What you’ll learn10 learning objectivesChoose one objective for a focused lesson, or study the complete topic.—AHL 5.9 (HL)—Further differentiation• Differentiate sin x, cos x, tan x, e^x, ln x and x^n for n in Q.• Use chain, product and quotient rules.• Solve related rates problems.Syllabus objective—AHL 5.10 (HL)—Second derivative and concavity• Use second derivative notation and second derivative test for maxima/minima.• Interpret concavity and points of inflexion in context.Syllabus objective—AHL 5.11 (HL)—Further integration• Integrate x^n, sin x, cos x, sec^2 x and e^x.• Use integration by inspection or substitution of the form integral f(g(x))g'(x) dx.Syllabus objective—AHL 5.12 (HL)—Areas and volumes• Find areas enclosed by curves and x- or y-axes, including negative integrals.• Find volumes of revolution about the x-axis or y-axis.Syllabus objective—AHL 5.13 (HL)—Kinematics• Use displacement, velocity and acceleration with v=ds/dt and a=dv/dt=d2s/dt2.• Use a=v dv/ds where appropriate.• Displacement is integral of velocity; total distance is integral of speed.Syllabus objective—AHL 5.14 (HL)—Differential equation modelling• Set up differential equations from contexts.• Solve separable differential equations; exponential models solve dy/dx=ky.• Know the term general solution.Syllabus objective—AHL 5.15 (HL)—Slope fields• Use and interpret slope fields and their diagrams.Syllabus objective—AHL 5.16 (HL)—Euler's method• Use Euler's method for numerical solutions of first-order differential equations.• Use technology/spreadsheets for approximations.• Numerically solve coupled first-order systems such as predator-prey models.Syllabus objective—AHL 5.17 (HL)—Phase portraits• Use phase portraits for coupled systems dx/dt=ax+by, dy/dt=cx+dy.• Analyse future paths using real, complex and imaginary eigenvalues.• Identify equilibrium points, stable populations, saddle points, spirals and circles/ellipses.Syllabus objective—AHL 5.18 (HL)—Second-order differential equations• Solve second-order differential equations numerically with Euler's method.• Rewrite as coupled first-order equations dx/dt=y and dy/dt=f(x,y,t).• Use phase portrait ideas for suitable second-order systems.Syllabus objective