Chapter 5

Suppose that \(f(x)\) is differentiable on \(\mathbf{R}\), with \(f(x)>0\) for \(x \in \mathbf{R}\). Show that if \(f(x)\) has a local maximum at \(x=c\), then \(g(x)=\ln f(x)\) also has a local maximum at \(x=c\).

Use the first derivative test and the second derivative test to determine where each function is increasing, decreasing, concave up, and concave down. You do not need to use a graphing calculator for these exercises. $$ y=\frac{1}{1+e^{-x}}, x \in \mathbf{R} $$

In Problems , find c such that \(f^{\prime}(c)=0\) and determine whether \(f(x)\) has a local extremum at \(x=c .\) $$ f(x)-(x-4)^{2} $$

Protein Binding We previously met Hill's function for describing cooperative binding of proteins to ligands (chemicals that have biological function) when we discussed the binding of hemoglobin to oxygen. If the concentration \(x\) of ligand is written in the right units, then Hill's function can be written in the form. $$f(x)=\frac{x^{k}}{1+x^{k}}, x \geq 0$$ where you should assume that \(k\) is a positive constant greater than 1 . (a) What are the roots of \(f(x)\) ? (b) Determine where \(f(x)\) is increasing and where it is decreasing. (c) Where is the function concave up and where is it concave down? Find all inflection points of \(f(x)\). (d) Find \(\lim _{x \rightarrow \infty} f(x)\) and decide whether \(f(x)\) has a horizontal asymptote. (e) Sketch the graph of \(f(x)\) together with its asymptotes and inflection points (if they exist). (f) Now assume that \(k\) is less than 1 . For definiteness, let \(k=1 / 2\). Is \(f(x)\) incrcasing or decrcasing? Show that \(f(x)\) still has a horizontal asymptote, but that it is concave down for all \(x \geq 0\), and use this information to make a sketch of \(f(x)\).

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow 0} \frac{e^{x}-1}{3 e^{2 x}-x} $$

Find the general antiderivative of the given function. $$ f(x)=2 e^{2 x} $$

Suppose that \(f(x)\) is differentiable on \(\mathbf{R}\). Show that if \(f(x)\) has a local maximum at \(x=c\), then \(g(x)=e^{f(x)}\) also has a local maximum at \(x=c\).

Sketch the graph of (a) a function that is increasing at an accelerating rate; and (b) a function that is increasing at a decelerating rate. (c) Assume that your functions in (a) and (b) are twice differentiable. Explain in each case how you could check the respective properties by using the first and the second derivatives. Which of the functions is concave up, and which is concave down?

In Problems , find c such that \(f^{\prime}(c)=0\) and determine whether \(f(x)\) has a local extremum at \(x=c .\) $$ f(x)=-x^{2} $$

We previously met the Michaelis-Menten rate function as a model for the rate at which a reaction occurs as a function of the concentration \(x\) of one of the reactants: $$f(x)=\frac{x}{a+x}, x \geq 0$$ where \(a\) is a positive constant. (a) Determine where \(f(x)\) is increasing and where it is decreasing. (b) Where is the function concave up and where is it concave down? Find all inflection points of \(f(x)\). (c) Find \(\lim _{x \rightarrow \infty} f(x)\) and decide whether \(f(x)\) has a horizontal asymptote. (d) Sketch the graph of \(f(x)\) together with its asymptotes and inflection points (if they exist). (e) Describe in words how the graph of the function \(f(x)\) changes if \(a\) is increased.

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow \infty} \frac{(\ln x)^{2}}{x^{2}} $$

A circular sector with radius \(r\) and angle \(\theta\) has area \(A\). Find \(r\) and \(\theta\) so that the perimeter is smallest when (a) \(A=2\) and (b) \(A=10\). (Note: \(A=\frac{1}{2} r^{2} \theta\), and the length of the arc \(s=r \theta\), when \(\theta\) is measured in radians; see Figure \(5.57\).)

A commonly used model for the density-dependent dynamics of a population is the recurrence equation: $$N_{t+1}=R N_{t}^{b}$$ where \(b, R>0\) are constants that take different values depending on the species of organism that is being modeled and its habitat. When \(b=1\), the model predicts that the population will grow exponentially. (a) Show that the population has a non-trivial equilibrium point (that is \(N \neq 0\) ), that you should determine. (b) Show that if \(-11\) ? Let \(b=2\) and \(R=1\), so \(N_{t+1}=N_{t}^{2}\). Find all of the equilibria of the recursion relation, and determine which (if any) are stable. (d) Calculate the first ten terms of the recursion relation \(N_{t+1}=\) \(N_{t}^{2}\) if (i) \(N_{0}=0.5\) and (ii) \(N_{0}=2 ?\) (e) If \(N_{t+1}=N_{t}^{2}\), what are the possible behaviors of the population as \(t \rightarrow \infty\) ?

