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Calculus/dagster calculus problem

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Expert: - 3/27/2011


Question
The velocity of a dragster t seconds after leaving the starting line is
v (t )=100t e−0.2t
ft/sec.
a. Estimate the distance covered by the dragster in the first 10 sec of its run.
Use 4 subintervals of equal length and use the midpoint of each
subinterval to construct rectangles.
b. Find the distance exactly.

Answer
Since we know velocity, distance is the integral of velocity.
That is, d(t) = integral(v(t).
Since v(t) = 100te^(-t/5), this needs to be done with the u-v method.
Let u = 100t and dv = e^(-t/5)dt.  Doing this lets du = 100 dt and v = -5e^(-0,2t).

Now that we know what we need, the answer is uv - integral(v du).
That is -500te^(-0.2t) - integral(-5e^(-t/5))dt.
This becomes -500te^(-0.2t) + 25e^(-t/5).
Assuming we call the starting line t=0 and d=0, this makes C=0.
Remember this to answer (b).

(a). To approximate the distance, it asks you to use the midpoint of  4 intervals over 10 sec.
The intervals would be (0,2.5), (2.5,5), (5,7.5), and (7.5,10).
The function is evaluated at the middle of each, then multiplied by the length of each, which is 2.5.  The width of each is 2.5.  This makes the approximation going to
2.5((f(1.25) + f*3.75) + f(6.25) + f(9.75)).  You can crunch the numbers.

(b) Now that (a) is done, (b) is done by completing the interal done as one that has limits.

Calculus

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