AP Precalculus / Unit 4: Functions Involving Parameters, Vectors, and Matrices
4.3 Parametric Functions and Rates of Change
Notes
Average rates of change:
- Horizontal: $\displaystyle \frac{x(b)-x(a)}{b-a}$; vertical: $\displaystyle \frac{y(b)-y(a)}{b-a}$.
- Slope of the path between those points: $\displaystyle \frac{y(b)-y(a)}{x(b)-x(a)}$.
Practice
$x(t)=t^2$, $y(t)=3t$. Average rate of change of $x$ on $[1,3]$:
$\displaystyle \frac{9-1}{2}$.
For the same function, the slope of the path between $t=1$ and $t=3$:
$\displaystyle \frac{9-3}{9-1}=\frac{6}{8}$.
If $y(t)$ is decreasing, the object is moving:
Vertical position falls.
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