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Problem 36 Hard Difficulty

The family of functions $ f(t) = C(e^{-at} - e^{-bt}) $, where $ a $, $ b $, and $ C $ are positive numbers and $ b > a $, has been used to model the concentration of a drug injected into the bloodstream at time $ t = 0 $. Graph several members of this family. What do they have in common? For fixed values of $ C $ and $ a $, discover graphically what happens as $ b $ decreases. Then use calculus to prove what you have discovered.

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