Chapter 5

Suppose that \(f(x)=x^{3}\). Explain why there exists a point \(c\) in the interval \((-1,1)\) such that \(f^{\prime}(c)=1\).

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow 0} \frac{\sqrt{x^{2}+x^{4}}-x}{x} $$

Assume that a is a positive constant. Find the general antiderivative of the given function. $$ f(x)=\frac{e^{(a+1) x}}{a} $$

We are not always given the function of interest in explicit form. In each of Problems \(41-44 y\) is related to \(x\) by an implicit equation. Determine using implicit differentiation and the first derivative test whether \(y\) is an increasing or a decreasing function of \(x\) \(x^{2}+y^{2}=1, \quad 00\)

Suppose that \(f(x)=x(2-x)\). Explain why there exists a point \(c\) in the interval \((0,2)\) such that \(f^{\prime}(c)=0\).

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

Assume that a is a positive constant. Find the general antiderivative of the given function. $$ f(x)=\sin ^{2}(a x+1) $$

We are not always given the function of interest in explicit form.y\( is related to \)x\( by an implicit equation. Determine using implicit differentiation and the first derivative test whether \)y\( is an increasing or a decreasing function of \)x\( \)x^{2}-y^{2}=1, \quad x>1, y>0$

Suppose that \(f(x)=x^{4}(5-x)\). Explain why there exists a point \(c\) in the interval \((0,5)\) such that \(f^{\prime}(c)=0\).

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

Assume that a is a positive constant. Find the general antiderivative of the given function. $$ f(x)=\frac{1}{a x+3} $$

We are not always given the function of interest in explicit form.y\( is related to \)x\( by an implicit equation. Determine using implicit differentiation and the first derivative test whether \)y\( is an increasing or a decreasing function of \)x\( \)\ln y=1-\frac{y}{x}, \quad x>0, y>0$

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

Assume that a is a positive constant. Find the general antiderivative of the given function. $$ f(x)=\frac{a}{a+x} $$

We are not always given the function of interest in explicit form.y\( is related to \)x\( by an implicit equation. Determine using implicit differentiation and the first derivative test whether \)y\( is an increasing or a decreasing function of \)x\( \)x y=e^{-y}, \quad x>0$

Assume that \(f\) is continuous on \([a, b]\) and differentiable on \((a, b) .\) Show that if \(f(a)

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

Assume that a is a positive constant. Find the general antiderivative of the given function. $$ f(x)=e^{a x} $$

Many fish join with others of their species to form schools, large groups that swim together in a coordinated way. It is thought that schooling helps the fish evade predators (predators are unable to pick out a single individual in the school to prey upon), and may also allow the fish to swim more efficiently by slip-streaming off each other. There is a lot of interest in how individual fish within the school interact to produce the complex swimming patterns seen in real schools. (a) Fish tend to prefer not to be too close or too far from their neighbors. D'Orsogna et al. (2006) propose that interactions between neighbors can be modeled by incorporating an energy of interaction \(U(r)\), that depends on the distance \(r\) between the fish. The force between the two fish can be derived from this energy from the formula: \(F(r)=-\frac{d U}{d r} .\) A positive force means that the fish repel each other, and a negative force means that they attract each other. D'Orsogna et al. assume the following form for the energy of interaction: $$ U(r)=a_{1} e^{-k_{1} r}-a_{2} e^{-k_{2} r} $$ Where \(a_{1}, a_{2}, k_{1}\), and \(k_{2}\) are all positive constants. Let's assume that for one particular species of fish \(a_{1}=3, a_{2}=2, k_{1}=2\), \(k_{2}=1 .\) Show that the fish repel each other when \(r<\ln 3\) and that they attract each other with \(r>\ln 3\). (b) Based on your answer to (a) why do you think that \(r=\ln 3\) is referred to as the equilibrium spacing of the fish? (c) In addition to maintaining their distance from each other, fish must also follow the movements of their neighbors.

. Assume 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(a)

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow \infty}\left(1+\frac{3}{x}\right)^{x} $$

Assume that a is a positive constant. Find the general antiderivative of the given function. $$ f(x)=\frac{e^{-a x}+e^{a x}}{2 a} $$

When trying to understand the processes by which proteins are organized through a cell, it is helpful to compare where the proteins are located in the cell to what would be expected if they were just placed at random (see for example \mathrm{\\{} C a m e r o n , ~ R o p e r ~ a n d ~ \(\mathrm{Zam}\) - bryski, 2012). One way to make this comparison is to measure the real distances between each protein and its nearest neighbor. For randomly placed proteins, the likelihood that two proteins are within distance \(d\) of each other is given approximately by a function: $$ P(d)=1-e^{-d / \mu}, \quad d \geq 0 $$ where \(\mu>0\) is a coefficient that depends on the size and geometry of the cell, and on how many proteins it contains. (a) Show that no matter what the value of \(\mu\) is, \(P(d)\) is an increasing function of \(d\). (b) Is \(P(d)\) concave up or concave down? Explain in words what \(P(d)\) being concave up or down means for the distance between proteins.

