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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: