Question

the graph shows the position s = f(t) of a car t hours after 5:00 pm 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 is the derivative of position. The car is speeding up when the velocity and acceleration (derivative of velocity) have the same sign. On a position - time graph, the slope of the tangent line to the curve represents velocity. The car is speeding up when the slope of the tangent line to the position - time graph is increasing.

Step2: Analyze the graph for speeding - up intervals

By observing the graph of \(s = f(t)\), we look for the intervals where the curve is getting steeper in the positive - slope direction. If the position function \(s=f(t)\) is given, and its derivative \(v(t)=f^\prime(t)\) represents velocity and \(a(t)=f^{\prime\prime}(t)\) represents acceleration. We find the intervals where \(v(t)\) and \(a(t)\) have the same sign. Looking at the graph, we estimate the intervals.

Step3: Find maximum and minimum velocity

The maximum velocity occurs where the slope of the position - time graph is the steepest (in magnitude). The minimum velocity occurs where the slope of the position - time graph is the least (in magnitude, could be close to zero). We estimate these values by looking at the steepness of the tangent lines to the curve at different points.

Answer:

(a) Without seeing the actual graph precisely, assume we observe the graph and find that the car is speeding up on the interval \((t_1,t_2)\) (you need to estimate \(t_1\) and \(t_2\) from the graph, for example, if from visual inspection \(t_1 = 2\) and \(t_2=5\), the answer in interval notation is \((2,5)\)).
(b) The car is traveling the fastest when the slope of the position - time graph is the steepest (either positive or negative slope with the largest magnitude). The car is traveling the slowest when the slope of the position - time graph is close to zero. Estimate the times from the graph. For example, if the steepest slope occurs at \(t = 8\) and the slope is close to zero at \(t = 12\), the answers are \(t = 8\) (fastest) and \(t = 12\) (slowest).
(c) The maximum velocity is the largest value of the slope of the tangent line to the position - time graph. The minimum velocity is the smallest value of the slope of the tangent line (could be zero or close to zero). Estimate these values from the graph. For example, if the maximum slope (maximum velocity) is approximately \(15\) miles per hour and the minimum slope (minimum velocity) is approximately \(0\) miles per hour, the answers are \(15\) miles per hour (maximum) and \(0\) miles per hour (minimum).