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- Let f(x) = (x − 3)^5 and x0 is not equal to 3. For each n ≥ 0, determine xn+1 from xn by using Newton’s method for finding the root of the equation f(x) = 0. Show that the sequence {xn} converges to 3 linearly with rate 4/5.The sales of a book publication are expected to grow according to the functionS = 300000(1 − e−0.06t),where t is the time, given in days. (i) Show using differentiation that the sales never attains an exact maximum value.(ii) What is the limiting value approached by the sales function? A town has a population of 5000 persons, but is expected to grow by 2% every year.(i) What would be the population size in 7 years? (ii) Find the sum of the first eight terms of the sequence 1/8, -1/4, 1/2..a) On the number line, mark the critical value(s) of f(x) and the sign of the first derivative between them as obtained from the graph. b) Where is f(x) increasing on it's domain? c) Where is f(x) decresing on it's domain? e) At what x-value(s) would f(x) have a relative maximum? d) At what x-value(s) would f(x) have a relative minimum?
- Suppose that we are interested in the amount of salt (in kg) at a certain moment in time, denoted by A(t), in a well-mixed, infinitely large tank with 400-L of water initially. Further, suppose we have set-up the experiment such that a salt solution containing 16 kg/L flows into the tank at a rate of 9 L/sec. If we allow the solution to flow out at 6 L/sec, then what is the volume of water in the tank at time t? V(t) = _____? ( (DO NOT use commas in your answer) Construct a model for the change of salt in the tank at time t. [To ensure you are marked correct, please use A to represent the amount of salt in the tank at time t not A(t)]. dA/dot=_________?5x−12/x^2−x−42 has vertical asymptote(s) at x=A tank contains 50-liters of brine in which 20 grams of salt is dissolved. Pure water is being pumped into the tank at a rate of 5 liters/min., the solution is continuously stirred and the well-mixed solution is pumped out at 4 liters/min. Which of the following initial-value problems must be solved to find the amount of salt x= x(t) in the tank at time t?
- An industrial food-manufacturing operation wishes to expand, and it requires an improved water supply. They propose to construct a fully penetrating well in an infinite, horizontal, confined, homogeneous, isotropic aquifer, and they plan to pump at a constant rate of 20 L/s. If T = 1.3 x 10-2 m2/s, and S is 2.0 x 10-4, make the following calculations. a. Calculate the drawdown that would occur in an observation well 60 m from a pumping well at times of 1, 5, 10, 50 and 210 min after the start of pumping. The environmental regulator has imposed a limitation on the drawdown from the well in order to prevent impacts to surrounding groundwater users. If the drawdown limit is set to a maximum of 1.5 m at a distance of 250 m, after ten days of pumping, is this well likely to be in compliance with the regulation?find the absolute maximum and minimumvalues of each function on the given interval. y = 10x(2 - ln x), (0, e2)Solve the initial value problem for x as a function of t