Tangent

You are interviewing for a position with an association representing the tire industry. The federal government mandates safety testing of all tires manufactured in the United States. Recently there has been concern that the rubber used in the tires could deteriorate while in store inventories. In September 2003, a safety group asked the U.S. government to require expiration dates for tires. Representatives for the manufacturing industry test the tires, collect data, and create graphs of functions to test tire durability. As part of the interviewing process, you are asked to solve the following problems. Two cars with new tires are driven at an average speed of 60 mph for a test drive of 2... click for more

Proofs

1- Given triangle as shown (infigure 241W) with AF = 1/4 AB, AE = 1/3 and BE Intersect CF = O (BE has ray bar on top and CF has distance bar on top) prove that AO hits BC At its midpoint. 2-use Ceva's theorem to prove that the internal bisctores of the angles of a triangle are concurrent (cevas theorem: given triangle ABC and 3 cevian lines L1, L2, L3 for each vertex hitting the opposite side at D, E,F respectively then L1,L2,L3 are concurren <=> AF/FB * BD/DC * CE/EA = 1) 3- Suppose that A', B', C' are midpoints of the sides of a triangle ABC and are also midpoints of KK' subset BC, LL' subset AC, MM' subset AB. Show that if AK', BL', CM' are also concurrent. (figure 241W-5)

Differential geometry/chain rule

(See attached file for full problem description) Please notice that this is a Differential Geometry class. It is different than simple things in baby rudin's book.

Differential Geometry/Inverse Function Theorem

(See attached file for full problem description) Please justify all your steps and claims. And notice that this is differential geometry class, graduate level.

Geometry

(See attached file for full problem description)