Differential Equations unit test
A test on this unit alone, marked as a percentage and a letter grade — for the test your class is actually sitting, rather than for May. Answer everything, then submit once: seeing the answer to question 3 before attempting question 4 makes the final percentage meaningless.
Identifying the carrying capacity
Solutions cannot cross
Verify a candidate solution twice
Choose the branch from the initial condition
Finding concavity without the solution
Exponentiating a sum gives a product
Resolve C before exponentiating
Euler recomputes the slope
Fastest logistic growth at M/2
Match a slope field by structure
Show the Euler table
Logistic solution behavior
Short answer 1. Define or explain: Logistic equilibria
3 ptsShort answer 2. Define or explain: Separation of variables with initial conditions
3 ptsShort answer 3. Define or explain: Logistic differential equation
3 ptsShort answer 4. Define or explain: Fastest growth in a logistic model
3 ptsFree response
9 ptsNO CALCULATOR. A population of fish in a lake is modeled by the differentiable function P, where P(t) is measured in hundreds of fish and t is measured in years. The population satisfies the logistic differential equation dP/dt = 0.5P(1 − P/200), with initial condition P(0) = 50.
A. State the carrying capacity of the lake for this population, and find lim (t→∞) P(t).
B. Find the value of P at which the population is growing most rapidly. Justify your answer.
C. Use Euler’s method, starting at t = 0 with two steps of equal size 1, to approximate P(2). Show the work that leads to your answer.
D. Determine whether the approximation in part C is an overestimate or an underestimate of the true value of P(2). Justify your answer.