Averages use more than one input
An interval containing a chosen instant uses values from multiple inputs. When available, inputs on both sides help compare the interval’s secant slope with the tangent slope.
Secant slope 12.64Tangent slope 12
Drag h toward zero to see the secant across [2−h,2+h] approach the tangent at x=2.
Compare average velocities around a time
A moving object’s position is s(t)=t³ meters. Compare average velocities on intervals surrounding t=2.
- 1Choose a wide intervalThe interval [1,3] surrounds time 2, using one input before and one after it.
- 2Find its averages(1)=1 and s(3)=27, so (27−1)/(3−1)=13 meters per second across the interval.
- 3Narrow the intervals(1.5)=3.375 and s(2.5)=15.625, so the average is (15.625−3.375)/1=12.25 meters per second.
- 4Narrow it agains(1.9)=6.859 and s(2.1)=9.261, giving (9.261−6.859)/0.2=12.01 meters per second.
- 5Interpret the patternThe averages move toward a value near 12, but each still summarizes its own interval, not every instant within it.
Interpret an interval’s average velocity
For s(t)=t³ meters, the average velocity on [1,3] is 13 meters per second. Which conclusion is justified by that average alone?
- AThe object’s velocity is 13 meters per second at every instant from 1 to 3.
- BThe object’s instantaneous velocity at time 2 is 13 meters per second.
- CThe object’s position at time 2 is 13 meters.
- DNet position change divided by elapsed time is 13 meters per second; the average alone gives no specific instant’s velocity.