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5 Calculus

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  1. 5.1 Calculus - SL content

    1. SL 5.1Limits and derivative concept

      • Estimate limits from tables or graphs; formal analytic limit methods are not required. • Interpret derivative as gradient function and rate of change. • Use notation dy/dx, f'(x), dV/dr and ds/dt.

    2. SL 5.2Increasing and decreasing functions

      • Interpret f'(x)>0, f'(x)=0 and f'(x)<0 graphically. • Identify intervals where functions are increasing or decreasing.

    3. SL 5.3Derivative of powers

      • Differentiate f(x)=ax^n to f'(x)=anx^(n-1), n in Z. • Differentiate sums of integer-power terms.

    4. SL 5.4Tangents and normals

      • Find tangents and normals at a given point and their equations. • Use analytic methods and technology.

    5. SL 5.5Integration as anti-differentiation

      • Integrate polynomial-type functions as anti-derivatives. • Use boundary conditions to determine constants. • Connect anti-derivatives, definite integrals and area under curves.

    6. SL 5.6Stationary points

      • Find x-values where gradient is zero by solving f'(x)=0. • Use technology to generate f'(x) and find local maxima/minima. • Local extrema may not be absolute extrema on a domain.

    7. SL 5.7Optimization

      • Solve optimization problems in context. • Examples include maximizing profit, minimizing cost and maximizing volume. • SL examinations do not include kinematics questions.

    8. SL 5.8Trapezoidal rule

      • Approximate areas using the trapezoidal rule. • Use equal-width intervals with a table of data or a function.

  2. 5.2 Calculus - AHL content

    1. • 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.

    2. • Use second derivative notation and second derivative test for maxima/minima. • Interpret concavity and points of inflexion in context.

    3. • 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.

    4. • Find areas enclosed by curves and x- or y-axes, including negative integrals. • Find volumes of revolution about the x-axis or y-axis.

    5. • 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.

    6. • Set up differential equations from contexts. • Solve separable differential equations; exponential models solve dy/dx=ky. • Know the term general solution.

    7. • Use and interpret slope fields and their diagrams.

    8. • 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.

    9. • 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.

    10. • 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.