7. Antiderivatives & Indefinite Integrals

Initial Value Problems

7. Antiderivatives & Indefinite Integrals

Initial Value Problems

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Multiple Choice

Using the acceleration function below, find the velocity function, if the velocity is v = 5 at time t = 2.
$a\left(t\right)=-20$

$v\left(t\right)=-20t+10$

$v\left(t\right)=-20t+5$

$v\left(t\right)=-20t+45$

$v\left(t\right)=-20t-35$

Verified step by step guidance

Start by understanding that the acceleration function a(t) = -20 is a constant. This means the velocity function v(t) is the antiderivative of a(t).

To find the velocity function v(t), integrate the acceleration function a(t) with respect to time t. The integral of a constant -20 with respect to t is -20t plus a constant of integration, C.

The general form of the velocity function after integration is v(t) = -20t + C, where C is a constant that we need to determine.

Use the initial condition given in the problem: v = 5 at t = 2. Substitute these values into the velocity function to solve for C: 5 = -20(2) + C.

Solve the equation 5 = -40 + C to find the value of C. Once C is determined, substitute it back into the velocity function to get the specific velocity function v(t).