Dynamic Calculus

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Average gradient of a curve

The graph below shows a section of the curve \(y = \frac{{{x^2}}}{4} + 2\). The average gradient of the curve between \(P\) and \(Q\) is the gradient of the secant \(PQ\). Investigate the average gradient of the curve by changing the position of the points \(P\) and \(Q\).
[You can reset the applet at any time by clicking the 'circular arrows' icon in the top right corner]

 

1. Reposition the points \(P\) and \(Q\) then calculate the gradient of the secant joining them. Check your answer by clicking the 'checkbox'.
2. What do you observe about the position of the secant as \(P\) approaches \(Q\)?
3. For the curve \(y = f(x)\), show that the gradient of the secant passing through the points \((x, f(x))\) and \((c, f(c))\) is \(\frac{{f(x) - f(c)}}{{x - c}}\).

State of New South Wales, Department of Education and Training 2008, Created with GeoGebra