GCSE · Maths · Edexcel · Spec 1MA1 · Higher
Gradients and areas under curves
A straight road climbs at one steady rate. A hill doesn't: gentle, then steep, then flat on top. Curves are hills, so 'what's the gradient?' needs another question: where?
Ride the curve y = x²
The red line is a tangent. It touches the curve at the small dot and has the same gradient as the curve there. Left of the turning point the curve falls, so the gradient is negative. At the turning point the tangent is flat: gradient 0. At x = 1 it is 2, and at x = 2 it is 4 — the curve keeps getting steeper.
The big dot marks the turning point, where the curve stops falling and starts rising. The number beside it is the gradient of the red tangent, wherever you have dragged the small dot to.
Predict, then check
No tangent this time — just straight lines joining two points on the curve. Those are called chords.
On y = x², the chord from (0, 0) to (3, 9) has gradient 9 ÷ 3 = 3. Now split it into three shorter chords: x = 0 to 1, x = 1 to 2 and x = 2 to 3. What gradients will those three chords give?
Reading a tangent you've drawn
Problem
A curved distance–time graph shows a journey, with distance in metres and time in seconds. A tangent has been drawn by hand at t = 3. Estimate the speed at t = 3.
What does this bit of the graph tell you?
Pick a feature, then pick what it tells you.
Still to sort
Average speed (0)
Speed over a whole interval.
Where the line is: An average needs two points (a chord). An instant needs one point (a tangent).
Speed at an instant (0)
Speed at one moment.
Where the line is: Only on a distance–time graph. The same tangent on a speed–time graph means something else.
Acceleration (0)
How fast the speed is changing.
Where the line is: Check the y-axis: if it shows speed, a gradient is an acceleration, not a speed.
Distance travelled (0)
How far, not how fast.
Where the line is: On a speed–time graph, distance comes from the area under the line, not from a gradient.
Same maths, different graphs. Sort each feature into what it actually measures.
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
A straight line has one gradient. A curve has a different gradient at every point. Drag the dot along the curve below and watch the red line turn.
What you need to know
- Gradient = change in y ÷ change in x, moving in the positive x-direction. If y falls as x increases, the gradient is negative.
- A straight line has one gradient everywhere; a curve's gradient changes from point to point.
- A chord gives an estimate of a curve's gradient between two points. The closer the points, the better the estimate.
- The gradient at a single point is the gradient of the tangent there. At a turning point it is 0.
- Distance–time graph: gradient = speed. Speed–time graph: tangent gradient = acceleration, and area under the graph = distance travelled.
The big picture
A curve's gradient changes from point to point. A chord between two points gives an estimate of the gradient between them, and the tangent at a point gives the gradient at that point. On a distance–time graph the gradient is a speed; on a speed–time graph a tangent's gradient is the acceleration and the area underneath is the distance travelled — exact for straight-line graphs, estimated with trapezia for curves.
Key points
Worked example
Problem
Estimate the gradient of the curve y = 5 − x² between x = 0 and x = 2.
⚠ Watch out
Treating one long chord as 'the gradient of the curve'. A chord only gives an average between two points, and across a turning point it can even give 0. For the gradient at a point, use the tangent at that point.
Memory hook
Slope says how fast, area says how far. A chord gives the average; a tangent gives the instant.
Check yourself
On a speed–time graph, what does a tangent's gradient tell you, and what does the area underneath tell you? And why can a long chord on a curve give a gradient of 0?
Flashcards
(15)What does the gradient of a graph measure?
Why do any two points on a straight line give the same gradient?
How do you tell the sign of a gradient before calculating?
How do you estimate a curve's gradient between two points?
Why are chords between close points better estimates?
What can a chord between far-apart points hide?
What is a tangent to a curve at a point?
Can a tangent meet the curve again somewhere else?
What is the gradient of the tangent at a turning point?
How do you get a gradient from a hand-drawn tangent?
On a distance–time graph, what do a chord and a tangent gradient give?
On a speed–time graph, what does the gradient of a tangent give?
Why is the area under a speed–time graph a distance?
How do you estimate the area under a curved speed–time graph?
When is an area estimate an underestimate, and when an overestimate?
Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.
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