Definition of Gradient

To estimate the gradient geometrically, one must first carefully draw a tangent line at the desired point using a ruler, ensuring it just 'skims' the curve.

Once the tangent is drawn, identify two distinct points on that line, $(x_1, y_1)$ and $(x_2, y_2)$, to calculate the slope.

Apply the standard linear gradient formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

The resulting value $m$ is the estimated gradient of the curve at the point of tangency.

Check for Smoothness: Always verify if the curve is smooth at the point in question; gradients are undefined at sharp corners or 'kinks' in a graph.

Tangent Accuracy: When drawing a tangent by hand, ensure the angles between the curve and the line are balanced on both sides of the point of contact.

Scale Awareness: When calculating $\frac{\Delta y}{\Delta x}$ from a graph, always check the axis scales, as they may not be 1:1, which can lead to calculation errors if you only count grid squares.

Sign Check: Perform a quick sanity check on the sign of your gradient; if the curve is sloping upwards from left to right, the gradient must be positive.

The 'No-Crossing' Myth: A common mistake is believing a tangent can never cross the curve. While it doesn't cross at the point of tangency, it can cross the curve elsewhere.

Confusing Average and Instantaneous: Students often calculate the gradient between two points on the curve (average) instead of drawing a tangent at a single point (instantaneous).

Undefined Gradients: Gradients do not exist at points where the function is discontinuous or has a vertical tangent (where the change in $x$ is zero).