If y = g(x) satisfies the initial value problem (IVP) 2-x = 1, y(1) = 4, dy dx then (a) the slope of y at x = 3 is (b) lim g(x) = x→1+
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- Does a Limiting Value Occur? A rocket ship is flying away from Earth at a constant velocity, and it continues on its course indefinitely. Let D(t) denote its distance from Earth after t years of travel. Do you expect that D has a limiting value?Show that the solution of initial value problem: d^2x/dt^2 + w^2x = F0cosAt, x(0)=0, x'(0)=0 is x(t)= F0(cosAt-coswt)/(w^2-A^2). b) Evaluate lim,y>w F0(cosAt-coswt)/(w^2-A^2).14x(e^(1/x))−14x limit x approaches infinity using lhospitals rule
- 15) Annual U.S. imports from a certain country in the years 1996 through 2005 could be approximated by I(t) = t2 + 3.5t + 48 (1 ≤ t ≤ 9) billion dollars, where t represents time in years since 1995. Annual U.S. exports to the country in the same years could be approximated by E(t) = 0.5t2 − 1.4t + 13 (0 ≤ t ≤ 10) billion dollars. Assuming that the trends shown in the above models continue indefinitely, calculate the limits lim t→+∞ I(t) and lim t→+∞ I(t)/E(t) algebraically. (If an answer does not exist, enter DNE.) lim t→+∞ I(t) = lim t→+∞ I(t) E(t) = Interpret your answers. In the long term, U.S. imports from the other country will (select) (be rounded or rise without bound) and be times U.S. exports to the other country. Could the given models be extrapolated far into the future? Yes or NoThe limit of (xy-2y) / (x2+y2-4x+4) as (x,y) approaches (2,0) is solved and found out that it does not exist. Can we make this continuous by defining f(2,0) = k for some real value k? If yes, what could this value be? If no, why not?a) Evaluate in terms of Gamma function ∫e^(−y^2) y^13 dy limit 0 to ∞ b) Find ∫f(x) dx limit: 1 to 38 if ∫f(x) dx=−17 limit -19 to 1 and ∫f(x) dx=10. limit : -19 to 38
- Consider the following initial value problem. a) Find the solution y(t) in an explicit form. b) Use the theory of limits to find the behavior of y(t) as t approaches -infinity and +infinity.A manufacturing company owns a major piece of equip -ment that depreciates at the (continuous) rate f= f(t) ,where is the time measured in months since its last overhaul.Because a fixed cost is incurred each time themachine is overhauled, the company wants to determine theoptimal time T (in months) between overhauls.(a) Explain why ∫0t f(s) ds represents the loss in value ofthe machine over the period of time t since the lastoverhaul. (b) Let C =C(t) be given by C(t) =1/t[A + ∫0t f(s) ds]What does C represent and why would the companywant to minimize C ?(c) Show that has a minimum value at the numbers t =Twhere C(T) =f(T).Solve for A and B so that F(x) has a limit at both x=2 and x=4
- A tank initially contains s0 lb of salt dissolved in 100 gal of water, where s0 is some positive number. Starting at t = 0, water containing 0.5 lb of salt per gallon enters the tank at a rate of 2 gal/min, and the well-stirred solution leaves the tank at the same rate. Letting c(t) be the concentration of salt at time t, show that the limiting concentration–i.e., limt→∞ c(t)–is 0.5 lb/gal. (1) Set up an initial value problem using the situation described above. It may help to draw a diagram. (2) Solve the differential equation, and use the initial value to solve the initial value problem, and use this function to write an explicit formula for c(t). (3) Show that limt→∞ c(t) = 0.5. (4) Discuss what this limit implies about the importance of the unknown quantity s0.What value of a makes ƒ(x) = x2 + (a/x) have a. a local minimum at x = 2? b. a point of inflection at x = 1?Show that the function f(x,y)=8x^2 y subject to 3x−y=9 does not have an absolute minimum or maximum. (Hint: Solve the constraint for y and substitute into f.) Solve the constraint for y. y = ? Substitute into f. f(x,y)= ? Determine the behavior of f as x approaches −∞. limx→−∞f(x,y)= ? Determine the behavior of f as x approaches ∞. limx→∞f(x,y)= ? Does this show that f does not have an absolute maximum or minimum? 1. No 2. Yes