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modern mathematical statistics with applications

Mathematics And Statistics For Science 1st Edition James Sneyd, Rachel M. Fewster, Duncan McGillivray - Solutions

Show that, for the ideal gas equation,∂P∂V∂V∂T∂T∂P= −1.
The ideal gas equation is PV = nRT. Calculate ∂P∂V and ∂P∂n.What are the scientific interpretations of these partial derivatives? Do they make scientific sense?
For each of the following functions show that the second-order mixed partial derivatives are equal (i.e., the order of differentiation doesn’t matter). If you need something extra to do, check the same thing for the third-order mixed partial derivatives.You should probably use a computer to do
If one of your fellow students tells you that they discovered a function, p(b, g), such that pb = 2b + 4g and pg = 3b + 5g, do you believe them? If that’s the correct expression for pb, what would be a possible expression for pg? For your choice of pg, what is the functio p?
If u(x) = x 2, v(t) = e tand G(u, v) = sin(u(v)2 + v 2), calculate dG dt .
If u(t) = t 2, v(t) = t + e tand G(u, v) = sin(u 2 + v 2), calculate dG dt .
If u(y) = sin(4y), v(y) = e y + y 2and G(u, v) = e ue v, calculate dG dy .
When an advantageous gene appears in a population, or when an infectious person appears in a population of susceptibles, a traveling wave can often appear. In the first case, the advantageous gene can spread through the population, while in the second case it is the infection that spreada. First,
Consider the contour map of Rangitoto Island in Fig. 18.9.How can you tell where the peak of the island is? How can you tell which are the steep parts? Can you sketch the basic shape of the island just by looking at the contour map?
In the previous question you plotted a Gaussian function G(x, y). In this question we look at contour plots of G.a. Set µ = ν = 0 and σx = σy = 1. Show that the contour plot of G is a series of circles.b. Change the values of µ and ν and draw the contour plot of G again. What do µ and ν
The function By the way, µ is the Greek letter, mu, not the Latin letter u, and ν is the Greek letter nu, not the Latin letter v.It’s easy to confuse them.G(x, y) = e−(x−µ)2 2σ2 x e−(y−ν)2 2σ2 yis a Gaussian function.a. Set µ = ν = 0 and σx = σy = 1, and plot G as a surface in
Suppose you take a bright red dye, and release a drop (at time t = 0) in a long, thin, horizontal tube of water. You then watch the drop spread to the right and to the left. You intuitively expect that the dye will initially be highly concentrated in a small area, giving a small but bright red dot,
We saw in Section 4.1 how the logistic equation is used to describe a variety of things. Another example of logistic growth is shown in Fig. 17.14, which plots the density of a population of E. coli bacteria (Ram et al, 2019) as a function of time.N(t) = 0.060 +0.59 1 + e −(t−4.2)1.6 , where N
The Lennard–Jones potential describes the intermolecular potential energy, V, as a function of the distance,r, between a pair of particles. It’s probably the most widely used simple model of intermolecular interactions. The equation is V(r) = 4σr12−σr6, for some constants and σ.a.
When a population (such as a fish species) is harvested, the maximal sustainable yield, or MSY, is the harvesting rate that gives the greatest yield, without driving the population to extinction. If you take out more than the MSY, then the population crashes; if you take out less than the MSY than
Ice floats on water because it has a lower density than water.If d and T are, respectively, the water density and temperature, This is a really unusual property.Other substances whose density decreases when they freeze are gallium, germanium, silicon and acetic acid.then, close to the freezing
Optimal foraging theory is a theory that animals spend the optimal amount of time collecting food from a single food patch. The theory is not without controversy, but it does seem to be a valuable way of interpreting animal behaviour in some situations.The word "optimal" means different things for
The rate, V, of an enzyme reaction is often described by the Michaelis–Menten equation, V =VmaxS K + S, where Vmax and S are constants. This is an equation we’ve used a lot in this book. However, the rate of an enzyme reaction is also commonly described by the Hill equation, V =VmaxS nKn + S n,
On page 305 we briefly considered a solution to the diffusion equation, which models how a pollutant, for example, diffuses from a single concentrated release. If you measure the concenIn other words, the pollutant is dumped into the environment only once, and then allowed to diffuse from the
It is shown in the theory of gravitational attraction that a wire bent in the form of a circle of radius a exerts a force, f , upon a particle in the axis of the circle (i.e., in the line through the centre of the circle perpendicular to the plane of the circle), where f (h) ∝ h(a 2 + h 2)3/2,
An undergraduate student is tossed into the air by a rampaging bull in Pamplona. The distance of the student from the ground is described by the function s(t) = 1 + 14t − 5t 2, where t is time in seconds, and s is distance in metres.a. Plot s(t).b. How high does the student go?c. What is the
Nuclear magnetic resonance (NMR) is a common method for determining the chemical composition of samples. Different compounds absorb electromagnetic radiation at a frequency characteristic of the compound. Thus, if the amount of radiation absorbed is plotted against frequency, the peak positions
A Gaussian function, G(x), has the form G(x) = e−(x−µ)2 2σ2.Find the first derivative of this function and hence show that there is a stationary point at x = µ.
If you present a light flash to a salamander rod then the electrical current generated by the rod first rises and then decays, as shown in Fig. 17.12. The red curves are a very simple model.If R is the response (i.e., the current), the simple model is R(t) = R0t n e −kt , for some constants R0 >
Hankinson’s equation describes the compressive strength of wood, as a function of the angle, relative to the grain of the wood, at which that stress is applied.Suppose that σ0 is the compressive strength of wood when the stress is applied parallel to the wood grain, and σ90 < σ0 is the
Is it possible to have a function that has no maximum or minimum?
Where are the critical points of the quadratic z(u) = au2 +bu+c? What kind of critical points are they?
Find a function, M(u), that has M00(0) = 0 but where u = 0 is not an inflection point.
In Exercise 4.22 we saw the equation, due to Fisher et al (1943), S = α ln 1 +Nα, where S is the number of species in a random sample from a population, N is the number of individuals in the sample, andα is a constant.Experimental data, and a comparison to the theoretical equation, are shown
If you connect two resistors in parallel, the total resistance, Rt, over the circuit is given by 1Rt=1 R1+1 R2.If R1 is decreasing by 2 Ω/s but R2 is increasing by 3 Ω/s, what is the rate of change of the total resistance when R1 = R2 = 10 Ω?
Suppose that an ideal gas (for which PV = nRT, remember)is changing its pressure and volume at a constant temperature and number of moles. That is, P and V are changing while n and T are held fixed.In this case PV = constant, which is called Boyle’s law.a. Derive an expression for dP/dt dV/dt.b.
When hydrogen ions (H+) enter a cell they quickly bind to various proteins, called buffers. This is so that the cell can control its pH precisely.Interestingly, the same is true of Ca2+ions, whose concentration also needs to be controlled carefully, but not true for Na+ or K+ions.If we assume that
The head length at age x, H(x) (in m), of the blue whale is related to the total length at age x, T(x), (also in m) by the equation This is an example of an allometric equation, which are discussed in more detail in Section 6.4. In Fig.6.11 we saw some similar data from humans, where organ weight
Consider the curve defined implicitly by xy 2 + y − 2 = 0.Calculate dy dx when x = 1. Check your answers by plotting the curve.
What is the relationship between the rate of growth of the area, A, of a circle, and the rate of growth of the radius, r, of that circle? When r = 1 what is A 0(t)r 0(t)?
The Folium of Descartes is described by the equation x3 + y 3 − 3axy = 0, for some constanta. Just for convenience, set a = 1. Find all This is one of the famous equations in mathematics, and actually has a role to play in the history of implicit differentiation. The mathematician Descartes
Find all the points on the graph s 2 + 2st − 3t 2 = 1 where the tangent line is horizontal. Check your answer by plotting the graph.
Find all the points on the graph h 3 − 27h = b 2 − 90 where the tangent line is vertical. Check your answer by plotting the graph.
A circle of radius 1 has the equation p2 + e 2 = 1.What is the slope of the circle when p =1 2? Sketch the graph and its tangents at p =1 2.
Saccharomyces cerevisiae is a species of yeast that has been used in baking and brewing for thousands of years, and so it has been the subject of much scientific research over the last 100 years or so. In particular, scientists are interested in how fast it grows. Typical data from a paper
In Fig. 12.1 at the beginning of Chapter 12 we saw a graph showing how the rate of a calcium ATPase pump varies as a function of calcium concentration. If we extract some of the data we get the following points, where we let c denote the calcium concentration, and we let R denote the rate.a. Use
A strong base was titrated against 10 mL of strong acid and the following table contains part of the data read by a pH probe.a. Plot the data.b. Numerically calculate the first derivative of pH at each of the data points.c. Numerically calculate the second derivative of pH at v = 9.9 and v = 10.d.
The rate of growth of a population was measured every year for 10 years, and the following data were obtained.a. Use the data points at y = 3 and y = 5 to estimate R 0 (y) at y = 4.b. Similarly, estimate R 0 at y = 5, y = 6 and y = 7.c. Calculate R 00(y) at y = 5 and y = 6. What do we conclude
In 2002, Krahn et al published a report for the US Department of Commerce, studying the number of Southern Resident Killer Whales in the US Northwest (Krahn et al, 2002). Their data, shown in Fig. 15.6, showed a significant decline in orca numbers from 1996 to 2001. The data from the final 10
Consider again the distance/time plots of the two sprinters shown in Fig. 13.5 at the beginning of Section 13.2. If we let x denote distance (in m), and t denote time (in s) then each sprinter’s path can be described by the general function x = vmt − vm(1 − e −βt )/β, for some constants
The Weber–Fechner law says that the relationship between Gustav Fechner was a student of Ernst Weber and named the law after his mentor, in recognition of the experimental work done by Weber that led to the formulation of the law. It is wonderful to see a student behaving with such respect to
The normal distribution, or Gaussian, is G(x) =1√2πσ2 e−(x−µ)2 2σ2, where σ > 0 and µ are constants. Where is G(x) increasing?Where is G(x) decreasing?
The Hill equation for the rate, V, of a cooperative enzyme reaction is V(S) =VmaxS nKn + S n, where n > 1 is usually an integer.a. What is V(0)? What is V(Kn)?b. Show that V 0(S) is sometimes increasing, sometimes decreasing, but that, like the Michaelis–Menten equation, V 0(S) → 0 as S →
The Michaelis–Menten equation for the rate, V, of an enzyme reaction is V(S) =VmaxS K + S Show that V 0 (S) is always decreasing and that V 0 (S) → 0 as S → ∞.
Consider again the equation for queue length we first saw at the beginning of Section 12.4. Just to remind you, if λ is the average rate of arrivals at the queue, and µ is the average serving time, then the average length of a queue, L, is given by L(λ) =λ2µ(µ − λ).Show that both L(λ) and
Suppose that when you throw a stone upwards its distance, x(t), measured in metres from the ground after t seconds is given bya. What is the stone’s average velocity between t = 0 and t = 2 seconds?b. Assuming that x(t) = a + bt + ct2 , wherea, b and c are constants, write a system of equations
The Gompertz equation is N(t) = N0e Aα(1−e−αt), for some (positive) constants N0, A and αa. Calculate N 0 (t) and N 00(t).b. As you will learn in Chapter 17, the maximum of N 0 is at a place where N 00 = 0. Use this fact to determine when the maximal rate of growth occurs. Show that the
The logistic equation is often used to model population growth(see Section 4.1.1); the number, N, in the population grows as a function of time according to N(t) =N0 1 + e−kt .a. Choose nice simple values for N0 and k (put them both equal to 1, for example) and plot N(t). Why does it make sense
A simple model for a quantum particle is the time-independent one-dimensional Schrödinger equation~2 2m d 2ψ(x)dx2 = Eψ(x).Here x represents distance, and ~ is equal to h 2π where h is Planck’s constant. We want to find a function ψ(x) that satisfies this equation. Although we don’t learn
When you take a single dose of an antibiotic, or other kind of medication, the concentration, C (in µg/mL), of drug in your blood stream is often well described by the equation C(t) = C0te−kt, for some constants C0 and k. Here, t usually has the units of hours.Furthermore, the rate, r, at which
What is the nth derivative of ln(x)?
What is the nth derivative of sin(2x)?
What is the nth derivative of e 2x?
In Fig. 13.5 we showed the data for how many seconds it took two sprinters to run 10 m segments of a 100 m sprint. The data for Florence Griffith-Joyner are given in Table 13.12.a. From these data calculate the average speed of GriffithJoyner over the entire race.b. Calculate the average speed of
When you take a single dose of a drug (such as aspirin or paracetamol), the concentration of the drug in your blood decays along an exponential curve, at least approximately. Thus, if C(t) is the blood concentration of the drug as a function of time, we know that a reasonable model is C( cise
Suppose that z(t) =√4 t. Using the limit definition of the derivative, show that z 0(t) does not exist at t = 0.
