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[8] 1. The length of a rectangle is fixed at 12 meters, while its width is decreasing at the rate of 2 m/sec. How fast is a diagonal of the rectangle changing at the moment when the width is 5 meters ?

We see from the picture that , where

y is the length of the diagonal and x is the width of

the rectangle. We are given that (the

negative sign since x decreases).

We want at the indicated moment. We find (by

the chain rule) that . At the moment when x=5 meters we get m/sec.

[9] 2. Evaluate the following integrals.

(a)

(b) 

(a) is one antiderivative of (that could be checked by differentiating ). It follows that .

(b) Here we try and check by differntiating it is indeed an antiderivative of . It follows that

[8] 3. An object travels along a straight line with a velocity of 5t m/sec, where t is the elapsed time. If at the initial moment the object is at a point A, how far it is from A after 3 seconds ?

Denote the distance from the object to the point A by . We are given that the velocity of the object is is . It follows that . At the initial time (when

) the distance is (given). On the other hand we find from the above formula for that at the same initial moment . It follows that . So, the distance at any moment is . After 3 seconds we have meters.