Consider an object moving along a line with the following velocity and initial position. \[ v(t)=9-3 t \text { on }[0,5] ; s(0)=0 \] Determine the position function for $t \geq 0$ using both the antiderivative method and the Fundamental Theorem of Calculus. Check for agreement 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. A. The position function is the antiderivative of the velocity function. B. The position function is the absolute value of the antiderivative of the velocity function. C. The position function is the derivative of the velocity function. D. The velocity function is the antiderivative of the absolute value of the position function. Which equation below will correctly give the position function according to the Fundamental Theorem of Calculus? A. $s(t)=s(0)+\int_{a}^{a} v(t) d t$ B. $s(0)=s(t)+\int_{0}^{1} v(x) d x$ C. $s(t)=\int_{a}^{b} v(t) d t$ D. $s(t)=s(0)+\int_{0}^{t} v(x) d x$ Determine the position function for $t \geq 0$ using both methods. Select the correct choice below and fill in the answer box(es) to complete your choice. A. The same function is obtained using each method. The position function is $s(t)=$

Step 1 ：The position function is the antiderivative of the velocity function. This is because the velocity function represents the rate of change of the position function. Therefore, to find the position function, we need to find the antiderivative of the velocity function.

Step 2 ：The Fundamental Theorem of Calculus states that the definite integral of a function can be found by evaluating the antiderivative of the function at the upper and lower limits of integration. In this case, the position function can be found by evaluating the antiderivative of the velocity function from 0 to t and adding the initial position.

Step 3 ：Therefore, the correct answer to the first question is A. The position function is the antiderivative of the velocity function.

Step 4 ：The correct equation to give the position function according to the Fundamental Theorem of Calculus is D. \(s(t)=s(0)+\int_{0}^{t} v(x) d x\).

Step 5 ：Now, let's calculate the position function using both methods.

Step 6 ：The position function obtained using both the antiderivative method and the Fundamental Theorem of Calculus method is \(s(t) = -\frac{3}{2}t^2 + 9t\). This function represents the position of the object at any time \(t \geq 0\).

Step 7 ：Final Answer: The position function is \(s(t) = -\frac{3}{2}t^2 + 9t\). Therefore, the correct choice is A. The same function is obtained using each method. The position function is \(s(t) = -\frac{3}{2}t^2 + 9t\). \(\boxed{-\frac{3}{2}t^2 + 9t}\).