Estimate the rate of change at a point on a curve from a tangent, compare it with an average rate from a chord, and (OCR J560) use the kinematics formulae v = u + at, s = ut + ½at² and v² = u² + 2as, which link to the gradient and area of a speed-time graph.
What you will learn
- Estimate a gradient at a point by drawing a tangent and using two points on it (Higher)
- Interpret the gradient of a tangent as a speed, an acceleration or a rate of cooling, with units (Higher)
- Tell an average rate (chord) from a rate at an instant (tangent) (Higher)
- Choose and use the kinematics formulae v = u + at, s = ut + ½at² and v² = u² + 2as (OCR J560)
A tangent shows the gradient at one point (Higher)
A curve's steepness changes as you move along it.
The gradient at a point is the gradient of the tangent there: the straight line that just touches the curve at that point and has its direction. Draw it with a ruler, then pick two points on the tangent that are far apart, ideally where it crosses grid lines.
- tangent at t = 4 through (2, 4) and (7, 24)
- gradient = (24 − 4) ÷ (7 − 2) = 20 ÷ 5 = 4
- acceleration at t = 4 ≈ 4 m/s²
It is an estimate because the tangent is drawn by eye.
The rest of this lesson and its 10 practice questions are locked.
In this lesson
- A tangent shows the gradient at one point (Higher)
- What the gradient means (Higher)
- The kinematics formulae (OCR J560)
- Try it: First question
10 practice questions, marked as you go; about 30 minutes in all.
Key terms
Tangent, Acceleration, Chord, Average
In this unit: Graphs and coordinates: shapes on axes, regions, areas and tangents
More graphs and coordinate geometry: shapes on coordinate axes, gradients and perpendicular lines in shapes, regions with strict and inclusive boundaries, the area under a curve from trapezia, and the gradient at a point from a tangent.
- Shapes on coordinate axes
- Gradients and lines in shapes
- Regions in depth
- Area under a curve
- Gradient at a point, and the kinematics formulae
All of Graphs and coordinates: shapes on axes, regions, areas and tangents