Instantaneous Rates of Change
The speed shown on your car’s speedometer at a specific moment is an instantaneous rate of change. This guide explores how to estimate this from a curved graph by finding the gradient of a tangent — a key higher-tier skill.
Average vs. Instantaneous Rate of Change
Average Rate of Change
Calculated over an interval. On a graph it’s the gradient of the chord — a straight line connecting two points on the curve.
Instantaneous Rate of Change
The rate at a specific moment. On a graph it’s the gradient of the tangent — a straight line that just touches the curve at exactly one point.
How to Estimate the Gradient of a Curve
- Locate the point on the curve where you need the rate of change.
- Draw a tangent at that point using a ruler. It should “kiss” the curve and match its steepness there.
- Choose two clear points on your tangent line that are easy to read from the grid.
- Calculate the gradient: $\dfrac{\text{change in }y}{\text{change in }x}$.
- Interpret your answer in context, including the correct units.
Worked Example
Estimating Speed from a Distance-Time Graph
Estimate the speed of the car at $t = 6$ seconds.
- Draw a tangent to the curve at the point where $t = 6$.
- Read two clear points on the tangent: $(4, 20)$ and $(8, 36)$.
- Calculate the gradient: $\dfrac{36 – 20}{8 – 4} = \dfrac{16}{4} = 4$.
- Interpret: The gradient of a distance-time graph is speed.
Answer: The speed at $t = 6$ s is approximately $4 \text{ m/s}$.
Tutor Insights
🤔 Common Misunderstandings
- Drawing a chord instead of a tangent. A chord connects two points; a tangent just touches the curve at one.
- Reading points off the curve to calculate the gradient, rather than off the tangent line you’ve drawn.
📝 Common Exam Mistakes
- Inaccurate tangent drawing. A poorly drawn tangent will give an answer outside the accepted range.
- Not showing the tangent line on the graph — the examiner needs to see it.
- Forgetting units (e.g., m/s for speed, m/s² for acceleration) in the final answer.
Practice Questions
- Explain the difference between a chord and a tangent on a curve.
- The gradient of a velocity-time graph represents a specific quantity. What is it?
- On the distance-time graph in the worked example, is the car speeding up or slowing down at $t = 6$ seconds? How can you tell?
Show Answers
- A chord connects two different points on a curve. A tangent touches the curve at a single point without crossing it.
- The gradient of a velocity-time graph represents acceleration.
- The car is slowing down. The curve is becoming less steep over time, meaning the gradient (speed) is decreasing.
Need Help with Rates of Change?
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