Gradients and areas under graphs
This is a Higher topic on Edexcel 1MA1, examined regularly on Papers 2H and 3H. The skill: read or estimate a gradient or area from a graph and interpret the result.
Gradient under a curve
The gradient at a point on a curve = slope of the tangent at that point. See R15 for the tangent technique.
Area under a graph
For straight-line segments, decompose the region into rectangles, triangles, and trapezia.
- Rectangle: A = base × height.
- Triangle: A = (1/2) × base × height.
- Trapezium: A = (1/2) × (a + b) × h, where a and b are the parallel sides and h is the perpendicular distance.
Estimating area under a curve — trapezium rule (informal)
Divide the area into vertical strips of equal width h. For each strip, draw a trapezium connecting the curve at the left and right ends. Sum the trapezia:
A ≈ (h/2) × [y₀ + 2(y₁ + y₂ + ... + yₙ₋₁) + yₙ]
This is an estimate — narrower strips give a closer approximation.
Contextual interpretations
- Velocity-time graph area = distance travelled.
- Force-distance graph area = work done (in mechanics, IGCSE/A-Level only).
- Rate-of-flow vs time area = total volume flowed.
- Power-time area = energy.
Common Edexcel mark-scheme phrasing
- M1 for valid decomposition into known shapes.
- M1 for each shape's area calculation.
- A1 for the total.
- B1 for contextual interpretation with units.
⚠Common mistakes— Common errors
- Counting squares without using the axis scales.
- Forgetting to multiply by (1/2) for a triangle.
- Including only one strip in the trapezium rule (need at least 3–5 for a usable estimate).
- Stating "distance travelled = 80" instead of "80 m" — units matter.
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