AP Calculus BC / Unit 1: Limits and Continuity
1.1 Introducing Calculus: Can Change Occur at an Instant?
Notes
Calculus studies how things change. Average rate of change is change over an interval; instantaneous rate of change is the rate at one exact moment. We get it by taking average rates over smaller and smaller intervals, which is what a limit does.
- Average rate of change: $\displaystyle \frac{f(b)-f(a)}{b-a}$, the slope of a secant line.
- Instantaneous rate of change: the value average rates approach as the interval shrinks to zero.
Practice
A car travels 120 miles in 2 hours. What does this tell you?
120/2 is an average over the interval. The speed at any instant could be different.
The average rate of change of $f(x)=x^2$ on $[1,3]$ is:
$\displaystyle \frac{9-1}{3-1}=4$.
Which idea lets calculus find a rate at a single instant?
Shrinking the interval toward zero gives the instantaneous rate.
Ready for more? Take the Unit 1 test.