Suppose thatU(i) =√i. Using the limit definition of the derivative, calculate U 0(i).
Suppose that C(y) = cos(y). Using the limit definition of the derivative, calculate C 0(y).
Suppose that A(s) = s +1 s. Using the limit definition of the derivative, calculate A 0(1).
Suppose that M(v) =−5 v. Using the limit definition of the derivative, calculate M0(4).
Suppose that J(p) = 3p 2. Using the limit definition of the derivative, calculate J 0(4).
In this question we will show that the derivative of g(x) =cos(ax) is g 0(x) = −a sin(ax), using the limit definition of the derivative, i.e., using g0(x) = lim h→0 g(x + h) − g(x)h.a. First, consider the two functions G1(m) =cos(m) − 1 mand G2(m) =sin(m)m.Notice that G1(0) and G2(0) are
Follow these steps to calculate the value of the derivative of H(r) = e rat the point r = 1.a. Calculate the gradient H(r+h)−H(r)h(with r = 1) for different values of h, and complete the following table.b. Plot the points in the table, and thus estimate the value of lim h→0 H(1 + h) − H(1)h
In Exercise 4.22 we saw the equation, due to Fisher et al (1943), S = α ln 1 +Nα, where S is the number of species in a random sample from a population, N is the number of individuals in the sample, andα is a constant. Experimental data, and a comparison to the theoretical equation, are shown
Imagine using your muscle power to lift a mass. If the mass is small, you can lift it quickly, as the force exerted on the mass by gravity is small. If the mass is larger, you would only be able to lift it more slowly, while if the mass was very large you might not be able to lift it at all, or it
When you take a single dose of a drug such as paracetamol, the concentration of the drug in your blood is described by some function of time, P(t), say. Without specifying P(t) exactly, sketch P(t) and describe any asymptotes it must have.Hint: have a look at Fig. 4.11.
A hydrogen molecule comprises two hydrogen atoms held together by their bond; if the molecule receives enough energy to overcome that bond the molecule will dissociate. The amount of energy needed is called the dissociation energy.The energy potential can be approximated as an exponential function
The behaviour of an electron in a hydrogen atom can be described by the radial function R(r) = Nr a02 e−r a0 , where N > 0 is a constant and a0 > 0 is the Bohr radius(another constant). For simplicity, set a0 = 1.a. Calculate limr→0 R(r).b. What is R(10)? What is R(100)? What can you say
As we saw in Exercise 4.23 the Gompertz equation, N(t) = N0e Aα(1−e−αt),can be used to model growing populations. Here, N is the number in the population, and A, N0 and α are positive constants.a. Calculate the maximal population in the Gompertz equation.b. Now, compare the shape of the
Population density, N(t), as a function of time, t, can be described by the logistic equation (see Section 4.1.1)N(t) =K 1 +K N(0)− 1e−r t.The initial population is N(0), and K and r are parameters that describe the population’s dynamics. Find lim t→∞N(t) and use this to explain what K
Photoreceptors are cells in the retina that change their voltage in response to light. These changes in voltage are relayed to the brain via the optic nerve, and thus we see things. In response to a bright but short pulse of light (a bright flash, say), the voltage response of a photoreceptor can
Calcium ATPase pumps are proteins that sit in the membrane of a cell and use energy in the form of molecules of adenosine triphosphate (ATP) to remove Ca2+from the cell, against a steep concentration gradient. They are vital components of all cells; if cells cannot control their intracellular
Suppose that a cell membrane receptor is stimulated (by a hormone, say, or a neurotransmitter) at time t = 0, and that the concentration of activated receptor, R∗, is given by the following function of time R∗(t) =VRt KR + t.This receptor then activates another protein inside the cell (such as
The rate,r, of many enzyme reactions is related to the substrate concentration, S, by the Hill equation r =V Sn Kn + S n, for some constants n (a positive integer), V and K. What is the horizontal asymptote? Does this horizontal asymptote depend on n? On K?What are the answers to those questions if
What is lim f →∞ke−Af 2 eB f ?You will have to consider four different cases, as A and B can each be either positive or negative.
