91Ó°ÊÓ

To find the position function from the given velocity function, integrate the velocity function with respect to time: $$s(t) = \int v(t) dt$$ Integrate the given velocity function: $$v(t) = 2 \cos t$$

After integrating, you will get a constant of integration, say C. Apply the initial condition s(0) = 0 to determine this constant: $$s(0) = 0$$

Write the position function, s(t), by combining the result of the integration and the determined value of the constant of integration.

Using a graphing tool, plot the given velocity function, $$v(t) = 2 \cos t$$, and the found position function, s(t), on the same graph for visualization. Now let's go through the steps: Step 1: Integrate the velocity function: $$s(t) = \int 2 \cos t\, dt = 2 \int \cos t \, dt = 2 (\sin t) + C$$ Step 2: Apply the initial condition: $$s(0) = 2 (\sin 0) + C = 0$$ $$C = 0$$ Step 3: Write the position function: $$s(t) = 2(\sin t) + 0 = 2\sin t$$ Step 4: Graph the Velocity and Position Functions: To graph these functions, you can use a graphing tool like Desmos or GeoGebra, and plot both $$v(t) = 2 \cos t$$ (velocity) and $$s(t) = 2\sin t$$ (position) functions on the same graph. You should label the two functions to help distinguish between them and observe the relationship between the velocity and position functions.