Question 5 please Difference equations 5. The self-replicating robot from lectur

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Question 5 please

Difference equations 5. The self-replicating robot from lecture takes one hour to assemble a copy of itself, and the copy takes another hour to calibrate its software before it is ready to make more copies. We start with one assembled (but not calibrated) robot. We argued in class that the difference equation for the number of assembled robots at the beginning of hour t is with initial conditions i 1 and N2 1. (a) Suppose that each robot assembles two copies in an hour. (The two copies are con- structed in parallel, and are completed simultaneously at the end of the hour.) What is the new difference equation for N? State the order of this difference equation, and whether it is autonomous or nonautonomous (b) Suppose that the robots get more efficient at assembling copies as time passes. Specif- ically: during hour t, each calibrated robot assembles t new copies in parallel. What is the new difference equation for N? State the order of this difference equation, and whether it is autonomous or nonautonomous (c) Challenge. Suppose that each robot assembles one copy in an hour, and it wears out after it has assembled two copies. A worn out robot falls apart and is no longer assembled What is the new difference equation for N,? State the order of this difference equation, and whether it is autonomous or nonautonomous Hint: the formula N Ni-1 + N-2 overcounts Nt. Determine why each of the terms Ni-1 and N-2 is too big, then try to track both corrections back to the same time step.

Explanation / Answer

a.) Caliberated robot can asseble two robots every hour

Here, N1=1, N2=1, N3= 3, N4=5, N5=11, N6=21, N7=43, N8=85

Nt = Nt-1 + 2*Nt-2, t >=3

b.) Caliberated robot can asseble t robots during hour t.

Here, N1=1,

N2=1,

N3= 1+1(3) = 4,

N4=4+1(4) = 8,

N5=8+4(5) = 28,

N6=28+8(6) = 76,

Nt = Nt-1 + (t)*(Nt-2), t >=3.

c.) Caliberated robot can asseble two robots every hour

Here, N1=1, N2=1, N3= 2, N4=2, N5=3, N6=4, N7=5, N8=7, N8=9

Nt = Nt-3 + Nt-4  + Nt-5, t >=6

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