Chapter 8

Let \(P(t)=\sqrt{t}\) and \(a\) be a positive number. Give reasons for the following steps to find \(P^{\prime}(a)\). $$ \begin{aligned} P^{\prime}(a) &=\lim _{b \rightarrow a} \frac{P(b)-P(a)}{b-a} \\ &=\lim _{b \rightarrow a} \frac{\sqrt{b}-\sqrt{a}}{b-a} \\ &=\lim _{b \rightarrow a} \frac{1}{\sqrt{b}+\sqrt{a}} \\ &=\frac{1}{\sqrt{a}+\sqrt{a}}=\frac{1}{2 \sqrt{a}} \end{aligned} $$

Exercise 8.6 .1 Suppose 100 fish in a lake are caught, marked, and returned to the lake. Suppose that ten days later 100 fish are caught, among which 5 were marked. How many fish would you estimate were in the lake? Another method to estimate population size is the 'Catch per Unit Effort Method', based on the number of whale caught per day of hunting. As the population decreases, the catch per day of hunting decreases. In the Report of the International Whaling Commission (1978), J. R. Beddington refers to the following model of Sei whale populations. $$ N_{t+1}=0.94 N_{t}+N_{t-8}\left[0.06+0.0567\left\\{1-\left(\frac{N_{t-8}}{N_{*}}\right)^{2.39}\right\\}\right]-0.94 C_{t} $$ \(N_{t}, N_{t+1}\) and \(N_{t-8}\) represent the adult female whale population subjected to whale harvesting in years \(t, t+1,\) and \(t-8,\) respectively. \(C_{t}\) is the number of female whales harvested in year \(t\). There is an assumption that whales reach sexual maturity and are able to reproduce at eight years of age and become subject to harvesting the same year that they reach sexual maturity. The whales of age less than 8 years are not included in \(N_{t} . N_{*}\) is the number of female whales that the environment would support with no harvesting taking place.

Try to solve \(x e^{-x}=0.2\) by iteration of \(x_{n+1}=0.2 e^{x_{n}}\) beginning with \(x_{0}=2.5 .\) This is easily done if your calculator has the ANS key. Enter 2.5. Then type \(0.2 \times e^{\text {ANS }}\), ENTER, ENTER, \(\cdots\). Describe the result. Try again with \(x_{0}=2.6 .\) Describe the result. An alternate procedure is to solve for \(x\) as follows. $$ \begin{aligned} x e^{-x} &=0.2 \\ e^{-x} &=\frac{0.2}{x} \\ -x &=\ln \frac{0.2}{x} \\ x &=-\ln \frac{0.2}{x} \quad=\quad \ln (5 x) \end{aligned} $$ Now let \(x_{0}=2.5\) and iterate \(x_{n+1}=\ln \left(5 x_{n}\right)\) and describe your results.

A pebble dropped into a pond makes a circular wave that travels outward at a rate 0.4 meters per second. At what rate is the area of the circle increasing 2 seconds after the pebble strikes the pond?

Find the maximum value of \(H(V)=2 V V^{-1 / 3} /\left(V+V^{-1 / 3}\right)\) for \(V>0\).

Suppose you have a 3 meter by 4 meter sheet of tin and you wish to make a box that has tin on the bottom and on two opposite sides. The other two sides are of wood that is in plentiful supply. You are going to make a rectangular box by folding up panels of width \(x\) across the ends that are 3 meters wide as shown in Figure \(8.2 .1 .\) What value of \(x\) will maximize the volume of the box and what is the volume?

Argue that \(e^{x}\) is an increasing function.

Suppose \(P(t)=t^{4}\). a. Write an expression for \((P(b)-P(a)) /(b-a)\), the difference quotient of \(P\) on the interval, \([a, b] .\) b. Simplify your expression. c. Use your simplified expression to show that the rate of change of \(P\) at \(a\) is \(4 a^{3}\).

