Chapter 4

In 1774 , Captain James Cook left 10 rabbits on a small Pacific island. The rabbit population is approximated by $$ P(t)=\frac{2000}{1+e^{5.3-0.4 t}} $$ with \(t\) measured in years since \(1774 .\) Using a calculator or computer: (a) Graph \(P\). Does the population level off? (b) Estimate when the rabbit population grew most rapidly. How large was the population at that time? (c) Find the inflection point on the graph and explain its significance for the rabbit population. (d) What natural causes could lead to the shape of the graph of \(P ?\)

You run a small furniture business. You sign a deal with a customer to deliver up to 400 chairs, the exact number to be determined by the customer later. The price will be \(\$ 90\) per chair up to 300 chairs, and above 300 , the price will be reduced by \(\$ 0.25\) per chair (on the whole order) for every additional chair over 300 ordered. What are the largest and smallest revenues your company can make under this deal?

Find the exact global maximum and minimum values of the function. The domain is all real numbers unless otherwise specified. $$ f(t)=\frac{t}{1+t^{2}} $$

Find constants \(a\) and \(b\) so that the minimum for the parabola \(f(x)=x^{2}+a x+b\) is at the given point. [Hint: Begin by finding the critical point in terms of \(a\). $$ (3,5) $$

(a) Water is flowing at a constant rate (i.e., constant volume per unit time) into a cylindrical container standing vertically. Sketch a graph showing the depth of water against time. (b) Water is flowing at a constant rate into a cone-shaped container standing on its point. Sketch a graph showing the depth of the water against time.

A warehouse selling cement has to decide how often and in what quantities to reorder. It is cheaper, on average, to place large orders, because this reduces the ordering cost per unit. On the other hand, larger orders mean higher storage costs. The warehouse always reorders cement in the same quantity, \(q\). The total weekly cost, \(C\), of ordering and storage is given by \(C=\frac{a}{q}+b q, \quad\) where \(a, b\) are positive constants. (a) Which of the terms, \(a / q\) and \(b q\), represents the ordering cost and which represents the storage cost? (b) What value of \(q\) gives the minimum total cost?

Find the exact global maximum and minimum values of the function. The domain is all real numbers unless otherwise specified. $$ f(t)=\left(\sin ^{2} t+2\right) \cos t $$

Find constants \(a\) and \(b\) so that the minimum for the parabola \(f(x)=x^{2}+a x+b\) is at the given point. [Hint: Begin by finding the critical point in terms of \(a\). $$ (-2,-3) $$

A business sells an item at a constant rate of \(r\) units per month. It reorders in batches of \(q\) units, at a cost of \(a+b q\) dollars per order. Storage costs are \(k\) dollars per item per month, and, on average, \(q / 2\) items are in storage, waiting to be sold. [Assume \(r, a, b, k\) are positive constants.] (a) How often does the business reorder? (b) What is the average monthly cost of reordering? (c) What is the total monthly cost, \(C\) of ordering and storage? (d) Obtain Wilson's lot size formula, the optimal batch size which minimizes cost.

Find the value(s) of \(x\) that give critical points of \(y=\) \(a x^{2}+b x+c\), where \(a, b, c\) are constants. Under what conditions on \(a, b, c\) is the critical value a maximum? A minimum?

Sketch several members of the family \(y=x^{3}-a x^{2}\) on the same axes. Discuss the effect of the parameter \(a\) on the graph. Find all critical points for this function.

(a) A cruise line offers a trip for \(\$ 2000\) per passenger. If at least 100 passengers sign up, the price is reduced for all the passengers by \(\$ 10\) for every additional passenger (beyond 100 ) who goes on the trip. The boat can accommodate 250 passengers. What number of passengers maximizes the cruise line's total revenue? What price does each passenger pay then? (b) The cost to the cruise line for \(n\) passengers is \(80,000+400 n .\) What is the maximum profit that the cruise line can make on one trip? How many passengers must sign up for the maximum to be reached and what price will each pay?

What value of \(w\) minimizes \(S\) if \(S-5 p w=3 q w^{2}-6 p q\) and \(p\) and \(q\) are positive constants?

