Math 19. Lecture 30Estimating Elapsed TimeT. JudsonFall 20051 Estimating TimeSuppose thatdsdt= f(s), (1)where s(0) = s0. If s0≤ s ≤ s1and f(s) > 0, we wish to estimate how longit takes to reach s1.• Step 1. Let t = t1be the time where s first takes on the value s1.Suppose that– fmaxbe the maximum value of f(s) on [s0, s1].– fminbe the minimum value of f(s) on [s0, s1].• Step 2. From equation (1),fmin≤dsdt≤ fmax(2)on [s0, s1].• Step 3. Integrating (2) from 0 to t1and using the Fundamental Theo-rem of Calculus, we havefmin· t1≤Zt10dsdtdt ≤ fmax· t1orfmin· t1≤ s(t1) − s(0) = s1− s0≤ fmax· t1.1• Step 4. Therefore,s1− s0fmax≤ t1≤s1− s0fmin.2 Some Examples• Letdxdt= 2 + sin(πx)with initial condition x(0) = 0. We wish to find upper and lower boundsfor t when x(t) = 1. We can replace sin πx by 1 to get a maximum for2 + sin(πx) and by 0 to get a minimum for 2 + sin(πx). Therefore,13≤ t ≤12.• Letdxdt= 2x4− x + 2with initial condition x(0) = 0. We wish to find upper and lower boundsfor t when x(t) = 1. The f unction 2x4− x + 2 has a critical point atx = 1/2. Sincef(0) = 2f(1/2) = 13/8f(1) = 3,we know that13≤ t ≤813.Readings and References• C. Taub es. Modeling Diffe rential Equations in Biology. Prentice Hall,Upper Saddle River, NJ, 2001. Chapter
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