Show that if there is no harvest \(\left(C_{t}=0\right)\) and both \(N_{t}\) and \(N_{t-8}\) are equal to \(N_{*}\) then \(N_{t+1}=N_{*}\) If we divide all terms of Equation 8.14 by \(N_{*},\) we get $$ \frac{N_{t+1}}{N_{*}}=0.94 \frac{N_{t}}{N_{*}}+\frac{N_{t-8}}{N_{*}}\left[0.06+0.0567\left\\{1-\left(\frac{N_{t-8}}{N_{*}}\right)^{2.39}\right\\}\right]-0.94 \frac{C_{t}}{N_{*}} $$ We might define new variables, \(D_{t}=\frac{N_{t}}{N_{*}}\) and \(E_{t}=\frac{C_{t}}{N_{*}}\) and have the equation $$ D_{t+1}=0.94 D_{t}+D_{t-8}\left[0.06+0.0567\left\\{1-D_{t-8}^{2.39}\right\\}\right]-0.94 E_{t} $$ Equation 8.15 is simpler by one parameter \(\left(N_{*}\right)\) than Equation 8.14 and yet illustrates the same dynamical properties. Rather than use new variables, it is customary to simply rewrite Equation 8.14 with new interpretations of \(N_{t}\) and \(C_{t}\) and obtain $$ N_{t+1}=0.94 N_{t}+N_{t-8}\left[0.06+0.0567\left\\{1-N_{t-8}^{2.39}\right\\}\right]-0.94 C_{t} $$ \(N_{t}\) now is a fraction of \(N_{*},\) the number supported without harvest, and \(C_{t}\) is a fraction of \(N_{*}\) that is harvested. Equation 8.14 says that the population in year \(t+1\) is affected by three things: the number of female whales in the previous year \(\left(N_{t}\right),\) the "recruitment" of eight year old female whales into the population subject to harvest, and the harvest during the previous year \(\left(C_{t}\right)\)

A boat is pulled toward a dock by a rope through a pulley that is 5 meters above the water. The rope is being pulled at a constant rate of 15 meters per minute. At the instant when the boat is 12 meters from the dock, how fast is the boat approaching the dock?

The island body size rule states that when an ecosystem becomes isolated on an island, say by rising sea levels, the species of large body size tend to evolve to a smaller body size, and species of small body size tend to increase in size. Craig McClain et al \(^{2}\) found a similar contrast in the sea between shallow water where nutrient levels are high and deep water (depth greater than 200 meters) where nutrient levels are low. Gastropod genera that have large shallow-water species tend to have smaller deep-water representatives; those that have small shallow water species tend to have larger deep-water species. Suppose the ratio of nutrient conversion to body size \(v s\) body size is similar to the graph in Figure 8.3 .2 and that at high nutrient concentrations, nutrient conversion is not the limiting factor predation and mate finding, for example, may be more important. Suppose further that a low nutrient concentrations, nutrient conversion becomes more important and nutrient conversion/body size must be greater than that for high nutrient levels. How is the graph consistent with the observed differences in body size?

Argue that \(\sin x\) is an increasing function on \(0 \leq x \leq \pi / 2\).

Use the definition of rate of change to find the rate of change of \(P(t)=\frac{1}{t}\) at \(a=5\). Repeat for \(a\) unspecified. Complete the formula $$ P(t)=\frac{1}{t} \quad \Rightarrow \quad P^{\prime}(t)= $$ The functions \(F\) and \(G\) of the next two exercises present interesting challenges.

Begin with \(x_{0}=0.25917110182\) and compute the iteration steps \(\left(x_{n+1}=x_{n} e^{-x_{n}}-0.2\right) .\) Describe your results.

Corn is conveyed up a belt at the rate of \(10 \mathrm{~m}^{3}\) per minute and dropped onto a conical pile. The height of the pile is equal to twice its radius. At what rate is the top of the pile increasing when the volume of the pile is \(1000 \mathrm{~m}^{3} ?\) (Note: Volume of a cone is \(\frac{1}{3} \pi r^{2} h\) where \(r\) is the radius of the base and \(h\) is the height of the cone.)

