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- a)Evaluate y at position x=0.5 b)Evaluate the derivative dy/dx at position x=0.5. c)Using your values from a) and b) form the Taylor's expansion of y at x=0.5, truncated to include the first 2 terms.a differential equation is classified as a homogeneous DE if one can directly isolate all functions of x with its derivative, dx, and isolate all functions of y with its derivative, dy.Solve the differential equation y'=5y(1000-y) subject to the initial condition y(0)=500 . From your solution, find the valueof the limit lim t →infinity of y(t)
- In the function: f(x)= (3x^2)ln(x) , x>0 What are the critical numbers of the function?Given the function f(x) = e-3x cos 3x, x = [0, 2π]. When is the rate of change of f(x) a maximum?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.
- find a linearization at a suitably chosen integernear a at which the given function and its derivative are easy to evaluate. ƒ(x) = e-x, a = -0.1In 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)(a) Write the differential equation for the amount of salt present at any time for both tanks (b) Construct a mathematical model for the number of kilograms x1(t) and x2(t) of salts in tank A and B respectively, at time t by using Laplace transform