Lower and Upper estimate

Question

The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table below. Find lower and upper estimates of the distance she covered during the first three seconds of the race.

The lower estimate is:

The upper estimate is:

At 0 seconds the velocity is at 0 feet per second.

At 0.5 seconds the velocity is at 6.6 feet per second.

At 1 second the velocity is at 11.7 feet per second.

At 1.5 seconds the velocity is at 15.7 feet per second.

At 2 seconds the velocity is at 19 feet per second.

At 2.5 seconds the velocity is at 22.5 feet per second.

At 3 seconds the velocity is at 23.3 feet per second

Answer 1

Distance = ∫³_o |v| dt

We will use Riemann sums to estimate it.

Since the it is given that the velocity is increasing steadily, the Left Riemann Sum will produce the lower estimate and the Right Riemann Sum - the upper estimate.

All the time intervals are same, so

LRS=∧t ( v(0)+v(0.5)+v(1)+...+v(2.5)) and RRS=∧t ( v(0.5)+v(1)+...+v(2.5)+v(3))

LRS=0.5(0+6.6+11.7+15.7+19+22.5) = 37.75

RRS=0.5(6.6+11.7+15.7+19+22.5+23.3) = 49.4

The lower estimate is: 37.75 feet

The upper estimate is: 49.4 feet

Hope it helps!