Find the general antiderivative of the given function. $$ f(x)=-3 e^{-4 x} $$

Determine all inflection points. \(f(x)=x^{3}-2, x \in \mathbf{R}\)

Show that if \(f(x)\) is the linear function \(y=m x+b\) where \(m\) and \(b\) are constants, then increases in \(f(x)\) are proportional to increases in \(x .\) That is, suppose initially that \(x=x_{0}\), and \(y=y_{0}=\) \(m x_{0}+b .\) Then we increase \(x\) by \(\Delta x\) to \(x=x_{0}+\Delta x .\) Calculate the increase in \(y .\) Show that the increase in \(y\) depends on \(\Delta x\) but does not depend on \(x_{0}\). This means that the same increment in \(x\) always produces the same increment in \(y\), independently of the starting value of \(x\). Contrast this behavior with a concave down function.

In Problems , find c such that \(f^{\prime}(c)=0\) and determine whether \(f(x)\) has a local extremum at \(x=c .\) $$ f(x)=e^{-x^{2}} $$

The logistic function defined by: $$N(t)=\frac{1}{a+e^{-t}}, t \geq 0$$ represents the growth of a population. We will derive this function by solving a differential equation model in Chapter \(8 .\) Assume that \(a\) is a positive constant. (a) Determine where \(N(t)\) is increasing and where it is decreasing. (b) Find and classify any local extrema that the function has. (c) Where is the function concave up and where is it concave down? Find all inflection points of \(N(t)\). (d) Find \(\lim _{t \rightarrow \infty} N(t)\) and decide whether \(N(t)\) has a horizontal asymptote. (e) Sketch the graph of \(N(t)\) together with its asymptotes and inflection points (if they exist). (f) Describe in words how the graph of the function changes if \(a\) is increased.

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow \infty} \frac{x^{7}}{e^{x}} $$

A circular sector with radius \(r\) and angle \(\theta\) has area \(A .\) Find \(r\) and \(\theta\) so that the perimeter is smallest for a given area \(A .\) (Note: \(A=\frac{1}{2} r^{2} \theta\), and the length of the arc \(s=r \theta\), when \(\theta\) is measured in radians; see Figure 5.57.)

The Ricker model was introduced by Ricker (1954) as an alternative to the discrete logistic equation to describe the density-dependent growth of a population. Under the Ricker model the population \(N_{t}\) sampled at discrete times \(t=0,1,2, \ldots\) is modeled by a recurrence equation $$N_{t+1}=R_{0} N_{t} \exp \left(-a N_{t}\right)$$ where \(R_{0}\) and \(a\) are positive constants that will vary between different species and between different habitats. (a) Explain why you would expect \(R_{0}>1\) (Hint: consider the population growth when \(N_{t}\) is very small.) (b) Show that the recursion relation has two equilibria, a trivial equilibrium (that is, \(N=0\) ) and another equilibrium, which you should find. (c) Show that if \(R_{0}>1\) then use the stability criterion for equilibria to show that the trivial equilibrium point is unstable. (d) Use the stability criterion for equilibria to show that the nontrivial equilibrium point is stable if \(0<\ln R_{0}<2\). (e) If \(R_{0}>1\) then \(\ln R_{0}>0\), so most populations will meet the first inequality condition. What happens if \(\ln R_{0}>2 ?\) Let's try some explicit values: \(R_{0}=10, a=1, N_{0}=1 .\) Calculate the first ten terms of the sequence, and describe in words how the sequence behaves.

Find the general antiderivative of the given function. $$ f(x)=\frac{1}{e^{2 x}} $$

Determine all inflection points. \(f(x)=(x-3)^{5}, x \in \mathbf{R}\)

You must solve Problem 23 before attempting any of Problems \(24-26\) We frequently must solve equations of the form \(f(x)=0\). When \(f\) is a continuous function on \([a, b]\) and \(f(a)\) and \(f(b)\) have opposite signs, the intermediate-value theorem guarantees that there exists at least one solution of the equation \(f(x)=0\) in \([a, b]\). Explain in words why there exists exactly one solution in \((a, b)\) if. in addition, \(f\) is differentiable in \((a, b)\) and \(f^{\prime}(x)\) is either strictly positive or strictly negative throughout \((a, b)\).

In Problems , find c such that \(f^{\prime}(c)=0\) and determine whether \(f(x)\) has a local extremum at \(x=c .\) $$ f(x)=x^{3} $$

The Gompertz function is used in mathematical models for the rate of growth of certain tumors. The mass \(M(t)\) of a tumor described by Gompertz's equation changes with time according to: $$M(t)=\exp \left(a e^{-t}\right), \quad t \geq 0$$ where you may assume that \(a>0\) is a positive coefficient. (a) Determine where \(M(t)\) is increasing and where it is decreasing. (b) Find and classify any local extrema that the function has. (c) Where is the function concave up and where is it concave down? Find all inflection points of \(M(t)\). (d) Find \(\lim _{t \rightarrow \infty} M(t)\) and decide whether \(M(t)\) has a horizontal asymptote. (e) Sketch the graph of \(M(t)\) together with its asymptotes and inflection points (if they exist). (f) Describe in words how the graph of the function changes if \(a\) is increased.

