Let c=ate^-bt represent a drug concentration curve.
Discuss the effect on peak concentration and time to reach peak concentration of varying the parameter b while keeping a fixed and suppose a=b, so C=ate^(-at).
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You need to remember that a peak occurs if `(dc)/(dt)=0` , hence, you need to differentiate the concentration function such that:
`(dc)/(dt) = ae^(-at) + at*e^(-at)*(-at)'`
`(dc)/(dt) = ae^(-at)- a^2*t*e^(-at)`
You need to factor out `ae^(-at)` such that:
`(dc)/(dt) = ae^(-at)*(1 - at)`
Solving the equation `(dc)/(dt) = 0` for t yields:
`ae^(-at) = 0 =gt a = 0`
`1 - at = 0 =gt -at = -1 =gt t = 1/a`
`c(t) = a*(1/a)*e^(-a/a) = e^(-1) = 1/e`
Hence, the concentration function reaches its peak, under given condition `a=b` , at`t = 1/a` .
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