the indicated point. Your answers will contain \(h\) as an unknown variable. \(f(x)=3 x^{2} ; c=1\)

Find the derivative with respect to the independent variable. $$ f(x)=\cos ^{2}\left(x^{2}-1\right) $$

Differentiate the functions with respect to the independent variable. \(f(x)=\exp [\cos (4 x)]\)

Apply the product rule to find the tangent line, in slope-intercept form, of \(y=f(x)\) at the specified point. \(f(x)=\left(3 x^{2}-2\right)(x-1)\), at \(x=1\)

Consider the chemical reaction $$ \mathrm{A}+\mathrm{B} \rightarrow \mathrm{AB} $$ Assume \(k\) is the rate constant for the reaction. (a) Explain why, if \([\mathrm{A}]\) is the amount of the chemical \(\mathrm{A}\) and \([\mathrm{B}]\) is the amount of the chemical B present, then $$ \frac{d[\mathrm{~A}]}{d t}=-k[\mathrm{~A}][\mathrm{B}] $$ (b) What is the corresponding differential equation describing the rate of change of \([\mathrm{B}]\) ?

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ h(s)=\left(\frac{2 s^{2}}{s+1}\right)^{4} $$

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=\log x\) at \(a=1\)

Compute \(f(c+h)-f(c)\) at the indicated point. Your answers will contain \(h\) as an unknown variable. \(f(x)=\sqrt{x} ; c=4\)

Differentiate the functions with respect to the independent variable. \(f(x)=\exp \left[\sin \left(x^{2}-1\right)\right]\)

Apply the product rule to find the tangent line, in slope-intercept form, of \(y=f(x)\) at the specified point. \(f(x)=4\left(2 x^{4}+3 x\right)\left(4-2 x^{2}\right)\), at \(x=-1\)

Differentiate the functions given with respect to the independent variable. $$ f(x)=20 x^{3}-4 x^{6}+9 x^{8} $$

Consider the chemical reaction $$ \mathrm{A}+\mathrm{B} \longrightarrow \mathrm{AB} $$ If \(x(t)\) denotes the concentration of \(\mathrm{AB}\) at time \(t\), and \(k\) is the rate constant for of the reaction, explain why: $$ \frac{d x}{d t}=k(a-x)(b-x) $$ where \(k\) is a positive constant and \(a\) and \(b\) denote the concentrations of \(\mathrm{A}\) and \(\mathrm{B}\), respectively, at time 0 .

Assume that the side length \(x\) and the volume \(V=x^{3}\) of a cube are differentiable functions of \(t\). Express \(d V / d t\) in terms of \(d x / d t\)

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ f(r)=\left(r^{2}-r\right)^{3}\left(r+3 r^{3}\right)^{-4} $$

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=\log \left(1+x^{2}\right)\) at \(a=0\)

Compute \(f(c+h)-f(c)\) at the indicated point. Your answers will contain \(h\) as an unknown variable. \(f(x)=\frac{1}{x} ; c=-2\)

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

Differentiate the functions with respect to the independent variable. \(f(x)=\exp \left[\cos \left(1-2 x^{3}\right)\right]\)

Apply the product rule to find the tangent line, in slope-intercept form, of \(y=f(x)\) at the specified point. \(f(x)=\left(3 x^{3}-3\right)\left(2-2 x^{2}\right)\), at \(x=0\)

Differentiate the functions given with respect to the independent variable. $$ f(x)=\frac{x^{3}}{15}-\frac{x^{4}}{20}+\frac{2}{15} $$

Chemicals \(\mathrm{A}\) and \(\mathrm{X}\) react through an autocatalytic reaction: $$ \mathrm{A}+\mathrm{X} \longrightarrow 2 \mathrm{X} $$ (a) If the rate constant for the reaction is \(k\), and the amounts of A and \(X\) present are denoted by \([\mathrm{A}]\) and \([\mathrm{X}]\), then explain why $$ \frac{d[\mathrm{~A}]}{d t}=-k[\mathrm{~A}][\mathrm{X}] $$ (b) Find a differential equation that describes the rate of change of the amount of \([\mathrm{X}]\).

Assume that the radius \(r\) and the area \(A=\pi r^{2}\) of a circle are differentiable functions of \(t\). Express \(d A / d t\) in terms of \(d r / d t\).

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ h(s)=\frac{2(3-s)^{2}}{s^{2}+(7 s-1)^{2}} $$

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=e^{x}\) at \(a=0\)

(a) Use the formal definition of the derivative to find the derivative of \(y=2 x^{2}\) at \(x=-1\). (b) Show that the point \((-1,2)\) is on the graph of \(y=2 x^{2}\), and find the equation of the tangent line at the point \((-1,2)\). (c) Graph \(y=2 x^{2}\) and the tangent line at the point \((-1,2)\) in the same coordinate system.

Find the derivative with respect to the independent variable. $$ f(x)=4 \cos x^{2}-2 \cos ^{2} x $$

Differentiate the functions with respect to the independent variable. \(f(x)=\sin \left(e^{x}\right)\)

Apply the product rule to find the normal line, in slope-intercept form, of \(y=f(x)\) at the specified point. \(f(x)=(2 x+1)\left(3 x^{2}-1\right)\), at \(x=1\)

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

21\. Chemical Reaction Chemical A spontaneously decomposes into chemicals \(\mathrm{B}\) and \(\mathrm{C}: \mathrm{A} \rightarrow \mathrm{B}+\mathrm{C} .\) The rate constant for this reaction is \(k\). (a) Explain why, if the amounts of \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\) present are denoted by \([\mathrm{A}],[\mathrm{B}]\), and \([\mathrm{C}]\) respectively, then \([\mathrm{B}]\) obeys the differential equation $$ \frac{d[\mathrm{~B}]}{d t}=k[\mathrm{~A}] $$ (b) What differential equations represent the rate of change of \([\mathrm{C}]\) and \([\mathrm{A}] ?\) (c) Suppose that at time \(t=0\) the initial amount of \(\mathrm{A}\) present is \(a\), and that there is no \(\mathrm{B}\) or \(\mathrm{C}\) present. Explain why $$ [\mathrm{A}]=a-[\mathrm{B}] $$

Assume that the radius \(r\) and the surface area \(S=4 \pi r^{2}\) of a sphere are differentiable functions of \(t\). Express \(d S / d t\) in terms of \(d r / d t\).

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ h(x)=\sqrt[5]{3-x^{4}} $$

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=e^{2 x}\) at \(a=0\)

(a) Use the formal definition to find the derivative of \(y=\) \(x^{2}+1\) at \(x=1\) (b) Show that the point \((1,2)\) is on the graph of \(y=x^{2}+1\), and find the equation of the tangent line at the point \((1,+2)\). (c) Graph \(y=x^{2}+1\) and the tangent line at the point \((1,2)\) in the same coordinate system.

Find the derivative with respect to the independent variable. $$ f(x)=-5 \cos \left(2-x^{3}\right)+2 \cos ^{3}(x-4) $$