Calculate f (x) = x 10e−x for the following values of x.a. x = 1.b. x = 10.c. x = 100.d. x = 1000.Using your results, infer the value of limx→∞ x 10e−x. Can you guess the value of limx→∞ x ne−x, for any n > 0? Check your guess by computing some values, or by using a computer package
Calculate f (x) = xe−x for the following values of x.a. x = 1.b. x = 10.c. x = 100.Using your results, infer the value of limx→∞ xe−x.
Calculate f (x) = e−x for the following values of x.a. x = 1.b. x = 10.c. x = 100.Using your results, infer the value of limx→∞ e−x.
What happens to the function sin(1 x) as x → 0? (Watch out.The answer to this is a bit tricky.)
Can you explain to yourself and to others what the following would mean.a. lim x→∞f (x) = ∞.b. lim x→∞f (x) = −∞.c. lim x→−∞f (x) = −∞.
Can a function have 3 horizontal asymptotes? Can a function have 3 vertical asymptotes?
For each of the following scenarios give an example function f (x) which satisfies the conditions provided, and sketch f (x)around this limiting behaviour.a. lim x→−∞f (x) → ∞, where f (x) is not linear.b. lim x→∞f (x) = 3, where f (x) is not constant.c. lim x→5 f (x) → −∞.d.
Consider the reaction of ferric nitrate with ammonium carbonate, i.e.,α Fe(NO3)3 + β (NH4)2CO3 −→ γ Fe2(CO3)3 + δ NH4NO3.a. Write an equation that relates α and γ such that there are the same number of iron atoms on both sides of this chemical equation.b. Do the same for each remaining
Methane burns in oxygen to form carbon dioxide and water.The reaction is CH4 + x1O2 −→ x2CO2 + x3H2O for some undetermined coefficients x1, x2, and x3.a. By balancing the number of H, C or O atoms on each side of the equation, write a system of equations that can be solved to determine the
The reaction of dichromate (Cr2O 2−7) with hydrogen ions (H+)and ferrous ions (Fe2+) is given by the reaction x1Cr2O 2−7 + x2Fe2+ + x3H+ −→ x4Cr3+ + x5Fe3+ + x6H2O.We have to determine the unknown coefficients x1, · · · , x6.a. By requiring that the reaction has the same number of Cr
Suppose a student performed an experiment to determine the concentration of a substrate as a function of time (with units of s). Denote the substrate concentration by S, with units of mol.The student believed that S could be approximately described by S(t) = S0e−kt.The data they collected include
Suppose we have a biochemical reactiion where the concentration, s, of substrate is known to be s(t) = ke−At2 eBt, for some constants k, A, and B. Here, t is time (in units of seconds) and s has units of µM.Now suppose you do an experiment to determine the three unknown constants. In this
The absorbance of a molecule, A, is proportional to its moNote that µg/mL is equivalent to ppm; this is a common unit in atomic absorption spectroscopy(AAS) which is usually done at low concentrations.lar absorptivity, , (with units of mL cm−1 µg−1), the path length of the cell,b, (measured
For our final problem using Leslie matrices we use some real data on a wild rabbit population in England (Smith and Trout, 1994). Their motivation for the study was to determine how best rabbit populations can be controlled. The rabbit population was Rabbits (Oryctolagus cuniculus) are native to
Suppose that a particular endangered species of fish has three distinct stages of life, each lasting one year. Each fish begins life as a hatchling (H), then becomes a juvenile (J), then an adult (A). At year n we denote the number of fish in each stage by Hn, Jn and An.• Each year, each adult
Suppose that we have a population that, in year n, has xn population of juvenile females and yn population of adult females, but this time we assume that the juvenile females can also reproduce, just not as fast the adult females. Hence, we suppose This is another Leslie matrix question, with only
Suppose that we have a population that, in year n, has xn population of juveniles and yn population of adult females.The units of xn and yn could be, for example, in millions of individuals (which would be typical for insect populations).Also suppose that in year n + 1, the number of juveniles is
The PageRank method is a lot more complex than simply ranking all the pages according to the number of incoming links.Illustrate this using the network in Fig. 11.7. What ranking would you get if you ranked only according to the number of incoming links? What is the ranking using the PageRank
Suppose you have a very small internet of 4 web pages, each of which links exactly once to each other web page. Which web page has the highest page rank? What about if you have a slightly larger internet of 5 web pages, each of which links exactly once to each other page; which page has the highest
Solve the page ranking problem in the previous chapter (Exercise 10.8) to rank the web pages in order of popularity.

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