Exercise 8.3 .3 Some small song birds intermittently flap their wings and glide with wings folded between flapping sessions. Why? R. M. Alexander \(^{3}\) suggests the following analysis. The power required to propel an airplane at a speed \(u\) is $$ P=A u^{3}+B L^{2} / u $$ where \(A\) and \(B\) are constants specific to the airplane and \(L\) is the upward force that lifts the plane. \(A u^{3}\) represents the drag on the airplane due largely to the air striking the front of the craft. a. For what speed is the required power the smallest? b. The energy required to propel the airplane is \(E=P / u\). For what speed is the energy required to propel the plane the smallest. c. How does the speed of minimum energy compare with the speed of minimum power? d. Suppose a bird has drag coefficient \(A_{b}\) with wings folded and \(A_{b}+A_{w}\) with wings extended and flapping, and let \(x\) be the fraction of time the bird spends flapping its wings. Suppose that the speed of the bird while flapping its wings is the same as the speed when the wings are folded and that all of the lift is provided when the wings are flapping. The lift over one complete cycle should be $$(1-x) L_{\text {folded }}+x L_{\text {flapping }}=x L_{\text {flapping }}=m g$$ where \(m\) is the mass of the bird. Then the power required while flapping is $$ P_{\text {flapping }}=\left(A_{b}+A_{w}\right) u^{3}+B\left(\frac{m g}{x}\right)^{2} \frac{1}{u} $$ Write an expression for \(P_{\text {folded }}\). e. The average power over a whole cycle should be $$ \bar{P}=(1-x) P_{\text {folded }}+x P_{\text {flapping }}=A_{b} u^{3}+x A_{w} u^{3}+B \frac{m^{2} g^{2}}{x u} $$ Find the value of \(x\) for which the average power over the whole cycle is minimum. f. The average energy over a whole cycle is \(\bar{E}=\bar{P} / u\). For what value of \(x\) is the average energy a minimum? This problem is continued in Exercise 13.2 .12 Alexander further notes that it may be necessary to consider also the efficiency of muscle contraction at different flapping rates.

Suppose penicillin concentration is given by \(C(t)=8 e^{-0.2 t}-8 e^{-0.4 t} \mu \mathrm{gm} / \mathrm{ml} t\) hours after ingestion of a penicillin pill. For what time period is the concentration increasing? What is the maximum penicillin concentration?

A light house beacon makes one revolution every two minutes and shines a beam on a straight shore that is one kilometer from the light house. How fast is the beam of light moving along the shore when it is pointing toward the point of the shore closest to the light house? How fast is the beam of light moving along the shore when it is pointing toward a point that is one kilometer from the closest point of the shore to the light house?

Suppose penicillin concentration is given by \(C(t)=0.4 t e^{-0.5 t} \mu \mathrm{gm} / \mathrm{ml} t\) hours after ingestion of a penicillin pill. For what time period is the concentration decreasing? What is the maximum penicillin concentration?

Two planes are traveling at the same altitude toward an airport. One plane is flying at 500 kilometers per hour from a position due North of the airport and the other plane is traveling at 300 kilometers per hour from a position due East of the airport. At what rate is the distance between the planes decreasing when the first plane is \(8 \mathrm{~km}\) North of the airport and the second plane is \(5 \mathrm{~km}\) East of the airport?

