onsider an object moving along a line with the following velocity and initial position.
on
eetermine the position function for using both the antiderivative method and the Fundamental Theorem of Calculus. Check for
greement between the two methods.
To determine the position function for 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.

Ask by Carter Horton. in the United States

Jan 23,2025

Question

onsider an object moving along a line with the following velocity and initial position.
on
eetermine the position function for using both the antiderivative method and the Fundamental Theorem of Calculus. Check for
greement between the two methods.
To determine the position function for 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.

Ask by Carter Horton. in the United States

Jan 23,2025

UpStudy AI · Accepted Answer

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.
The position function is $$ s(t) = 9t - \frac{3t^2}{2} $$.

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