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How Fast is Fast Right Now?
How Fast is Fast Right Now?
A car's speedometer does not show the average speed over a journey — it shows your speed RIGHT NOW at this instant. Mathematically, that instantaneous speed is the gradient of a distance-time graph at that precise moment. But on a curved graph, the gradient is constantly changing. To find it at any single point, we draw a tangent — a straight line that just touches the curve — and calculate its gradient. This is the bridge between graphs and calculus, and it is a powerful Higher tier skill.
What is a tangent to a curve?: A straight line that touches the curve at exactly one point and has the same gradient as the curve at that point.
It does NOT cross through the curve at that point — it only touches it.
The gradient of the tangent = the gradient of the curve at that point.
Used to estimate the rate of change at an instant.
Mathematics glossary
- What is a tangent to a curve?
- A straight line that touches the curve at exactly one point and has the same gradient as the curve at that point. It does NOT cross through the curve at that point — it only touches it. The gradient of the tangent = the gradient of the curve at that point. Used to estimate the rate of change at an instant.
Earn the mark scheme marks
💡 Exam Tips for Gradients of Curves
- Choose points far apart on the tangent for the gradient calculation — nearby points introduce larger percentage errors from drawing inaccuracy
- Label your tangent clearly and show the two points you used for the gradient calculation
- Always include units: gradient has units formed from the axis labels (m/s, °C/min, etc.)
- Zero gradient = turning point: state "the rate of change is zero" or "the maximum/minimum occurs here"
- Mark scheme tolerance: gradient estimates from tangents are usually accepted within a range — check the question; answers within ±1 of the expected value typically earn the mark
Now try it yourself
Quiz · Question 1 of 9
How do you find the gradient of a curve at a specific point?
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