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Calculus II Test
#1 March
5, 2003
Name____________________ R. Hammack Score
______
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(1) Find the following integrals.
(a)
(b)
(c)
(d)
(e)
(f)
(2) Find the following definite integrals.
(a)
(b)
(c)
= ln(8) - ln(7) = ln(8/7)
(d) =1/5
(3) The expression represents
a definite integral over the interval [3, 5]. Write the definite
integral. (You do not need to find its value.)
(4) Find the average value
of
on the interval [0, 3].
(5) Find the following integrals. You
may find it easiest to consider the area under the graphs.
(a)
Region is one fourth of a circle of radius 4.
(b) 1/2(4)(4)
+ 1/2(2)(2) = 10.
Region is two triangles.
(6) Find the derivative of the function
(7) A train,
moving with constant acceleration, travels 25 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 half-hour period?
The information says:
s(0)
= 0
s(1/2)
= 25
v(0)
= 10
Let a
be the constant acceleration.
Know:
Then 1,
meaning C = 10.
Thus .
If we could just find a, then we would
have the formula for velocity, and the answer to the problem would be v(1/2).
The information that we have not used yet is
s(0)
= 0 and s(1/2)
= 25. That is information about position, so to use it we must make the position
function.
Know:
Then ,
meaning C = 0.
Thus
Now, 25 =
So 25 =
And 20 =
So a = 160
Finally, we now have the velocity function as .
The velocity at the end of the half-hour period is v(1/2) = 160(1/2) + 10 =
90mph.
(8) Suppose that and .
(a)
(b)
12, as follows: