Calculus II 
Quiz #4

March 6, 2002

Name____________________ 
R. Hammack

Score ______

(1) Evaluate the definite integrals. Simplify your answer as much as possible
(a) ==
u=πx
du=πdx
(b) =ln(e+5)
 ln(6) = ln((e+5)/6)
u =
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 halfhour period, it has a
velocity of 10 miles per hour. What is its velocity at the end of the
halfhour period?
Let's set the clock so that t=0 represents the beginning of the halfhour
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
= 10.
Therefore the velocity is v(t) = at
+10. If we can just find a, the answer to the question will
be v(1/2).
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 halfhour
period is
v(1/2) = 120(1/2)+10 = 70 miles per hour.