Question

Use the position function 13- 472 + 12t + 5, which models the motion of a particle moving along the x-axis on the closed interval [0,7] to answer the following question

Find v(4) and explain what it means in the problem.

Answer 1

To find v(4), calculate the derivative of the position function which is 'velocity function' and substitute t=4. The velocity v(4) is -20, indicating the particle's velocity is -20 units per second in the negative x-axis direction at t = 4 seconds.

Step-by-step explanation:

To find v(4), which represents the velocity of the particle at t = 4 seconds, we need to take the derivative of the position function with respect to time, which will give us the velocity function. Since the provided position function is 13 - 4t2 + 12t + 5 (assuming the missing base of the t in the second term is a typo and should be 't'), the derivative of this function is -8t + 12. Then, we substitute 4 into this derivative to get v(4).

The calculation will look like this:

• The derivative of the position function x(t) = 13 - 4t2 + 12t + 5 is v(t) = -8t + 12.
• Substitute t = 4 into v(t) to find v(4) = -8(4) + 12 = -32 + 12 = -20.

Therefore, v(4) = -20. This means at t = 4 seconds, the particle is moving along the x-axis at a velocity of -20 units per second, in the negative direction of the x-axis.