Question

Here is the task:

You are tracking the velocity and position of a rocket-propelled object near the surface of Mars. The velocity is v(t) and the position is s(t), where t is measured in seconds, s in meters, and v in meters per second. It is known that the v(t) = ds/dt = 4.94 – 3.72t and s(0) = 5.

A. Explain why the condition “f is continuous over [a, b]” from the Evaluation Theorem is fulfilled by this scenario.

B. Explain why the condition “F is any antiderivative of f on [a, b]” from the Evaluation Theorem is fulfilled by this scenario.

C. Determine the position function s(t) using the initial condition that s(0) = 5.

D. Apply the Evaluation Theorem to this scenario to compute ∫v(t)dt31.

E. Explain why solving the rocket’s displacement in meters over the time interval from t = 1 to t = 3 is an application of the evaluation theorem.

For Part E I wrote this:

E. First, displacement is the change in position between two points. The statement displacement at t-3-t-1 will give us the change in displacement between these time periods. Now as per the definition of evaluation theorem, If y f(x) is a continuous function on [a bl then h Saf(x)dx F(b)F(a) Where fis any antiderivative of f First we need to look for a function that is continuous between the points t-1 and t-3 ds = 4.94 3.72t f(C) (assume) dt Now f(t) is a linear function on t. Linear functions are always continuous so f(t) will obviously be continuous for t-1 and t3. Now as per evaluation theorem definition if f(t) is continuous between these points we can write Sf(t)dx f(3)-f(1) Where fis the antiderivative of f(t) ds F(t) dt S f(t) s(t) So we can rewrite equation f()dx f(3)-f(1) As f(t)dx s(t 3) -f(t-1)

Here is what my professor wrote: "In part E your definition of displacement is mostly good, but what do you mean by “two points”? In the next sentence you have confusing notation with “t=3 – t=1”. Then in the rest of the work it almost seems like you are redoing what was done in previous parts of the task, but you haven’t explained how you know the work in part D computed displacement. That is the goal. You used the evaluation theorem in part D and the answer is the displacement of the rocket, but how do we know this? You need to use the definition of displacement to show how the work in part D found the displacement. I would explain this in words rather than in math symbols. You may be trying to do this with the symbols and equations, but it would be easier to explain if you gave the meaning of the computations, rather than re-showing the computations."

Please help me fix this part.

Answer 1

You should start with Displacement from one point to another is the net change in position from the starting point to the ending point

(t=3)-(t=1) looks like you're subtracting things. Rather say from t=1 to t=3. This also clears the starting and ending points

We know it because the rate of change of displacement is the velocity so v(t)=s'(t) And so is the integral of the velocity which is the displacement

s(t) v(t) dt

$\blacksquare$

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