_________________________________________________________________________________

Calculus II Test
#1 March
5, 2003

Name____________________ R. Hammack Score
______

_________________________________________________________________________________

(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: