Position, velocity, and acceleration Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.
b. Find and graph the velocity function. When is the object stationary, moving to the right, and moving to the left?
f(t) = 18t-3t²; 0 ≤ t ≤ 8

Verified step by step guidance

1. Step 1: Understand the relationship between position, velocity, and acceleration. The position function s = f(t) describes the location of an object at time t. The velocity function v(t) is the derivative of the position function, representing the rate of change of position with respect to time.
2. Step 2: Differentiate the position function f(t) = 18t - 3t^2 to find the velocity function v(t). Use the power rule for differentiation: if f(t) = at^n, then f'(t) = n*at^(n-1).
3. Step 3: Apply the power rule to each term in f(t). The derivative of 18t is 18, and the derivative of -3t^2 is -6t. Therefore, the velocity function is v(t) = 18 - 6t.
4. Step 4: Determine when the object is stationary by setting the velocity function equal to zero and solving for t. This will give the time(s) when the object is not moving.
5. Step 5: Analyze the sign of the velocity function to determine when the object is moving to the right (v(t) > 0) and moving to the left (v(t) < 0). This involves solving inequalities based on the velocity function.