MATH 296 PROBLEMS 1

IGOR KRIZ

1. Two hallways, of widths a and b, meet at a right angle. What is the greatest possible length of a ladder which can be carried horizontally around the corner? (see picture) [Hint: express the length of the dotted line as a function; find its minimum via derivative.]

2. Calculate:

lim_{x→0} \frac{cos^{2}(x) − 1}{x^{2}}.

[L’Hospital rule OK]

3. What is wrong with the following use of l’Hospital’s rule?

lim_{x→1} \frac{x^{3} + x − 2}{x^{2} − 3x + 2} = lim_{x→1} \frac{3x^{2} + 1}{2x − 3} = lim_{x→1} \frac{6x}{2} = 3.

(The limit is actually −4.)

4. Ecological Ed must cross a circular lake of radius 1 mile. He can row across at 2 mph or walk around at 4 mph, or he can row part way and walk the rest (see picture). What route should he take so as to:

(a) see as much scenery as possible?

(b) cross as quickly as possible?