GraphsStudy Notes

Worked Examples

Part of Gradients of Curves — GCSE Mathematics

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.

Topic position

Section 8 of 10

Practice

9 questions

Recall

10 flashcards

✏️ Worked Examples

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.

Show Solution

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: Finding the Exact Gradient Using Differentiation (Higher)

Question: Find the gradient of the curve y = x² + 3x at x = 2.

Show Solution

Step 1: Differentiate y with respect to x — using the power rule (multiply by power, reduce power by 1)

y = x² + 3x

dy/dx = 2x + 3

Step 2: Substitute x = 2

dy/dx = 2(2) + 3 = 4 + 3 = 7

Answer: Gradient at x = 2 is exactly 7

Practice Questions for Gradients of Curves

How do you find the gradient of a curve at a specific point?

A. Draw a chord joining two points on the curve and find its gradient
B. Draw a tangent to the curve at that point and find the gradient of the tangent
C. Find the average of the y-values on either side of the point
D. Divide the y-coordinate by the x-coordinate of the point

1 markfoundation

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.

2 markshigher

Quick Recall Flashcards

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.

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.