onsider an object moving along a line with the following velocity and initial position. \( v(t)=9-3 t \) on \( [0,5] ; s(0)=0 \) eetermine the position function for \( t \geq 0 \) using both the antiderivative method and the Fundamental Theorem of Calculus. Check for greement between the two methods. To determine the position function for \( t \geq 0 \) using the antiderivative method, first determine how the velocity function and the position function are related. Choose the correct answer below. B. The position function is the absolute value of the antiderivative of the velocity function. B. The velocity function is the antiderivative of the absolute value of the position function. C. The antiderivative of the velocity function. D. The position function is the derivative of the velocity function.

To determine the position function \( s(t) \) from the velocity function \( v(t) = 9 - 3t \), we need to understand the relationship between velocity and position. The correct answer is: **C. The antiderivative of the velocity function.** This means that the position function is obtained by integrating the velocity function. Let's proceed with the calculations step by step. ### Step 1: Find the Antiderivative of the Velocity Function The velocity function is given by: \[ v(t) = 9 - 3t \] To find the position function \( s(t) \), we need to compute the antiderivative of \( v(t) \): \[ s(t) = \int v(t) \, dt = \int (9 - 3t) \, dt \] ### Step 2: Calculate the Antiderivative Now, let's calculate the integral: \[ s(t) = \int (9 - 3t) \, dt = 9t - \frac{3t^2}{2} + C \] ### Step 3: Determine the Constant of Integration We know the initial condition \( s(0) = 0 \). We can use this to find the constant \( C \): \[ s(0) = 9(0) - \frac{3(0)^2}{2} + C = 0 \implies C = 0 \] Thus, the position function is: \[ s(t) = 9t - \frac{3t^2}{2} \] ### Step 4: Verify Using the Fundamental Theorem of Calculus According to the Fundamental Theorem of Calculus, if \( v(t) \) is continuous on \( [0, 5] \), then: \[ s(t) = s(0) + \int_0^t v(u) \, du \] Calculating the integral: \[ s(t) = 0 + \int_0^t (9 - 3u) \, du \] Now, let's compute the integral: \[ \int_0^t (9 - 3u) \, du = \left[ 9u - \frac{3u^2}{2} \right]_0^t = \left( 9t - \frac{3t^2}{2} \right) - 0 = 9t - \frac{3t^2}{2} \] ### Conclusion Both methods yield the same position function: \[ s(t) = 9t - \frac{3t^2}{2} \] Thus, we have verified that the position function obtained through both the antiderivative method and the Fundamental Theorem of Calculus is consistent.