About this topic
Before you begin
Differentiate and integrate algebraic, trigonometric and exponential functions, then apply rates, optimisation, signed areas and motion models.
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
- Differentiate exponential compositions on the stated input interval
- Differentiate logarithmic compositions on the stated input interval
- Differentiate mixed function sums on the stated input interval
- Find complete exponential antiderivatives on the stated interval
- Find antiderivatives within the permitted scope
- Differentiate expressions with tan(ax + b) exactly
- Integrate sec²(ax + b) exactly
- 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
- Evaluate definite integrals
- Relate a definite integral to area under a curve
- Model and evaluate a polynomial curve–line region extending below the x-axis
- Find motion values, rest and reversal times, displacement and distance
- Relate definite integrals to signed area and geometric area
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, sin x , cos x , tan x , e^x and ln x , 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, sin x , cos x , sec² x and e^x , together with constant multiples, sums and differences
- Integration of (ax + b)^n for any rational n, sin(ax + b), cos(ax + b) and e^(ax+b)
- 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)
- Finding areas of regions below the x-axis
- Application of differentiation and integration to problems involving displacement, velocity and acceleration of a particle moving in a straight line
- 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, sin x , cos x , tan x , e^x and ln x , 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, sin x , cos x , sec² x and e^x , together with constant multiples, sums and differences
- Integration of (ax + b)^n for any rational n, sin(ax + b), cos(ax + b) and e^(ax+b)
- 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)
- Finding areas of regions below the x-axis
- Application of differentiation and integration to problems involving displacement, velocity and acceleration of a particle moving in a straight line
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 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.
- Not startedSigned areas and motionNextUse sign intervals and initial conditions to distinguish signed area, position, displacement, distance and changing speed.