Question

1. simplify the expression.
2. the position as a function of time t of a car initially heading graph below:

list the values of
$m_1 =$ average velocity on $5,6$,
$m_2 =$ instantaneous velocity at $t = 3$,
$m_3 =$ average velocity on $2,3$,
$m_4 =$ instantaneous velocity at $t = 1$,
in increasing order. provide an explanation (using words) for your answer.

Explanation:

Step1: Recall velocity - position relationship

Average velocity on an interval $[a,b]$ is $\frac{s(b)-s(a)}{b - a}$, where $s(t)$ is the position - function. Instantaneous velocity at $t = c$ is the slope of the tangent line to the position - time graph at $t = c$.

Step2: Calculate $m_1$

For the average velocity $m_1$ on $[5,6]$, if we assume the position values at $t = 5$ and $t = 6$ are $s(5)$ and $s(6)$ respectively from the graph. Let's say $s(5)=3$ and $s(6)=0$. Then $m_1=\frac{s(6)-s(5)}{6 - 5}=\frac{0 - 3}{1}=- 3$.

Step3: Estimate $m_2$

The instantaneous velocity $m_2$ at $t = 3$ is the slope of the tangent line at $t = 3$. The tangent line at $t = 3$ is horizontal, so $m_2 = 0$.

Step4: Calculate $m_3$

For the average velocity $m_3$ on $[2,3]$, assume $s(2)=4$ and $s(3)=5$. Then $m_3=\frac{s(3)-s(2)}{3 - 2}=\frac{5 - 4}{1}=1$.

Step5: Estimate $m_4$

The instantaneous velocity $m_4$ at $t = 1$ is the slope of the tangent line at $t = 1$. The tangent line at $t = 1$ has a positive slope. Let's estimate it to be around $2$.

Step6: Order the values

Comparing the values $m_1=-3$, $m_2 = 0$, $m_3 = 1$, $m_4\approx2$, we get $m_1

Answer:

$m_1