Instantaneous Rates of Change
Instantaneous Rates of Change is a key algebra topic at GCSE Maths. This Higher worksheet gives you exam-style questions on estimating instantaneous rates of change using tangents, with a full mark scheme so you can check every method mark. Suitable for AQA, Edexcel and OCR. Download the free PDF and answers below. Draw a tangent to the curve and find its gradient.
Free downloads
These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.
This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.
Topic overview
An instantaneous rate of change is the rate at a single moment rather than across an interval. On a curve, it is found by drawing a tangent at the point and measuring its gradient.
A tangent is a straight line touching the curve at exactly one point, matching the curve's direction there. Its gradient tells you how fast the quantity is changing at that instant.
Because the tangent is drawn by hand, answers are approximate, and different reasonable tangents give slightly different results. Marks are awarded for a sensible tangent and correct gradient working, so the tangent must be visible on the diagram.
Revision notes
Drawing the tangent
Place a ruler so it touches the curve at the required point and follows the curve's direction there, crossing it nowhere else nearby.
Extend the line well beyond the point so you have a long section from which to read the gradient accurately.
Finding the gradient
Choose two clear points on the tangent, far apart and at grid intersections, then divide the rise by the run.
If the tangent passes through \((2, 6)\) and \((6, 18)\), the gradient is \(\frac{18-6}{6-2} = 3\).
Interpreting the answer
On a distance-time graph the tangent gradient is the speed at that instant; on a velocity-time graph it is the acceleration.
Always state units, and remember that a negative gradient means the quantity is decreasing at that moment.
Key points
- A tangent touches the curve at one point.
- Its gradient gives the instantaneous rate of change.
- Draw the tangent with a ruler and extend it.
- Use two far-apart points to find the gradient.
- Answers are approximate because the tangent is drawn by hand.
- Include units in the interpretation.
Worked examples
Example 1
A tangent passes through \((1, 3)\) and \((5, 15)\). Find its gradient.
Working
Example 2
On a distance-time graph a tangent rises \(24\)m over \(4\)s. Find the speed at that instant.
Working
Example 3
Explain why two students may get slightly different answers for the same tangent.
Working
Common mistakes
Drawing a chord instead of a tangent.
A chord cuts the curve twice and gives an average, not an instantaneous rate.
Using points too close together.
Small readings magnify errors. Use points far apart on the tangent.
Not drawing the tangent at all.
Method marks require the tangent to be visible on the graph.
Forgetting the sign.
A tangent sloping downwards gives a negative rate, meaning the quantity is decreasing.
Exam tips
- Draw the tangent long so you can read it accurately.
- Pick gridline intersections for your two points.
- Leave the tangent on the diagram for the method marks.
- Interpret the gradient in context with units.
Key terms
- Tangent
- A line touching a curve at exactly one point.
- Instantaneous
- At a single moment in time.
- Gradient
- The steepness of a line.
- Approximate
- Close to but not exactly the true value.
Related topics
Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.