About this topic
Before you begin
Interpret derivative graphs and solve problems about tangents, normals, optimisation and connected rates.
Learning goals
- Relate a function to its derivative graphs
- Find and interpret a tangent to a curve
- Find the complete tangent to a polynomial at a stated input
- Construct a polynomial tangent using its derivative and contact values
- Differentiate implicit and parametric relationships within stated scope
- Find and interpret a normal to a curve
- Classify stationary points using justified derivative tests
- Relate connected rates of change
- Model and solve an optimisation problem
- Form a polynomial model from a described situation and find its optimum
- Find every input where a polynomial curve is concave upward or downward
- Find the gradient and the tangent or normal at a point on an implicit curve
- Find dy/dx in terms of the parameter and the tangent or normal at a point
Syllabus statements covered
- Include: graphical interpretation of f′(x) > 0, = 0, < 0 and f″(x) > 0, < 0; relating f′ and f graphs; simple implicit and parametric differentiation; first or second derivative tests for local extrema and stationary points of inflexion in simple cases; calculator or software location of extrema and approximate derivative at a point; tangents and normals including implicit and parametric curves; local extrema and connected rates of change. Exclude non-stationary points of inflexion and second derivatives of parametric functions.
- Include: graphical interpretation of f′(x) > 0, = 0, < 0 and f″(x) > 0, < 0; relating f′ and f graphs; simple implicit and parametric differentiation; first or second derivative tests for local extrema and stationary points of inflexion in simple cases; calculator or software location of extrema and approximate derivative at a point; tangents and normals including implicit and parametric curves; local extrema and connected rates of change. Exclude non-stationary points of inflexion and second derivatives of parametric functions.
Choose a lesson
Work through them in order.
- Not startedFunction, gradient and local changeNextDistinguish function value from local rate of change and connect derivative tests to curve behaviour.
- Not startedPower, chain, product and quotient rulesNextChoose differentiation rules from the expression structure, retain real domains, and verify combined derivatives by another form.
- Not startedTrigonometric and exponential calculusNextUse radians, preserve composite domains and verify standard derivatives, antiderivatives and a recovered curve against its original conditions.
- Not startedImplicit and parametric tangentsNextFind branch-specific tangents and normals, handle zero coordinate rates, and verify slopes across curve representations.
- Not startedStationary points, tangents and normalsNextInterpret derivative signs, classify stationary points, and distinguish local behaviour from extrema on a stated domain.
- Not startedConnected rates and optimisationNextConnect changing dimensions and net rates, then justify constrained optima using the original model and feasible domain.