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Calculus

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  1. 14. Calculus

    1. 14.1. Understand derived functions

      • Understand the idea of a derived function. • Only an informal understanding of limits is expected; differentiation from first principles is not required.

    2. 14.2. Use differentiation notation

      • Use notation for derivatives and increments, including f'(x), f''(x), dy/dx, d^2y/dx^2, delta y / delta x and the limiting notation as delta x -> 0.

    3. 14.3. Differentiate standard functions

      • Know and use derivatives of x^n for any rational n, sin x, cos x, tan x, e^x and ln x. • Includes constant multiples, sums and composite functions using the chain rule. • For trigonometric functions, angles are always in radians.

    4. 14.4. Differentiate products and quotients

      • Differentiate products and quotients of functions.

    5. 14.5. Find gradients, tangents and normals

      • Use differentiation to find gradients, tangents and normals.

    6. 14.6. Find stationary points

      • Use differentiation to find stationary points. • Points of inflexion are not included.

    7. 14.7. Apply differentiation to rates and approximations

      • Apply differentiation to connected rates of change, small increments and approximations.

    8. 14.8. Apply differentiation to maxima and minima

      • Apply differentiation to practical problems involving maxima and minima.

    9. 14.9. Use derivative tests for maxima and minima

      • Use first and second derivative tests to discriminate between maxima and minima. • Points of inflexion are not included. • Give full justification of conclusions. • Explain how to distinguish between a maximum point and a minimum point when required. • Unless specified otherwise, any valid method is allowed.

    10. 14.10. Understand integration as reverse differentiation

      • Understand integration as the reverse process of differentiation. • Solutions for indefinite integrals should include an arbitrary constant.

    11. 14.11. Integrate powers of x

      • Integrate sums of terms in powers of x, including 1/x and 1/(ax + b). • Solutions for indefinite integrals should include an arbitrary constant.

    12. 14.12. Integrate standard composite functions

      • Integrate functions of the form (ax + b)^n for any rational n, sin(ax + b), cos(ax + b), sec^2(ax + b) and e^(ax+b). • Includes the case where n = -1. • For trigonometric functions, angles are always in radians. • Solutions for indefinite integrals should include an arbitrary constant.

    13. 14.13. Evaluate definite integrals and areas

      • Evaluate definite integrals and apply integration to plane areas. • Plane areas include areas between a line and a curve, between two curves, and a sum of two areas.

    14. 14.14. Apply calculus to kinematics

      • Apply differentiation and integration to kinematics problems involving displacement, velocity and acceleration of a particle moving in a straight line with variable or constant acceleration.

    15. 14.15. Use kinematics graphs

      • Use the relationships in 14.14 to draw and use displacement-time, distance-time, velocity-time, speed-time and acceleration-time graphs.