7) Consider the following function f(N) with parameters a, b, and call strictly

Question

7) Consider the following function f(N) with parameters a, b, and call strictly greater than 0 IN bl) NE b a N bl In f(N) N b a) Is the function f(N) continuous at b? Justify your answer. If continuous, show how f(N) satisfies the three conditions of Continuity. If not continuous, show the condition of continuity which fails. b) Find and classify all equilibrium values of the first order autonomous differential equation given by f (N) as either stable, unstable, or semi-stable. Answers may be in terms a, b, and c of Stable Value(s): N Unstable Value(s): N Semi-Stable Value(s): NE c Draw out the Phase Diagram of the system in the space to the right. d When we restrict the values of N to be strictly greater than b, we get the following: 1/3 dN a(N b) dt This differential equation is a variation of Gompertz Law, which was used to model the growth of cancerous tumors, where N(t is proportional to the number of cells in the tumor and a,b,c>0 are parameters. While this is a relatively simple model, it was used to predict tumor growth throughout the 1970s and 1980s, and is quite effective provided N is not too sma Find the general solution N(t) for this differential equation (use K as your unknown constant since c is already taken). General Solution: N (t 10

Explanation / Answer

The above equation is continous at N=b, because at N=b- and N=b+ the value of function is 0.

Hence Left hand lmit =Right hand limit =0.

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