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5 Calculus

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  1. 5.1 Calculus - SL content

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

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

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

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

    6. • Differentiate x^n (n in Q), sin x, cos x, e^x and ln x. • Use sum/multiple rules, chain rule, product rule and quotient rule.

    7. • Use second derivative notation d2y/dx2 and f''(x). • Relate graphs of f, f' and f''; use technology where useful.

    8. • Find local maxima/minima using first-derivative sign changes or second derivative tests. • Solve optimization problems in contexts such as profit, area and volume. • Identify points of inflexion and concavity.

    9. • Use displacement s, velocity v, acceleration a and total distance travelled. • v=ds/dt and a=dv/dt=d2s/dt2. • Displacement is integral of velocity; distance is integral of speed |v(t)|.

    10. • Integrate x^n, sin x, cos x, 1/x and e^x. • Use composites with linear functions and reverse chain rule/substitution by inspection.

    11. • Use definite integral analytically: integral_a^b g'(x) dx = g(b)-g(a). • Find areas under curves and between curves, accounting for sign. • Write correct integral expressions before calculating.

  2. 5.2 Calculus - AHL content

    1. AHL 5.12 (HL)—Continuity, differentiability and first principles

      • Understand continuity and differentiability informally. • Use convergence/divergence of limits. • Define derivative from first principles for polynomials; use higher-derivative notation.

    2. AHL 5.13 (HL)—L'Hopital's rule and limits

      • Evaluate limits of f(x)/g(x) as x approaches a value or infinity. • Use L'Hopital's rule or Maclaurin series for indeterminate forms 0/0 and infinity/infinity. • Repeated use of L'Hopital's rule may be required.

    3. AHL 5.14 (HL)—Implicit differentiation and related rates

      • Use implicit differentiation and related rates. • Solve optimization problems, including endpoint cases where appropriate.

    4. AHL 5.15 (HL)—Further derivatives and integrals

      • Differentiate tan x, sec x, cosec x, cot x, a^x, log_a x, arcsin x, arccos x and arctan x. • Integrate their derivative forms, including composites with linear functions. • Use partial fractions to rearrange integrands.

    5. AHL 5.16 (HL)—Advanced integration techniques

      • Use integration by substitution; substitutions are provided where not directly recognizable. • Use integration by parts, including repeated integration by parts.

    6. AHL 5.17 (HL)—Areas and volumes of revolution

      • Find area enclosed by a curve and the y-axis over an interval. • Find volumes of revolution about the x-axis or y-axis.

    7. AHL 5.18 (HL)—Differential equations

      • Solve first-order differential equations. • Use Euler's numerical method. • Solve separable, homogeneous and first-order linear equations with integrating factor.

    8. AHL 5.19 (HL)—Maclaurin series

      • Use Maclaurin series for e^x, sin x, cos x, ln(1+x) and (1+x)^p. • Obtain other series by substitution, products, integration and differentiation. • Develop Maclaurin series from differential equations.