and the critical points come from solving . Out of the two solutions of the last equation only is in the interval we consider (so we ignore ). Now we find:

; and . So the absolute minimum of is achieved at and we have an absolute maximum of when .

[7] 2. Find the dimensions of the rectangle with perimeter 40cm and largest possible area. Justify your answer.

Denote the dimensions by x and y, denote the perimeter by P and the area by A. Then we have and . Since we find that and so . Substituting in the area we get with x between 0 and 20. We compute so that x=10 is the only critical point. Since the area is 0 when x=0 or when x=20, and it is 100 when x=10, it follows that the absolute maximum is attained when x=10. The other dimension is therefore too.

[6] 3. Suppose a circle is expanding in such a way that the area of the circle increases at the rate of . How fast is the radius of the circle increasing at the moment when the area is square meters.

The area of a circle is , where is the radius. Notice here that when the area is square meters the radius is equal to 1 metre. We find (using the chain rule) that . At the specified moment we have (given), and (as noticed above). So from where we see that metre per second.

[6] 4. Use differentials to find an approximation of . Show your work.

is approximately equal to the differential of corresponding to the change starting at . We find that and when and we have . So , and thus .