Example 1: Estimating Gradient from a Tangent
Question: A tangent is drawn to a distance-time curve at t = 4 seconds. The tangent passes through the points (2, 10) and (6, 50). Find the instantaneous speed at t = 4 s.
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Step 1: Choose two well-separated points on the tangent — (2, 10) and (6, 50) are given (far apart — good for accuracy)
Step 2: Calculate the gradient
m = (y₂ - y₁) ÷ (x₂ - x₁) = (50 - 10) ÷ (6 - 2) = 40 ÷ 4 = 10
Step 3: Include units — x-axis is time (s), y-axis is distance (m), so gradient has units m/s
Answer: Instantaneous speed at t = 4 s is 10 m/s
Example 2: Interpreting Gradient as a Rate of Change
Question: A temperature-time graph has a tangent drawn at t = 3 minutes. The tangent passes through (1, 20) and (5, 44). Find the rate of temperature change at t = 3 minutes.
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Step 1: Calculate the gradient using two points on the tangent
m = (44 − 20) ÷ (5 − 1) = 24 ÷ 4 = 6
Step 2: State units — x-axis is time (min), y-axis is temperature (°C), so gradient has units °C/min
Answer: Temperature is increasing at 6 °C per minute at t = 3 minutes
This study notes covers Worked Examples 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 8 of 10 in this topic. Use this study notes to connect the idea to the wider topic before moving on to questions and flashcards.
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.