(BC only) Euler's method and logistic growth models

Learning objectives

Apply Euler's method with the update y_(n+1) = y_n + h f(x_n, y_n), stepping forward from an initial condition with a fixed step size h and tabulating the results

Explain that Euler's method follows a sequence of tangent lines, and decide whether the approximation is an overestimate or an underestimate from the concavity of the true solution

Describe how the error of Euler's method behaves as the step size is reduced, and carry out the method with a negative step size to move backwards from the initial point

Write the logistic differential equation dP/dt = kP(1 − P/L), identify L as the carrying capacity, and state that P → L as t → ∞ for any positive starting population

Show that logistic growth is fastest when P = L/2, by differentiating the right-hand side or by recognizing the vertex of the quadratic in P, and interpret that point as the inflection point of the solution curve

Read the carrying capacity and the constant k from a logistic equation written in any equivalent form, without solving the equation analytically