AP Calculus AB / Unit 2: Differentiation: Definition and Fundamental Properties
2.1 Defining Average and Instantaneous Rates of Change at a Point
Notes
- Average rate of change on $[a,b]$: $\displaystyle \frac{f(b)-f(a)}{b-a}$, the slope of the secant line.
- Instantaneous rate of change at $x=a$: $\displaystyle \lim_{h\to0}\frac{f(a+h)-f(a)}{h}$, the slope of the tangent line.
Practice
For $f(x)=x^3$, the average rate of change on $[0,2]$ is:
$\displaystyle \frac{8-0}{2-0}=4$.
The instantaneous rate of change of $f$ at $x=a$ is the slope of:
A secant slope becomes a tangent slope in the limit.
$V(t)$ is volume in liters and $t$ is minutes. The units of $V'(t)$ are:
Output units per input unit.
Ready for more? Take the Unit 2 test.