Estimating Gradients & Areas under Graphs (AQA GCSE Maths)

A tangent is a line that touches a curve at a point (and does not cross it).

The gradient of a curve at a point is defined to be the same as the gradient of the tangent to the curve at that point.

False.

Drawing a tangent line to a curve at a point will not always give the exact gradient of the curve at that point.

A tangent line is drawn by eye so it will give an estimate of the gradient.

To find an estimate for the gradient of a curve at a point:

• Draw a tangent to the curve at the point.

• Find the gradient of the tangent using

  • Gradient = rise ÷ run

  • or difference in y ÷ difference in x.

• The gradient of the tangent will give the estimate for the gradient of the curve at the point.

To find an estimate for the area under a graph:

• Split area into vertical strips.

• Draw straight lines at top of the strips.

• Find area of strips (trapeziums) using Area = ½(a + b)h.

• Add the areas.

When estimating the area under a graph, you can tell if your answer is an underestimate or overestimate by looking at the straight lines at the top of the trapeziums.

If the straight lines are below the curve, it is an underestimate.

If the straight lines are above the curve, it is an overestimate.