A company manufactures only one product. The quantity, \(q\), of this product produced per month depends on the amount of capital, \(K\), invested (i.e., the number of machines the company owns, the size of its building, and so on) and the amount of labor, \(L\), available each month. We assume that \(q\) can be expressed as a Cobb-Douglas production function: $$ q=c K^{\alpha} L^{\beta} $$ where \(c, \alpha, \beta\) are positive constants, with \(0<\alpha<1\) and \(0<\beta<1 .\) In this problem we will see how the Russian government could use a Cobb-Douglas function to estimate how many people a newly privatized industry might employ. A company in such an industry has only a small amount of capital available to it and needs to use all of it, so \(K\) is fixed. Suppose \(L\) is measured in man-hours per month, and that each man-hour costs the company \(w\) rubles (a ruble is the unit of Russian currency). Suppose the company has no other costs besides labor, and that each unit of the good can be sold for a fixed price of \(p\) rubles. How many man-hours of labor per month should the company use in order to maximize its profit?

Find the value of \(a\) so that the function \(f(x)=x e^{a x}\) has a critical point at \(x=3\).

A company can produce and sell \(f(L)\) tons of a product per month using \(L\) hours of labor per month. The wage of the workers is \(w\) dollars per hour, and the finished product sells for \(p\) dollars per ton. (a) The function \(f(L)\) is the company's production function. Give the units of \(f(L) .\) What is the practical significance of \(f(1000)=400 ?\) (b) The derivative \(f^{\prime}(L)\) is the company's marginal product of labor. Give the units of \(f^{\prime}(L) .\) What is the practical significance of \(f^{\prime}(1000)=2 ?\) (c) The real wage of the workers is the quantity of product that can be bought with one hour's wages. Show that the real wage is \(w / p\) tons per hour. (d) Show that the monthly profit of the company is $$ \pi(L)=p f(L)-w L $$ (e) Show that when operating at maximum profit, the company's marginal product of labor equals the real wage: $$ f^{\prime}(L)=\frac{w}{p} $$

The energy expended by a bird per day, \(E\), depends on the time spent foraging for food per day, \(F\) hours. Foraging for a shorter time requires better territory, which then requires more energy for its defense. \({ }^{4}\) Find the foraging time that minimizes energy expenditure if $$ E=0.25 F+\frac{1.7}{F^{2}}. $$

(a) If \(b\) is a positive constant and \(x>0\), find all critical points of \(f(x)=x-b \ln x\) (b) Use the second derivative test to determine whether the function has a local maximum or local minimum at each critical point.c

If you have 100 feet of fencing and want to enclose a rectangular area up against a long, straight wall, what is the largest area you can enclose?

(a) For \(a\) a positive constant, find all critical points of \(f(x)=x-a \sqrt{x} .\) (b) What value of \(a\) gives a critical point at \(x=5\) ? Does \(f(x)\) have a local maximum or a local minimum at this critical point?

Find formulas for the functions described A function of the form \(y=b x e^{-a x}\) with a local maximum at \((3,6)\).

A closed box has a fixed surface area \(A\) and a square base with side \(x\). (a) Find a formula for its volume, \(V\), as a function of \(x\). (b) Sketch a graph of \(V\) against \(x\). (c) Find the maximum value of \(V\).

Let \(g(x)=x-k e^{x}\), where \(k\) is any constant. For what value(s) of \(k\) does the function \(g\) have a critical point?

On the west coast of Canada, crows eat whelks (a shellfish). To open the whelks, the crows drop them from the air onto a rock. If the shell does not smash the first time, the whelk is dropped again. \({ }^{5}\) The average number of drops, \(n\), needed when the whelk is dropped from a height of \(x\) meters is approximated by $$n(x)=1+\frac{27}{x^{2}} \text { . }$$ (a) Give the total vertical distance the crow travels upward to open a whelk as a function of drop height, \(x\). (b) Crows are observed to drop whelks from the height that minimizes the total vertical upward distance traveled per whelk. What is this height?

If \(a\) and \(b\) are nonzero constants, find the domain and all critical points of $$ f(x)=\frac{a x^{2}}{x-b} $$

During a flu outbreak in a school of 763 children, the number of infected children, \(I\), was expressed in terms of the number of susceptible (but still healthy) children, \(S\), by the expression $$I=192 \ln \left(\frac{S}{762}\right)-S+763 .$$ What is the maximum possible number of infected children?