Exercise 8.3.5 Sickle cell anemia is an inherited blood disease in which the body makes sickle-shaped red blood cells. It is caused by a single mutation from glutamic acid to valine at position 6 in the protein Hemoglobin B. The gene for hemoglobin \(\mathrm{B}\) is on human chromosome \(11 ;\) a single nucleotide change in the codon for the glutamic acid, GAG, to GTG causes the change from glutamic acid to valine. The location of a genetic variation is called a locus and the different genetic values (GAG and GTG) at the location are called alleles. People who have GAG on one copy of chromosome 11 and GTG on the other copy are said to be heterozygous and do not have sickle cell anemia and have elevated resistance to malaria over those who have GAG on both copies of chromosome 11 . Those who have GTG on both copies of chromosome 11 are said to be homozygous and have sickle cell anemia \(-\) the hydrophobic valine allows aggregation of hemoglobin molecules within the blood cell causing a sickle- like deformation that does not move easily through blood vessels. Let \(A\) denote presence of \(\mathrm{GAG}\) and \(a\) denote presence of GTG on chromosome \(11,\) and let \(A A,\) \(A a\) and \(a a\) denote the various presences of those codons on the two chromosomes of a person (note: \(A a=a A) ; A A, A a\) and aa label are the genotypes of the person with respect to this locus. It is necessary to assume non-overlapping generations, meaning that all members of the population are simultaneously born, grow to sexual maturity, mate, leave offspring and die. Let \(P, Q\) and \(R\) denote the frequencies of \(A A, A a\) and aa genotypes in a breeding population and let \(p\) and \(q\) denote the frequencies of the alleles \(A\) and \(a\) among the chromosomes in the same population. The frequencies \(P, Q,\) and \(R\) are referred to as genotype frequencies and \(p\) and \(q\) are referred to as allele frequencies. In a population of size, \(N\), there will be \(2 N\) chromosomes and \(P \times 2 N+Q \times N\) of the chromosomes will be \(A\). In a mating of \(A A\) with \(A a\) adults, the chromosome in the fertilized egg (zygote) obtained from \(A A\) must be \(A\) and the chromosome obtained from \(A a\) will be \(A\) with probability \(1 / 2\) and will be \(a\) with probability \(1 / 2\). Therefore, the zygote will be \(A A\) with probability \(1 / 2\) and will be \(A a\) with probability \(1 / 2\) a. Show that the allele frequencies \(p\) and \(q\) in a breeding population with genotype frequencies \(P, Q\) and \(R\) are given by $$ p=P+\frac{1}{2} Q $$ and $$ q=\frac{1}{2} Q+R $$ b. Assume a closed population (no migration) with random mating and no selection. Complete the table showing probabilities of zygote type in the offspring for the various mating possibilities, the frequencies of the mating possibilities, and the zygote genotype frequencies. Include zeros with the zygote type probabilities but omit the zeros in the zygote genotype frequencies. Random mating assumes that the selection of mating partners is independent of the genotypes of the partners. c. When the table is complete, you should see that $$ \begin{aligned} \Sigma_{A a} &=\frac{1}{2} P Q+P R+\frac{1}{2} Q P+\frac{1}{2} Q^{2}+\frac{1}{2} Q R+R P+\frac{1}{2} Q R \\ &=P Q+2 P R+\frac{1}{2} Q^{2}+Q R=2 P\left(\frac{1}{2} Q+R\right)+Q\left(\frac{1}{2} Q+R\right) \\ &=(2 P+Q)\left(\frac{1}{2} Q+R\right) \quad=\quad 2\left(P+\frac{1}{2} Q\right)\left(\frac{1}{2} Q+R\right) \\ &=2 p q \end{aligned} $$ Show that $$ \Sigma_{A A}=p^{2} \quad \text { and } \quad \Sigma_{a a}=q^{2} $$ This means that under the random mating hypothesis, the zygote genotype frequencies of the offspring population are determined by the allele frequencies of the adults. This is referred to as the Hardy-Weinberg theorem. If the probability of an egg growing to adult and contributing to the next generation of eggs is the same for all eggs, independent of genotype, then the allele frequencies, \(p\) and \(q,\) are constant after the first generation. Random mating does not imply the promiscuity that might be imagined. It means that the selection of mating partner is independent of the genotype of the partner. In the United States, blood type would be a random mating locus; seldom does a United States young person inquire about the blood type of an attractive partner. In Japan, however, this seems to be a big deal, to the point that dating services arranging matches also match blood type. The major histocompatibility complex (MHC) of a young person would seem to be fairly neutral; few people even know their MHC type. It has been demonstrated, however, that young women are repulsed by the smell of men of the same MHC type as their own \(^{4}\). d. Show that in a closed random mating population with no selection, if the frequency of \(A\) in the adults in one generation is \(\hat{p},\) then the frequency of \(A\) in adults in the next generation will also be \(\hat{p}\). e. Suppose that because of malaria, an \(A A\) type egg, either male or female, has probability 0.8 of reaching maturity and mating and because of sickle cell anemia an aa type has only 0.2 probability of mating, but that an \(A a\) type has 1.0 probability of mating. This condition is called selection. Then the distribution of genotypes in the egg and the mating populations will be \(\begin{array}{lccc}\text { Genotype } & A A & A a & a a \\ \text { Egg } & p^{2} & 2 p q & q^{2} \\ \text { Adult } & 0.8 p^{2} / F & 2 p q / F & 0.2 q^{2} / F\end{array}\) $$ \text { where } \quad F=0.8 p^{2}+2 p q+0.2 q^{2} $$ Find the frequency of \(A\) in the adult population. Note: This will also be the frequency of \(A\) in the next egg population. f. We call \(F(p)\) the balance of the population, and because \(p+q=1\) $$ F=F(p)=0.8 p^{2}+2 p(1-p)+0.2(1-p)^{2} $$ You will be asked in Exercise 8.3 .8 to show that when the probability of reproduction depends on the genotype (selection is present), during succeeding generations, allele frequency, \(p,\) moves toward the value of local maximum of \(F\). 1\. Show that \(F(p)=1-0.2 p^{2}-0.8(1-p)^{2}\). 2\. Find the value \(\hat{p}\) of \(p\) that maximizes \(F(p)\).

