The critical points of a system of ODEs, classify them and its importance. Critical point: The critical point of a system y ' = A y defined as a point at which d y 2 d y 1 becomes undetermined. There are five types of critical points are improper nodes, proper nodes, saddle points, centers and spiral points. Critical points are classified depending on the geometric shape of the trajectories near them.

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3. Consider the following theorem: Theorem: If n is an odd integer, then n³ is an odd integer. Note: There is an implicit universal quantifier for this theorem. Technically we could write: For all integers n, if n is an odd integer, then n³ is an odd integer. (a) Explore the statement by constructing at least three examples that satisfy the hypothesis, one of which uses a negative value. Verify the conclusion is true for each example. You do not need to write your examples formally, but your work should be easy to follow. (b) Pick one of your examples from part (a) and complete the following sentence frame: One example that verifies the theorem is when n = We see the hypothesis is true because and the conclusion is true because (c) Use the definition of odd to construct a know-show table that outlines the proof of the theorem. You do not need to write a proof at this time.

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