Instantaneous Rate of Change [H]

Instantaneous Rate of Change

For a curve, the gradient changes from point to point. The instantaneous rate of change at a point equals the gradient of the tangent there. The average rate of change between two points equals the gradient of the chord.

\[ \text{average rate} = \frac{f(b)-f(a)}{b-a} \qquad \text{instantaneous rate} = \text{gradient of tangent at } P \] x y A B chord — average rate P tangent

Estimating the Gradient

  1. Draw the tangent at the required point.
  2. Read two widely spaced points on it.
  3. Compute gradient \( = \Delta y / \Delta x \).

As the chord shortens, its gradient approaches the tangent gradient.

Worked Examples

Height \( h \): at \( t=1 \), \( h=15 \); at \( t=3 \), \( h=7 \). Average rate of change:
\[ \frac{7-15}{3-1} = -4 \text{ m/s} \]
Tangent at \( t=2 \) through \( (0,18) \) and \( (4,10) \). Instantaneous rate:
\[ \frac{10-18}{4-0} = -2 \text{ m/s} \]

 Key Takeaways

  • Average rate = chord gradient \( = \Delta y / \Delta x \).
  • Instantaneous rate = tangent gradient at a single point.
  • Use two widely spaced points on the tangent for accuracy.
  • As the chord shortens, its gradient approaches the instantaneous rate.