Identify the intervals, if any, on which \(f(x)\) is increasing and intervals, if any, on which \(f^{\prime}\) is increasing. a. \(f(x)=x^{2}-1 \quad-2 \leq x \leq 2\) b. \(f(x)=x^{2}-x \quad-2 \leq x \leq 2\) c. \(f(x)=x^{3}-x^{2} \quad-2 \leq x \leq 2\) d. \(f(x)=\frac{x}{x^{2}+1} \quad-2 \leq x \leq 2\) e. \(f(x)=e^{-x} \quad-2 \leq x \leq 2\) f. \(f(x)=x e^{-x} \quad 0 \leq x \leq 3\) g. \(f(x)=e^{-x^{2}} \quad-2 \leq x \leq 2\) h. \(\quad f(x)=e^{-2 x}-e^{-x} \quad 0 \leq x \leq 4\) i. \(f(x)=\ln x\) \(0

Consider two alleles \(A\) and \(a\) at a locus of a random mating population and the fractions of \(A A, A a\) and aa zygotes that reach maturity and mate are in the ratio \(1+s_{1}: 1: 1+s_{2}\) where \(s_{1}\) and \(s_{2}\) can be positive, negative, or zero, but \(s_{1} \geq-1\) and \(s_{2} \geq-1 .\) The balance function is $$ \begin{aligned} F(p)=\left(1+s_{1}\right) p^{2}+2 p q+\left(1+s_{2}\right) q^{2} &=\left(1+s_{1}\right) p^{2}+2 p(1-p)+\left(1+s_{2}\right)(1-p)^{2} \\ &=1+s_{1} p^{2}+s_{2}(1-p)^{2} \end{aligned} $$ where \(p\) and \(q\) are the frequencies of \(A\) and \(a\) among the zygotes. a. Sketch the graphs of \(F\) and find the values \(\hat{p}\) of \(p\) in [0,1] for which \(F(p)\) is a maximum for 1\. \(s_{1}=0.2\) and \(s_{2}=-0.3\). 2\. \(s_{1}=0\) and \(s_{2}=-0.2\) 3\. \(s_{1}=-0.2\) and \(s_{2}=-0.3\). 4\. \(s_{1}=0.2\) and \(s_{2}=0.3\) b. Suppose that \(s_{1}+s_{2} \neq 0\) and \(0 \leq s_{2} /\left(s_{1}+s_{2}\right) \leq 1 .\) Is it true that \(\hat{p} s_{2} /\left(s_{1}+s_{2}\right)\) is the value of \(p\) in [0,1] for which \(F(p)\) is a maximum?

A woman 1.7 meters tall walks under a street light that is 10 meters above the ground. She is walking in a straight line at a rate of 30 meters per minute. How fast is the tip of her shadow moving when she is 5 meters beyond the street light?

Find the area of the largest rectangle that can be inscribed in a right triangle with sides of length 3 and 4 and hypotenuse of length 5 .

Trout, Moose, and Bear lakes are connected into a chain by a stream that runs into Trout Lake, out of Trout Lake into Moose Lake, out of Moose Lake and into Bear Lake and out of Bear Lake. The volumes of all of the three lakes are the same, and stream flow is constant into and out of all lakes. A load of waste is dumped into Trout Lake. With \(t\) measured in days and concentration measured in \(\mathrm{mg} / \mathrm{l}\), the concentration of wastes in the three lakes is projected to be $$ \begin{array}{ll} \text { Trout Lake: } & C_{\mathrm{T}}(t)=0.01 e^{-0.05 t} \\ \text { Moose Lake: } & C_{\mathrm{M}}(t)=0.005 t e^{-0.05 t} \\ \text { Bear Lake: } & C_{\mathrm{B}}(t)=0.000025 \frac{t^{2}}{2} e^{-0.05 t} \end{array} $$ For each lake, find the time interval, it any, on which the concentration in the lake is increasing.

Ricker's equation for population growth with proportional harvest is presented in Exercise 14.3 .4 as $$ P_{t+1}-P_{t}=\alpha P_{t} e^{-P_{t} / \beta}-R P_{t} $$ If a fixed number is harvested each time period, the equation becomes $$ P_{t+1}-P_{t}=\alpha P_{t} e^{-P_{t} / \beta}-H $$ For the parameter values \(\alpha=1.2, \beta=3\) and \(H=0.1,\) calculate the positive equilibrium value of \(P_{t}\).

