Question

the graph shows the position s = f(t) of a car t hours after 5:00 p.m. relative to its starting point s = 0, where s is measured in miles. (a) describe the velocity of the car. specifically, when is it speeding up and when is it slowing down? (b) at approximately what time is the car traveling the fastest? the slowest? (c) what is the approximate maximum velocity of the car? the approximate minimum velocity? (a) when is the car speeding up? (simplify your answer. type your answer in interval notation. round to the nearest grid - line as needed. use a comma to separate answers as needed.)

Explanation:

Step1: Recall the relationship between position - velocity

Velocity $v(t)$ is the derivative of position $s(t)$. The car is speeding up when the velocity and acceleration have the same sign. Acceleration $a(t)$ is the derivative of velocity $v(t)$. On the position - time graph, the slope of the tangent line to the curve $s = f(t)$ gives the velocity. The car is speeding up when the slope of the tangent line to the $s - t$ graph is increasing in magnitude.
Looking at the graph, we find the intervals where the curve is getting steeper in the positive - slope direction.

Step2: Analyze the graph for speeding - up intervals

We observe the graph of $s=f(t)$. The car is speeding up when the slope of the curve is increasing. By looking at the graph, we can estimate the intervals. The curve is getting steeper in the positive - slope direction approximately on the intervals where the graph is bending upwards.
Let's assume the $x$ - axis represents time $t$ (in hours) and the $y$ - axis represents position $s$ (in miles).
By visual inspection, the car is speeding up on the interval $(2,6)$ (approximate values based on the graph).

Step3: Analyze for maximum and minimum velocity

The maximum velocity occurs where the slope of the $s - t$ graph is the steepest (either positive or negative). The minimum velocity occurs where the slope of the $s - t$ graph is the least steep (closest to zero).
The maximum velocity occurs approximately at $t = 4$ (where the slope of the tangent line is the steepest) and the minimum velocity occurs approximately at $t=10$ (where the slope of the tangent line is closest to zero).

Answer:

(a) $(2,6)$
(b) The car is speeding up from approximately $t = 2$ hours to $t = 6$ hours, slowing down outside of this interval. It is fastest at approximately $t = 4$ hours and slowest at approximately $t = 10$ hours.
(c) The approximate maximum velocity occurs at $t\approx4$ hours and the approximate minimum velocity occurs at $t\approx10$ hours. The maximum velocity is the steepest slope of the tangent line to the $s - t$ graph and the minimum velocity is the slope closest to zero of the tangent line to the $s - t$ graph.