Assume \(f\) has a derivative everywhere and has just one critical point, at \(x=3 .\) In parts \((a)-(d)\), you are given additional conditions. In each case decide whether \(x=3\) is a local maximum, a local minimum, or neither. Explain your reasoning. Sketch possible graphs for all four cases. (a) \(f^{\prime}(1)=3\) and \(f^{\prime}(5)=-1\) (b) \(f(x) \rightarrow \infty\) as \(x \rightarrow \infty\) and as \(x \rightarrow-\infty\) (c) \(f(1)=1, f(2)=2, f(4)=4, f(5)=5\) (d) \(f^{\prime}(2)=-1, f(3)=1, f(x) \rightarrow 3\) as \(x \rightarrow \infty\)

An apple tree produces, on average, \(400 \mathrm{~kg}\) of fruit each season. However, if more than 200 trees are planted per \(\mathrm{km}^{2}\), crowding reduces the yield by \(1 \mathrm{~kg}\) for each tree over 200 . (a) Express the total yield from one square kilometer as a function of the number of trees on it. Graph this function. (b) How many trees should a farmer plant on each square kilometer to maximize yield?

(a) On a computer or calculator, graph \(f(\theta)=\theta-\sin \theta\). Can you tell whether the function has any zeros in the interval \(0 \leq \theta \leq 1 ?\) (b) Find \(f^{\prime}\). What does the sign of \(f^{\prime}\) tell you about the zeros of \(f\) in the interval \(0 \leq \theta \leq 1 ?\)

The quantity of a drug in the bloodstream \(t\) hours after a tablet is swallowed is given, in \(\mathrm{mg}\), by $$q(t)=20\left(e^{-t}-e^{-2 t}\right)$$ (a) How much of the drug is in the bloodstream at time \(t=0 ?\) (b) When is the maximum quantity of drug in the bloodstream? What is that maximum? (c) In the long run, what happens to the quantity?

The quantity of a drug in the bloodstream \(t\) hours after a tablet is swallowed is given, in \(\mathrm{mg}\), by $$q(t)=20\left(e^{-t}-e^{-2 t}\right)$$ (a) How much of the drug is in the bloodstream at time \(t=0 ?\) (b) When is the maximum quantity of drug in the bloodstream? What is that maximum? (c) In the long run, what happens to the quantity?

As an epidemic spreads through a population, the number of infected people, \(I\), is expressed as a function of the number of susceptible people, \(S\), by \(I=k \ln \left(\frac{S}{S_{0}}\right)-S+S_{0}+I_{0}, \quad\) for \(k, S_{0}, I_{0}>0\) (a) Find the maximum number of infected people. (b) The constant \(k\) is a characteristic of the particular disease; the constants \(S_{0}\) and \(I_{0}\) are the values of \(S\) and \(I\) when the disease starts. Which of the following affects the maximum possible value of \(I ? \mathrm{Ex}-\) plain. \- The particular disease, but not how it starts. \- How the disease starts, but not the particular disease. \- Both the particular disease and how it starts.

The hypotenuse of a right triangle has one end at the origin and one end on the curve \(y=x^{2} e^{-3 x}\), with \(x \geq 0\). One of the other two sides is on the \(x\) -axis, the other side is parallel to the \(y\) -axis. Find the maximum area of such a triangle. At what \(x\) -value does it occur?

A person's blood pressure, \(p\), in millimeters of mercury \((\mathrm{mm} \mathrm{Hg})\) is given, for \(t\) in seconds, by $$p=100+20 \sin (2.5 \pi t)$$ (a) What are the maximum and minimum values of blood pressure? (b) What is the interval between successive maxima? (c) Show your answers on a graph of blood pressure against time.

A chemical reaction converts substance \(A\) to substance \(Y\); the presence of \(Y\) catalyzes the reaction. At the start of the reaction, the quantity of \(A\) present is \(a\) grams. At time \(t\) seconds later, the quantity of \(Y\) present is \(y\) grams. The rate of the reaction, in grams/sec, is given by Rate \(=k y(a-y), \quad k\) is a positive constant. (a) For what values of \(y\) is the rate nonnegative? Graph the rate against \(y\). (b) For what values of \(y\) is the rate a maximum?

In a chemical reaction, substance \(A\) combines with substance \(B\) to form substance \(Y\). At the start of the reaction, the quantity of \(A\) present is \(a\) grams, and the quantity of \(B\) present is \(b\) grams. At time \(t\) seconds after the start of the reaction, the quantity of \(Y\) present is \(y\) grams. Assume \(a