March 6, 2002
(1) Evaluate the definite integrals. Simplify your answer as much as possible
- ln(6) = ln((e+5)/6)
du = dx
(2) Find the derivative of the function F(x)
Using the chain rule and Part 2 of the Fundamental Theorem of Calculus, we get
F '(x) = sin( -(x+ln(x))
)(1 + 1/x)
(3) A freight train, moving with constant acceleration, travels 20 miles
in half an hour. At the beginning of the half-hour period, it has a
velocity of 10 miles per hour. What is its velocity at the end of the
Let's set the clock so that t=0 represents the beginning of the half-hour
period and t = 1/2 represents the end.
Represent the constant acceleration as a, so the velocity is v(t)
= at +C.
Since it's given that 10 = v(0) = a(0) + C, we get C
Therefore the velocity is v(t) = at
+10. If we can just find a, the answer to the question will
To use the information about position (i.e. the fact that the train went 20 miles),
we must now construct the position function.
Notice that s(t) = =
and from 0 = s(0) =
we obtain K=0.
Therefore, position is s(t) =
Since the train went 20 miles in half an hour, we know that 20 = s(1/2)
which becomes 20 = a/8 +5. Solving for a, we get a = 120.
Therefore, we can finally write velocity as v(t)
= 120t +10.
Here is the answer to the question. The velocity at the end of the half-hour
v(1/2) = 120(1/2)+10 = 70 miles per hour.