Differentiation and integration
Build and connect the representations and methods in differentiation and integration.
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Before you begin
Build and connect the representations and methods in differentiation and integration.
Learning goals
- Interpret a derivative as gradient or rate of change
- Find and interpret a tangent to a curve
- Differentiate within the permitted function scope
- Apply the chain rule to composite functions
- Apply product and quotient differentiation rules
- Find antiderivatives within the permitted scope
- Classify stationary points using justified derivative tests
- Find and interpret a normal to a curve
- Relate connected rates of change
- Model and solve an optimisation problem
- Evaluate definite integrals
- Relate a definite integral to area under a curve
- 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
Lessons
Work through them in order.
About lesson progress
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- Function, gradient and local changeDistinguish function value from local rate of change and connect derivative tests to curve behaviour.
- Power, chain, product and quotient rulesChoose differentiation rules from the expression structure, retain real domains, and verify combined derivatives by another form.
- Trigonometric and exponential calculusUse radians, preserve composite domains and verify standard derivatives, antiderivatives and a recovered curve against its original conditions.
- Stationary points, tangents and normalsInterpret derivative signs, classify stationary points, and distinguish local behaviour from extrema on a stated domain.
- Connected rates and optimisationConnect changing dimensions and net rates, then justify constrained optima using the original model and feasible domain.
- Antiderivatives and positive areaRecover antiderivatives with their conditions, then select the correct boundaries and vertical separation for positive area.
- Signed areas and motionUse sign intervals and initial conditions to distinguish signed area, position, displacement, distance and changing speed.