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- At time t = 0, there is one individual alive in a certain population. A pure birth process then unfolds as follows. The time until the first birth is exponentially distributed with parameter λ. After the first birth, there are two individuals alive. The time until the first gives birth again is exponential with parameter λ, and similarly for the second individual. Therefore, the time until the next birth is the minimum of two exponential (λ) variables, which is exponential with parameter 2λ. Similarly, once the second birth has occurred, there are three individuals alive, so the time until the next birth is an exponential rv with parameter 3λ, and so on (the memoryless property of the exponential distribution is being used here). Suppose the process is observed until the sixth birth has occurred and the successive birth times are 25.8, 42.0, 52.0, 55.8, 59.2, 63.3 (from which you should calculate the times between successive births). Derive the mle of λ. [Hint: the likelihood is a…Find the absolute maximum and minimum values of ƒ(x) =10x (2 - ln x) on the interval [1, e2 ]Show mathematically that MRS is increasing for U=1-ex +e2y and is diminishing for U=lnX +LnY
- Suppose a tank contains 60 gallons of pure water. A mixture consisting of 1 pound of salt per gallon is flowing into the tank at a rate of 2 gallons per minute, and the mixture is continuously stirred. Meanwhile, the brine in the tank is allowed to empty out the tank at the same time at a rate of 3 gallons per minute. If the tank is completely empty after 1 hour, Önd the amount of salt in the tank at any time t.In 1934, the Austrian biologist Ludwig von Bertalanffy derived andpublished the von Bertalanffy growth equation, which continues tobe widely used and is especially important in fisheries studies. LetL(t) denote the length of a fish at time t and assume L(0) = L0. Thevon Bertalanffy equation isdL/dt = k (A - L),where A = limtSqL(t) is the asymptotic length of the fish and k is aproportionality constant.Assume that L(t) is the length in meters of a shark of age t years.In addition, assume A = 3, L(0) = 0.5 m, and L(5) = 1.75 m. Solve the von Bertalanffy differential equation.13.Find a sinusoidal function with a maximum value of 9 that occurs at x=4 , and a minimum value of -1 that occurs at x=11.
- The population in California p(t) (in million) can be approximated by the logistic growth function P(t)= 95.2/ 1+1.8e^-0.018t where t is the number of years since the year 2000. 1. Determine the population in the year 2000. 2. What is the limiting value of the population of California (i.e., as t---->+infinity) under this model?Solve the initial value problem for y as a function of xSolve the initial value problem for x as a function of t
- Suppose f is differential equation on (-inf, +inf), f(3)=12 and f'(3)=-2 use linear approximation to estimate f(3.3).2) Two strains of bacteria are growing in seperate petri dishes. Intitially, there are strain A 300 bacteria and, from a prior experiment, you know that the population should double every 20 minutes. Having never worked with strain B before, you monitor its growth over the first hour and notice that there are 200 bacteria after 30 minutes and 600 bacteria after 1 hour. a) Let t be the number of hours that have passed since the two populations of bacteria start growing. Express the number of strain A bacteria as a function of t. (Your answer should be of the form PA (t) = Ca bt for some numbers C, a and b.) b) Express the number of strain B bacteria as a function of t, (Again, your answer should be of the form P B (t) = Cabt for some numbers C, a and b, possibly different that those you found in (a)).) c) How many strain B bacteria were there intitially?In the Four-Step Rule, which among the steps is the most important, upon which the foundation of determining differential equations rests? a. Step 3 (divide the difference between f(x + Δx) and f(x) by Δx) b. Step 1 (write f(x) as f(x + Δx)) c. Step 4 (apply the limit Δx → 0) and simplify. d. Step 2 (subtract f(x) from f(x + Δx) and simplify)