Mathematics Cheat Sheet for Population Biology - Stanford University

f(x)E[f(x)]=[f(A)+f(B)]/2f(E[x])=f([A+B]/2)AE[x]=(A+B)/2BxFigure 1: Illustration of Jensen’s Inequality.e r ≈ 1 + r + r22 + r36 . . .Expanding log(1 + x) around a = 0 yields:2.3 Jacobianlog(1 + x) ≈ x − x22 + x33 − . . .For a system of equations, F (x) and G(λ), the Jacobian matrix is( )∂F/∂x ∂F/∂λJ =.∂G/∂x ∂G/∂λThis is very important **for** the analysis of stability of interacting models such as those **for**epidemics and predator-prey systems. The equilibrium of a system is stable if and only if thereal parts of all the eigenvalues of J are negative.2.4 IntegrationLinearity∫∫[af(x) + bg(x)] dx = a∫f(x)dx + bg(x)dxIntegration by Parts∫∫u · v ′ dx = u · v −v · u ′ dx3