Tangents, Stationary Points and Graphs
Use derivatives to determine tangent equations, monotonic intervals, stationary-point nature and a coherent curve sketch.
How to study Higher Mathematics
Practise a skill without a calculator first, connect it to neighbouring techniques, then apply it in unfamiliar contexts with complete working and interpretation.
Core concepts
Concept 1
tangent gradient
Exam cue: What is f′ at the specified point?
Concept 2
line equation
Exam cue: Where does f′ change sign?
Concept 3
increasing and decreasing
Exam cue: What features are needed for the full sketch?
Concept 4
stationary points
Concept 5
end behaviour
Risk pitfalls and guardrails
Using the point’s y-value as the tangent gradient
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Finding stationary x-values but not coordinates or nature
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Ignoring intercepts or end behaviour
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Memory anchors
Differentiate for gradient
Evaluate f′ at the point before forming the tangent line.
Point plus slope
Use y − y₁ = m(x − x₁).
Positive rises, negative falls
The sign of f′ controls increasing and decreasing intervals.
Zero, then nature
Solve f′ = 0 and test the sign around each stationary point.
Sketch needs anchors
Include intercepts, stationary points and large-|x| behaviour.
Coordinates need y
Substitute each stationary x-value into the original function.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
Find the gradient of y=x³−2x²+1 at x=2.
Find the tangent to y=x²+3x at x=1.
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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