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow 0} \frac{\sin x}{x} $$

Molecular Dynamics One popular model for the interactions between two molecules is the Leonard-Jones \(6-3\) potential. According to this model, the energy of interaction between two molecules that are distance \(r\) apart is given by a function: $$ V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}} r>0 $$ Molecules will attract or repel each other until they reach a distance that minimizes the function \(V(r)\). The coefficient \(A\) is a positive constant. (a) What is the behavior of \(V(r)\) as \(r \rightarrow 0 ?\) What is the behavior of \(V(r)\) as \(r \rightarrow+\infty\) ? (b) Explain why you expect there to be a value of \(r\) that minimizes \(V(r)\), and then calculate that value of \(r\) (it may help for your argument to determine the sign of \(V^{\prime}(r)\) for large \(\left.r\right)\). (c) Would you still expect there to be a spacing that minimizes \(V(r)\) if \(A\) were a negative number? Justify your answer.

Find the general antiderivative of the given function. $$ f(x)=\frac{3}{e^{-x}} $$

Determine all inflection points. \(f(x)=e^{-x^{2}}, x \geq 0\)

In Problems , find c such that \(f^{\prime}(c)=0\) and determine whether \(f(x)\) has a local extremum at \(x=c .\) $$ f(x)=e^{x^{3}} $$

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow 0} \frac{\sin x}{x^{2}} $$

Fish Schooling One model that is used for the interactions between animals, including fish in a school, is that the fish have an energy of interaction that is given by a Morse potential: $$ V(r)=e^{-r}-A e^{-a r} r>0 $$ The fish will attract or repel each other until they reach a distance that minimizes the function \(V(r) .\) The coefficients \(A\) and \(a\) are positive numbers. (a) Assume initially that \(a=1 / 2\) and \(A=1\), what is the behavior of \(V(r)\) as \(r \rightarrow 0\). What is the behavior of \(V(r)\) as \(r \rightarrow+\infty\) ? (b) Find the value of \(r\) that minimizes \(V(r)\). (c) Explain what happens to the spacing that minimizes the energy of interaction if \(a=1 / 2\) and \(A=4\) ?

Find the general antiderivative of the given function. $$ f(x)=\sin (2 x) $$

Determine all inflection points. \(f(x)=x e^{-x}, x \geq 0\)

In Problems , find c such that \(f^{\prime}(c)=0\) and determine whether \(f(x)\) has a local extremum at \(x=c .\) $$ f(x)=(x+1)^{2} $$

Find the general antiderivative of the given function. $$ f(x)=\cos (3 x) $$

Determine all inflection points. \(f(x)=\frac{x}{x+1}, x \geq 0\)

In Problems , find c such that \(f^{\prime}(c)=0\) and determine whether \(f(x)\) has a local extremum at \(x=c .\) $$ f(x)=(x+1)^{3} $$

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow \infty} x^{2} e^{-x} $$

The trait is said to be under-dominant if the fitness of individuals with two different genes is less than the fitness of individuals who have identical copies of the gene. Show that if \(w_{12}

If a patient takes ibuprofen every \(T\) hours, rather than every 6 hours then the concentration of ibuprofen in their blood one hour after each pill is taken (that is, after \(1,1+T, 1+2 T\), hours, and so on) is given by a recurrence equation: $$C_{n+1}=(0.7575)^{T} C_{n}+40$$ (a) Find the equilibrium point of this recurence equation, and show that it is locally stable for any value of \(T>0\). (b) Assume that \(T=1\) and \(C_{1}=40 .\) Make a cobweb plot to illustrate the behavior of the sequence \(C_{1}, C_{2}, C_{3}, \ldots .\)

Find the general antiderivative of the given function. $$ f(x)=\sin \left(\frac{x}{3}\right)+\cos \left(\frac{x}{3}\right) $$

Determine all inflection points. \(f(x)=\frac{x^{2}}{x^{2}+1}\)

Suppose that \(f\) is continuous on \([a, b]\) and differentiable on \((a, b) .\) Show that if \(f^{\prime}(x)<0\) for all \(x \in(a, b)\), then \(f\) is decreasing on \([a, b]\).

. Show that \(f(x)=|x|\) has a local minimum at \(x=0\) but \(f(x)\) is not differentiable at \(x=0\).

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow \infty} x^{5} e^{-x} $$

Find the general antiderivative of the given function. $$ f(x)=\cos \left(\frac{x}{5}\right)-\sin \left(\frac{x}{5}\right) $$

Determine all inflection points. \(f(x)=\tan x,-\frac{\pi}{2}

Show that \(f(x)=|x-1|\) has a local minimum at \(x=1\) but \(f(x)\) is not differentiable at \(x=1\).