17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.


a. Determine the position function, for t≥0, using the antiderivative method


v(t) = 9−t² on [0, 4]; s(0)=−2

Verified step by step guidance

1. Recall that the position function \(s(t)\) is the antiderivative (integral) of the velocity function \(v(t)\) plus the initial position constant. Mathematically, this is expressed as \(s(t) = \int v(t) \, dt + C\).
2. Given the velocity function \(v(t) = 9 - t^{2}\), set up the integral to find the position function: \(s(t) = \int (9 - t^{2}) \, dt + C\).
3. Integrate each term separately: the integral of 9 with respect to \(t\) is \$9t\(, and the integral of \)-t^{2}\( with respect to \)t$ is \(-\frac{t^{3}}{3}\). So, \(s(t) = 9t - \frac{t^{3}}{3} + C\).
4. Use the initial condition \(s(0) = -2\) to solve for the constant \(C\). Substitute \(t=0\) into the position function: \(s(0) = 9 \cdot 0 - \frac{0^{3}}{3} + C = C\). Since \(s(0) = -2\), it follows that \(C = -2\).
5. Write the final position function incorporating the constant: \(s(t) = 9t - \frac{t^{3}}{3} - 2\) for \(t \geq 0\).

Key Concepts

Velocity and Position Relationship

Velocity is the rate of change of position with respect to time. The position function s(t) describes the location of an object at time t, and its derivative s'(t) equals the velocity v(t). Understanding this relationship allows us to find position by integrating velocity.

Antiderivative (Indefinite Integral)

The antiderivative of a function is another function whose derivative is the original function. To find position from velocity, we compute the antiderivative of v(t), which gives s(t) plus a constant of integration. This constant is determined using initial conditions.

Initial Conditions and Constants of Integration

When integrating velocity to find position, an unknown constant appears. The initial condition, such as s(0) = -2, allows us to solve for this constant, ensuring the position function accurately reflects the object's starting location.