The problem provides velocity readings for a bicycle moving along a straight path at different times. The goal is to estimate the total distance covered using left endpoints.

1. Problem Description

The problem provides velocity readings for a bicycle moving along a straight path at different times. The goal is to estimate the total distance covered using left endpoints.

2. Solution Steps

To estimate the total distance using left endpoints, we need to sum the product of the velocity at the left endpoint of each interval and the length of the interval. The time intervals are of length $0.1$ seconds. The left endpoints are $0, 0.1, 0.2, 0.3, 0.4$. The corresponding velocities are $10, 10.5, 11, 11.5, 12$.

The formula for the total distance using left endpoints is:

$Distance \approx \sum_{i=0}^{n-1} v(t_i) \Delta t$

where $v(t_i)$ is the velocity at the left endpoint of the $i$-th interval, $\Delta t$ is the length of the time interval, and $n$ is the number of intervals.

In this case, $\Delta t = 0.1$ and $n = 5$. So,

$Distance \approx (10 \times 0.1) + (10.5 \times 0.1) + (11 \times 0.1) + (11.5 \times 0.1) + (12 \times 0.1)$

$Distance \approx 0.1 \times (10 + 10.5 + 11 + 11.5 + 12)$

$Distance \approx 0.1 \times (55)$

$Distance \approx 5.5$

3. Final Answer

The estimated total distance covered is $5.5$ meters.