Dynamic Calculus
Home > The gradient function
The limiting position of the secant
The applet below shows the graph of the function \(f(x) = \frac{{{x^2}}}{4}\). Investigate the gradient of the secant \(PQ\) for varying positions of \(P\) and \(Q\).
[You can reset the applet at any time by clicking the 'circular arrows' icon in the top right corner]
- Investigate the function of the 'slider' by moving it left and right. As \(Q\) approaches \(P\) (first from one side of \(P\), and then from the other), what do you observe about the position of the secant \(PQ\) in relation to the position of the tangent at \(P\)?
- Verify that the gradient of the secant through the two points \((x, f(x))\) and \((c, f(c))\) is \(\frac{{f(x) - f(c)}}{{x - c}}\). Comment on the significance of this formula when \(x = c\).
- Click the checkboxes for the gradient of the secant and the gradient of the tangent. As \(Q\) moves closer to \(P\), what do you observe in relation to the gradients? Confirm your observation by clicking the 'difference' checkbox.
- Change the position of the point \(P\) (and therefore the position of the tangent at \(P\)) by dragging the point. Repeat steps (1) and (3). Are your observations the same for other positions of the point \(P\)?
- What is the limiting value of the gradient of the secant \(PQ\)?