About this topic
Before you begin
Connect derivatives with tangents, stationary points, rates and optimisation, then recover antiderivatives and calculate positive areas.
Learning goals
- Find and interpret a tangent to a curve
- Find the complete tangent to a polynomial at a stated input
- Interpret a derivative as gradient or rate of change
- Differentiate power sums on the stated input interval
- Differentiate within the permitted function scope
- Differentiate composite functions, products and quotients exactly
- Apply the chain rule to composite functions
- Apply product and quotient differentiation rules
- Find a complete second derivative on the stated input interval
- Find and classify every stationary point of a polynomial
- Find the greatest polynomial value and every attaining input on a closed interval
- Find the least polynomial value and every attaining input on a closed interval
- Find the complete normal to a polynomial curve at a stated input
- Find every input where a polynomial is increasing or decreasing
- Classify stationary points using justified derivative tests
- Find and interpret a normal to a curve
- Find a connected rate of change from a stated relation
- Form a polynomial model from a described situation and find its optimum
- Relate connected rates of change
- Model and solve an optimisation problem
- Model and evaluate a nonnegative polynomial curve–line region with vertical strips
- Find complete power antiderivatives on the stated interval
- Evaluate a polynomial definite integral with linked exact working
- Find antiderivatives within the permitted scope
- Evaluate definite integrals
- Relate a definite integral to area under a curve
Syllabus statements covered
- Derivative of f (x) as the gradient of the tangent to the graph of y = f (x) at a point
- Derivative as rate of change
- Use of standard notations f′(x), f″(x), dy/dx, d²y/dx² = d/dx(dy/dx)
- Derivatives of x^n , for any rational n, together with constant multiples, sums and differences
- Derivatives of products and quotients of functions
- Use of Chain Rule
- Increasing and decreasing functions
- Stationary points (maximum and minimum turning points and stationary points of inflexion)
- Use of second derivative test to discriminate between maxima and minima
- Apply differentiation to gradients, tangents and normals, connected rates of change and maxima and minima problems
- Integration as the reverse of differentiation
- Integration of x^n for any rational n, (excluding n = −1 ), together with constant multiples, sums and differences
- Integration of (ax + b)n for any rational n, (excluding n = −1 )
- Definite integral as area under a curve
- Evaluation of definite integrals
- Finding the area of a region bounded by a curve and line(s) (excluding area of region between 2 curves)
- Derivative of f (x) as the gradient of the tangent to the graph of y = f (x) at a point
- Derivative as rate of change
- Use of standard notations f′(x), f″(x), dy/dx, d²y/dx² = d/dx(dy/dx)
- Derivatives of x^n , for any rational n, together with constant multiples, sums and differences
- Derivatives of products and quotients of functions
- Use of Chain Rule
- Increasing and decreasing functions
- Stationary points (maximum and minimum turning points and stationary points of inflexion)
- Use of second derivative test to discriminate between maxima and minima
- Apply differentiation to gradients, tangents and normals, connected rates of change and maxima and minima problems
- Integration as the reverse of differentiation
- Integration of x^n for any rational n, (excluding n = −1 ), together with constant multiples, sums and differences
- Integration of (ax + b)^n for any rational n, (excluding n = −1 )
- Definite integral as area under a curve
- Evaluation of definite integrals
- Finding the area of a region bounded by a curve and line(s) (excluding area of region between 2 curves)
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 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.
- Not startedAntiderivatives and positive areaNextRecover antiderivatives with their conditions, then select the correct boundaries and vertical separation for positive area.