This introduction covers How Fast is Fast Right Now? within Gradients of Curves for GCSE Mathematics. Revise Gradients of Curves in Graphs for GCSE Mathematics with 9 exam-style questions and 10 flashcards. This topic appears less often, but it can still be a useful differentiator on mixed-topic papers. It is section 1 of 10 in this topic. Use this introduction to connect the idea to the wider topic before moving on to questions and flashcards.
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.
Practice questions for Gradients of Curves
How do you find the gradient of a curve at a specific point?
Explain why the gradient of a chord between two points on a curve is only an estimate of the gradient at a point, and how this estimate can be improved.
Quick recall flashcards
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.
Steps to estimate the gradient of a curve at a point
1. Mark the point on the curve
2. Place a ruler so it just TOUCHES the curve at that point (tangent)
3. Make the tangent line extend well across the graph
4. Choose two clear points on the tangent line
5. Calculate: gradient = (y2 - y1)/(x2 - x1)
Tip: use points far apart on the tangent for greater accuracy.