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How does instantaneous rate of change differ from average rate of change?

Instantaneous rate of change is measured at one precise point, whilst average rate of change uses two points to average the slope between those two points.

In the tangent method, a line is drawn against the function that is the same angle or steepness of the function at that point. This is usually done by eye and is hence subjective and inaccurate compared to calculations of slope using algebraic differentiation. A right triangle can be formed against the tangent line, and two points used on this tangent line. These points are then used to calculate slope in the simple rise/run method. Here the red line is a tangent line to the curve. We then use two points on this red tangent line to calculate slope.

The average rate of change is found by connecting two points of the function and forming a right triangle that can also be used to calculate a slope. The difference is you use two points on the function itself, as opposed to two that are created by a tangent line. Here the green line is the secant line against which slope can be derived.

It's accuracy will especially depend on how much the curve changes between the two points. The closer the secant line 'fits' the curve the more accurately it will approximate slope between the two points used.

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