Assume that \(f\) is continuous on \([0,1]\) and differentiable on \((0,1)\). Assume that \(f^{\prime}(1 / 2)=0\), show by sketching the graph of a function \(f(x)\) that satisfies all of these conditions (you do not need to write down the equation of the function) that it is not necessary that \(f(0)=f(1)\).

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

Find the general solution of the differential equation. $$ \frac{d y}{d x}=\frac{2}{x}-x, x>0 $$

(Adapted from Reiss, 1989) Suppose that the rate at which body weight \(W\) changes with age \(t\) is $$ \frac{d W}{d t} \propto W^{a} $$ where \(a>0\) is a coefficient that takes different values for different species of animal. (a) The relative growth rate (percentage weight gained per unit of time) is defined as $$ G(W)=\frac{1}{W} \frac{d W}{d t} $$ Write down a formula for \(G(W) .\) For which values of \(a\) is the relative growth rate increasing, and for which values is it decreasing? (b) As fish grow larger, their weight increases each day but the relative growth rate decreases. If the rate of growth is described by \((5.7)\) explain what constraints must be imposed on \(a .\)

A car moves in a straight line. At time \(t\) (measured in seconds), its position (measured in meters) is $$ s(t)=\frac{1}{10} t^{2}, 0 \leq t \leq 10 $$ (a) Find its average velocity between \(t=0\) and \(t=10\). (b) Find its instantaneous velocity for \(t \in(0,10)\). (c) At what time is the instantaneous velocity of the car equal to its average velocity?

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

Find the general solution of the differential equation. $$ \frac{d y}{d x}=\frac{2}{x^{3}}-x^{3}, x>0 $$

The \(\mathrm{pH}\) value of a solution measures the concentration of hydrogen ions, denoted by \(\left[\mathrm{H}^{+}\right]\), and is defined as $$ \mathrm{pH}=-\log \left[\mathrm{H}^{+}\right] $$ Use calculus to decide whether the \(\mathrm{pH}\) value of a solution increases or decreases as the concentration of \(\mathrm{H}^{+}\) increases.

A car moves in a straight line. At time \(t\) (measured in seconds), its position (measured in meters) is $$ s(t)=\frac{1}{100} t^{3}, 0 \leq t \leq 10 $$ (a) Find its average velocity between \(t=0\) and \(t=10\). (b) Find its instantaneous velocity for \(t \in(0,10)\). (c) At what time is the instantaneous velocity of the car equal to its average velocity?

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

Find the general solution of the differential equation. $$ \frac{d y}{d x}=x(1+x), x>0 $$

Growth Allometric equations describe the scaling relationship between two measurements, such as tree height versus tree diameter or skull length versus backbone length. These equations are often of the form $$ Y=b X^{a} $$ where \(b\) is some positive constant and \(a\) is a constant that can be positive, negative, or zero. (a) Assume that \(X\) and \(Y\) are body measurements (and therefore positive) and that their relationship is described by an allometric equation of the form \((5.8) .\) For what values of \(a\) is \(Y\) an increasing function of \(X ?\) (b) For what values of \(a\) is \(Y\) an increasing function of \(X\) but \(Y / X\) is a decreasing function of \(X ?\) Is \(Y\) concave up or concave down in this case? (c) In vertebrates, we typically find [skull length \(] \propto\) [body length \(]^{a}\) for some \(a \in(0,1) .\) One measure of the animals' proportions is to calculate the ratio of skull length to total body length. Use your answer in (b) to explain what (5.9) means for the ratio of skull length to body length in juveniles versus adults. It may help to draw a picture!

Prof. Roper drives to work in stop-and-go traffic. His speed measured in miles per hour (mph) is given by the following function of time, \(t\), measured in minutes $$ v(t)=30+20 \sin (t / 5) $$ The total journey time is 1 hour. Explain using the MVT why the total distance that he travels in this hour is somewhere between 10 and 50 miles.

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

Find the general solution of the differential equation. $$ \frac{d y}{d x}=e^{x+1}, x>0 $$

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

Find the general solution of the differential equation. $$ \frac{d y}{d t}=t(1-t), t \geq 0 $$

Denote the size of a population at time \(t\) by \(N(t)\), and assume that \(N(0)=50\) and \(|d N / d t| \leq 20\) for all \(t \in[0,5] .\) What can you say about \(N(5) ?\) [Hint: Remember also that it is impossible for the number of organisms to become negative].

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

Find the general solution of the differential equation. $$ \frac{d y}{d t}=t^{2}\left(1+t^{2}\right), t \geq 0 $$

Denote the total biomass in a given area of soil at time \(t\) by \(B(t)\), and assume that \(B(0)=3\) and \(|d B / d t| \leq 1\) for all \(t \in[0,3]\). What can you say about \(B(3) ?\)

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

Find the general solution of the differential equation. $$ \frac{d y}{d t}=e^{-t / 2}, t \geq 0 $$

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

Find the general solution of the differential equation. $$ \frac{d y}{d t}=1-e^{-2 t}, t \geq 0 $$

Suppose that \(f\) is differentiable for all \(x \in \mathbf{R}\) with \(f(2)=3\) and \(f^{\prime}(x)=0\) for all \(x \in \mathbf{R}\). Find \(f(x)\).

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