GCSE · Maths · AQA · Spec 8300 · Higher
Instantaneous and average rate of change (Higher)
Your average speed on a trip isn't what the speedometer showed as you passed the school. Graphs ask both questions: how fast on average, and how fast right now?
A car pulls away from the lights. Slide through time and watch the red line tilt.
The red line is the tangent. It touches the curve at the moment you pick and runs in the same direction as the curve there. Its gradient is the car's speed at that instant. At t = 2 s it climbs 4 m over 2 s, so the speed is 2 m/s. At t = 6 s it climbs 12 m over 2 s, so the speed is 6 m/s. Same car, same curve, three times the speed.
The curve shows how far the car has gone after t seconds. Notice that the curve never moves: only the tangent does.
Watch it done: one curve, both kinds of rate
Problem
A ball rolls down a slope. The distance it has rolled, d metres, after t seconds is d = t² + t, for 0 ≤ t ≤ 5. (a) Find the average speed of the ball between t = 1 and t = 4. (b) Draw the graph and use a tangent to estimate the speed of the ball at t = 3.
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
On a curve, the rate of change never sits still. Here's how to catch it at one exact moment, and how that differs from an average.
What you need to know
- A chord is a straight line joining two points on a curve. Its gradient is the average rate of change between those points.
- A tangent is a straight line that just touches a curve at one point, going in the same direction as the curve there. Its gradient is the instantaneous rate of change at that point.
- Gradient = change in y ÷ change in x, for a chord or a tangent.
- To estimate a tangent's gradient: draw it with a ruler, choose two points on the tangent far apart, then divide the change in y by the change in x.
- The units of a rate are the y-units per x-unit, such as m/s or litres per minute. A negative rate means the quantity is going down.
The big picture
A rate of change tells you how fast one quantity changes compared with another, such as metres per second. On a curve, that rate is different at every point. The average rate of change between two points is the gradient of the chord joining them, and you can calculate it exactly from the two values. The instantaneous rate of change at one point is the gradient of the tangent there: draw the tangent, read two points far apart on it, and work out change in y ÷ change in x. Give the answer with units, and say what it means in the context.
Key points
Worked example
Problem
A cup of tea cools. The table shows its temperature, T °C, t minutes after it was made. t (min): 0, 5, 10, 15, 20 T (°C): 85, 62, 47, 37, 31 Find the average rate of change of temperature between t = 5 and t = 15, and say what it means.
⚠ Watch out
Counting squares instead of reading the scales. If each x-square is 2 and each y-square is 10, a tangent climbing 3 squares over 3 squares has gradient 30 ÷ 6 = 5, not 1. Read both changes from the axis numbers.
Memory hook
Two points, chord, average. One point, tangent, right now.
Check yourself
Chord or tangent? (a) A runner's average speed between 2 s and 5 s. (b) A balloon's speed at exactly 4 s. (c) How fast a drink cools at 10 minutes. (Answers: chord, tangent, tangent.)
Flashcards
(6)The gradient of a chord gives you…
The gradient of a tangent at a point gives you…
What is a tangent to a curve?
How do you estimate the gradient of a tangent you've drawn?
What are the units of a rate of change?
What does a negative rate of change mean?
Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.
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