PARTIAL DERIVATIVES AND CHAIN RULE

Consider a triangle with vertices A,B and C. Let a,b,c be the lengths of the sides opposite the corresponding vertices. If we know a,b,c then we can construct the triangle using a ruler and compass - so we can determine the angles of the triangle. In fact one may prove

with similar formulas for B and C. Note that a,b,c can be the lengths of the sides of a triangle only if a,b,c are positive and a+b>c  i.e. the sum of the lengths of two sides must exceed the length of the third.

We wish to study the following questions.

1. Does A increase or decrease when a increases (with b,c fixed)?
2. Does A increase or decrease when b increases (with a,c fixed) when ? When a=3,b=3,c=5?
3. It seems that as b increases (a,c fixed), A increases sometimes and decreases sometimes. Find a geometrical condition on one of the angles A, B, or C which is necessary and sufficient to guarantee that A increases as b increases.
4. Suppose a, b, and c are changing with time, simultaneously, and da/dt = 1, db/dt=2, dc/dt=1 i.e. b is increasing twice as fast as a and c. Is A increasing or decreasing with time when a=3,b=5,c=5 and how rapidly - compute dA/dt?

 
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