A gas in a perfectly insulated container and at constant temperature satisfies the gas law \(p v^{1.4}=\) constant. When the pressure is 20 Newtons per \(\mathrm{cm}^{2}\) the volume is 3 liters. The gas is being compressed at the rate of 0.2 liters per minute. How fast is the pressure changing at the instant at which the volume is 2 liters?

The program shown in Exercise 8.3 .7 assumes a population with an A-allele frequency of \(\mathrm{p}=0.2\) and computes future populations assuming that there is selection pattern AA: \(0.7 ;\) Aa: \(1.0 ;\) aa: \(0.5 .\) The heterozygote, \(\mathrm{Aa},\) is favored over either of the two homozygotes, \(\mathrm{AA}\) and aa; the condition is referred to as over dominance and occurs in sickle cell anemia. The A-allele frequency of \(\mathrm{p}=0.2\) is out of balance with the selection forces acting on the population, and in subsequent generations p moves toward the value \(\hat{p}\) that maximizes the balance function, $$ \begin{aligned} F(p)=\left(1+s_{1}\right) p^{2}+2 p q+\left(1+s_{2}\right) q^{2} &=\left(1+s_{1}\right) p^{2}+2 p(1-p)+\left(1+s_{2}\right)(1-p)^{2} \\ &=1+s_{1} p^{2}+s_{2}(1-p)^{2} \end{aligned} $$ This exercise proves that it always happens in the case of over dominance. a. Consider two alleles \(A\) and \(a\) at a locus of a random mating population with non-overlapping generations and the fractions of \(A A, A a\) and aa zygotes that reach maturity and mate are in the ratio \(1+s_{1}: 1: 1+s_{2}\) where \(-1 \leq s_{1}<0\) and \(-1 \leq s_{2}<0 .\) Show that the maximum value of \(F(p)=1+s_{1} p^{2}+s_{2}(1-p)^{2} \quad\) occurs at \(\quad \hat{p}=\frac{s_{2}}{s_{1}+s_{2}} \quad\) and \(\quad F(\hat{p})=1+\frac{s_{1} s_{2}}{s_{1}+s_{2}}<1\) b. Assume the egg allele frequencies in the first generation are \(A \quad p_{0}\) and \(a \quad q_{0}=1-p_{0}\) and that the egg genotype frequencies are \(A A \quad p_{0}^{2}, A a \quad 2 p_{0} q_{0}\) and \(a a \quad q_{0}^{2}\). After selection the adult genotype frequencies are $$ \begin{array}{ccc} A A & A a & a a \\ \frac{\left(1+s_{1}\right) p_{0}^{2}}{F\left(p_{0}\right)} & \frac{2 p_{0} q_{0}}{F\left(p_{0}\right)} & \frac{\left(1+s_{2}\right) q_{0}^{2}}{F\left(p_{0}\right)} & \text { where } F\left(p_{0}\right)=1+s_{1} p_{0}^{2}+s_{2}\left(1-p_{0}\right)^{2} \end{array} $$ Show that the frequency, \(p_{1}\) of \(A\) in the adult population (and therefore of the resulting egg population) is $$ p_{1}=\frac{\left(1+s_{1}\right) p_{0}^{2}+p_{0}\left(1-p_{0}\right)}{F\left(p_{0}\right)} $$ c. For the \(n^{\text {th }}\) generation $$ p_{n+1}=\frac{\left(1+s_{1}\right) p_{n}^{2}+p_{n}\left(1-p_{n}\right)}{F\left(p_{n}\right)} \quad \text { where } \quad F\left(p_{n}\right)=1+s_{1} p_{n}^{2}+s_{2}\left(1-p_{n}\right)^{2} $$ It is the sequence \(\left\\{p_{0}, p_{1}, p_{2}, \cdots\right\\}\) that we wish to show converges to \(\hat{p}\).

Find \(x^{\prime}\) at the instant that \(x=2\) if \(y^{\prime}=5\) and a. \(x y+y=9\) b. \(x e^{y}=2 e\) c. \(x^{2} y+x y^{2}=2\)

A dog kennel with four pens each of area 7 square meters is to be constructed. An exterior fence surrounding a rectangular area is to be built of fence costing $$\$ 20$$ per meter. That rectangular area is then to be partitioned by three fences that are all parallel to a single side of the original rectangle and using fence that costs $$\$ 10$$ per meter. What dimensions of pens will minimize the cost of fence used?

