Highway travel A state patrol station is located on a straight north-south freeway. A patrol car leaves the station at 9:00 A.M. heading north with position function s = f(t) that gives its location in miles t hours after 9:00 A.M. (see figure). Assume s is positive when the car is north of the patrol station. <IMAGE>
c. Find the average velocity of the car over the interval [1.75, 2.25]. Estimate the velocity of the car at 11:00 A.M. and determine the direction in which the patrol car is moving.

Verified step by step guidance

1. To find the average velocity of the car over the interval [1.75, 2.25], use the formula for average velocity: \( v_{avg} = \frac{f(b) - f(a)}{b - a} \), where \( a = 1.75 \) and \( b = 2.25 \).
2. Substitute the values of \( a \) and \( b \) into the formula: \( v_{avg} = \frac{f(2.25) - f(1.75)}{2.25 - 1.75} \).
3. To estimate the velocity of the car at 11:00 A.M., which corresponds to \( t = 2 \) hours after 9:00 A.M., find the derivative of the position function \( s = f(t) \) to get the velocity function \( v(t) = f'(t) \).
4. Evaluate the derivative at \( t = 2 \) to estimate the velocity: \( v(2) = f'(2) \).
5. Determine the direction of the patrol car by checking the sign of \( v(2) \). If \( v(2) > 0 \), the car is moving north; if \( v(2) < 0 \), the car is moving south.

Key Concepts

Average Velocity

Average velocity is defined as the change in position divided by the time interval over which that change occurs. Mathematically, it is calculated as (s(t2) - s(t1)) / (t2 - t1), where s(t) is the position function. In this context, it helps determine how fast the patrol car is moving on average between two specific times.

Position Function

The position function s = f(t) describes the location of an object at any given time t. In this scenario, it represents the distance of the patrol car from the station in miles, where t is measured in hours after 9:00 A.M. Understanding this function is crucial for calculating both average and instantaneous velocities.

Instantaneous Velocity

Instantaneous velocity refers to the velocity of an object at a specific moment in time, which can be found by taking the derivative of the position function with respect to time. This concept is essential for estimating the patrol car's speed at 11:00 A.M. and determining its direction of movement, as it indicates whether the car is moving north or south.