Calculus
- Syllabus
- 0606–2028–2029
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Topic 14
14. Calculus
Objectives in this topic
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.
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.
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.
14.4. Differentiate products and quotients
- Differentiate products and quotients of functions.
14.5. Find gradients, tangents and normals
- Use differentiation to find gradients, tangents and normals.
14.6. Find stationary points
- Use differentiation to find stationary points.
- Points of inflexion are not included.
14.7. Apply differentiation to rates and approximations
- Apply differentiation to connected rates of change, small increments and approximations.
14.8. Apply differentiation to maxima and minima
- Apply differentiation to practical problems involving maxima and minima.
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.
14.10. Understand integration as reverse differentiation
- Understand integration as the reverse process of differentiation.
- Solutions for indefinite integrals should include an arbitrary constant.
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.
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.
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. 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.
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.