A point is moving along the parabola \(y=x^{2}\) and its \(y\) coordinate increases at a constant rate of \(2 .\) At what rate is the distance from the point to (4,0) changing at the instant at which \(x=2 ?\)

A ladder is to be put against a wall that has a 2 meter tall fence that is 1 meter away from the wall. What is the shortest ladder that will reach from the ground to the wall and go above the fence?

Accept as true that if a particle moves along a graph of \(y=f(x)\) then the speed of the particle along the graph is \(\sqrt{\left(x^{\prime}\right)^{2}+\left(y^{\prime}\right)^{2}}\). (The notation means that \(x^{\prime}\) is the rate at which the \(x\) -coordinate is increasing, \(y^{\prime}\) denotes the rate at which the \(y\) -coordinate is increasing and the speed of the particle is the rate at which the particle is moving along the curve.) a. Suppose a particle moves along the graph of \(y=x^{2}\) so that \(y^{\prime}=2 .\) What is \(x^{\prime}\) when \(y=2 ?\) How fast is the particle moving? b. Suppose a particle moves along the circle \(x^{2}+y^{2}=1\) so that its speed is \(2 \pi\). Find \(x^{\prime}\) when \(x=1\) c. Suppose a particle moves along the circle \(x^{2}+y^{2}=1\) so that its speed is \(2 \pi\). Find \(x^{\prime}\) when \(x=0 .\) (Two answers.)

a. What is the length of the longest ladder than can be carried horizontally along a 2 meter wide hallway and turned a corner into a 1 meter hallway? Suppose the floor to ceiling height in both hallways is 3 meters. b. What is the longest pipe that can be carried around the corner?

A box with a square base and with a top and bottom and a shelf entirely across the interior is to be made. The total surface area of all material is to be \(9 \mathrm{~m}^{2}\). What dimensions of the box will maximize the volume?

A rectangular box with square base and with top and bottom and a shelf entirely across the interior is to have \(12 \mathrm{~m}^{3}\) volume. What dimensions of the box will minimize the material used?

A box with no top is to be made from a \(22 \mathrm{~cm}\) by \(28 \mathrm{~cm}\) piece of card board by cutting squares of equal size from each corner and folding up the 'tabs'. What size of squares should be cut from each corner to make the box of largest volume?

An orange juice can has volume of \(48 \pi \mathrm{cm}^{3}\) and has metal ends and cardboard sides. The metal costs 3 times as much as the card board. What dimensions of the can will minimize the cost of material?

The 'strength' of a rectangular wood beam with one side vertical is proportional to its width and the square of its depth. A sawyer is to cut a single beam from a 1 meter diameter log. What dimensions should he cut the beam in order to maximize the strength of the beam?

A rectangular wood beam with one side vertical has a 'stiffness' that is proportional to its width and the cube of its depth. A sawyer is to cut a single beam from a 1 meter diameter log. What dimensions should he cut the beam in order to maximize the stiffness of the beam?

A life guard at a sea shore sees a swimmer in distress 70 meters down the beach and 30 meters from shore. She can run 4 meters/sec and swim 1 meter per second. What path should she follow in order to reach the swimmer in minimum time?

You stand on a bluff above a quiet lake and observe the reflection of a mountain top in the lake. Light from the mountain top strikes the lake and is reflected back to your eye, the path followed, by Fermat's hypothesis, being that path that takes the least time. Show that the angle of incidence is equal to the angle of reflection. That is, show that the angle the beam from the mountain top to the point of reflection on the lake makes with the horizontal surface of the lake (the angle of incidence) is equal to the angle the beam from the point of reflection to your eye makes with the horizontal surface of the lake (the angle of reflection). Let \(v_{a}\) denote the velocity of light in air.

Two light bulbs of different intensities are a distance, \(d,\) apart. At any point, the light intensity from one of the bulbs is proportional to the intensity of the bulb and inversely proportional to the square of the distance from the bulb. Find the point between the two bulbs at which the sum of the intensities of light from the two bulbs is minimum.

Ricker's model for population growth is $$ P^{\prime}=R P e^{-\frac{P}{\alpha}} $$ where \(P\) is population size, \(R\) is the low density growth rate, and \(\alpha\) reflects the carrying capacity of the environment. For what value of \(P\) will the growth rate, \(P^{\prime}\) be greatest?

A comet follows the parabolic path, \(y=x^{2}\) and Earth is at \((3,8) .\) How close does the comet come to Earth?