5.2 Calculus - AHL content

• 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.

• Use second derivative notation and second derivative test for maxima/minima.

• Interpret concavity and points of inflexion in context.

• 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.

• Find areas enclosed by curves and x- or y-axes, including negative integrals.

• Find volumes of revolution about the x-axis or y-axis.

• 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.

• Set up differential equations from contexts.

• Solve separable differential equations; exponential models solve dy/dx=ky.

• Know the term general solution.

• Use and interpret slope fields and their diagrams.

• 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.

• 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.

• 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.