Algebra · Graphs of functions
Chapter 1 · 3
The idea
Rates of change from graphs (tangents, chords and the area under a graph)
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Algebra · Graphs of functions
Rates of change from graphs (tangents, chords and the area under a graph)
Read a rate off a curved graph: the gradient of a chord is the average rate of change over an interval, the gradient of a tangent is the rate at one instant, and the units on the axes say what that rate is (a speed, an acceleration, litres per minute). Then the other half: the area under a speed–time graph is the distance travelled, found exactly from straight pieces or estimated with strips, with a reason why the estimate is too big or too small.
Why it works
The tram's speed keeps changing
A tram leaves one stop and pulls into the next stop seconds later, m along the line. The graph shows how far it has travelled at each moment.Its average speed for the trip is m/s, yet the tram is hardly ever doing m/s. It starts at rest, speeds up, then slows down to stop. On a travel graph made of straight lines, the speed is the gradient of the line. This graph is a curve: gentle at both ends and steepest in the middle, so it has no single gradient. This page reads a rate off a curve, first over a stretch of time and then at one instant.Keep reading — free
The rest of the explanation, plus 4 worked examples you step through move by move.
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