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95 A herd of llamas is growing exponentially. At time t = 0 it has 1000 llamas in it, and at time t = 4 it has 2000 llamas. Write a formula for the number of llamas at arbitrary time t. 96 A herd of elephants is growing exponentially. At time t = 2 it has 1000 elephants in it, and at time t = 4 it has 2000 elephants. Write a formula for the number of elephants at arbitrary time t. 97 A colony of bacteria is growing exponentially. At time t = 0 it has 10 bacteria in it, and at time t = 4 it has 2000.

Such a relation between an unknown function and its derivative (or derivatives) is what is called a differential equation. Many basic ‘physical principles’ can be written in such terms, using ‘time’ t as the independent variable. Having been taking derivatives of exponential functions, a person might remember that the function f (t) = ekt has exactly this property: d kt e = k · ekt dt For that matter, any constant multiple of this function has the same property: d (c · ekt ) = k · c · ekt dt And it turns out that these really are all the possible solutions to this differential equation.

Then f (0) = +1 > 0 also, so we might doubt that there is a root in [0, 1]. Continue: f (−1) = 1 > 0 again, so we might doubt that there is a root in [−1, 0], either. Continue: at last f (−2) = −5 < 0, so since f (−1) > 0 by the Intermediate Value Theorem we do indeed know that there is a root between −2 and −1. 5 since this is the midpoint of the interval method gives: x1 x2 x3 x4 on which we know there is a root. 3247179572447898011 so right away we have what appears to be 5 decimal places accuracy, in 4 steps rather than 21.