14. Tangent Lines and Rate of Change
Dated: 30-10-2024
The secent line is the one which connects \(\overline{PQ}\).
The \(m_{sec}\) is the slope1
As the \(x_1 - x_0 \to 0\)
- The
secentline approaches thetangentline if \(x_1\) moves from right to \(x_0\). - The
tangentline approaches thesecentline if \(x_0\) moves from left to \(x_1\).
Therefore, the slope1 of tangent line is given by
\[m_{tan} = \lim_{x_1 \rightarrow x_0} \frac{f(x_1) - f(x_0)}{x_1 - x_0}\]
Average Velocity
The distance is defined to be a function2 of time.
Since we are taking the snapshots from a very large gap in time, we say this is average not instantaneous rate of change.
\[Average = \frac{f(t_1) - f(t_0)}{t_1 - t_0}\]
Instantaneous Velocity
\[instantaneous =\lim_{t_1 \rightarrow t_0} \frac{f(t_1) - f(t_0)}{t_1 - t_0}\]
References
Read more about notations and symbols.