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A drug used to treat cancer is effective at low doses with an efficacy that increases with the quantity of the drug. However, at sufficiently high doses, the drug becomes lethal. For positive values of the constants $k_1$ and $k_2$, the fraction of patients surviving cancer with this drug treatment is given by $$S(q) = \dfrac{q}{k_1+q}-\dfrac{q}{k_2+q}$$ where $q$ is the drug quantity in milligrams/day given to the patient. Assume that $k_1<k_2$ (think about why this assumption is necessary).

  1. Find $S(0)$ and $\lim_{q\to\infty}S(q)$ and in each case explain what your findings mean in medical terms.
  2. What is the optimal daily drug quantity to administer in terms of $k_1$ and $k_2$ so as to maximize the fraction of patients surviving?
  3. Suppose Health Canada has only approved the use of the drug of up to $45$ mg/day and suppose $k_1=25$ mg/day is the same for all patients but $k_2$ varies from patient to patient. To determine a personalized treatment strategy it would be useful for physicians to have a plot of the optimal daily drug quantity as a function of $k_2$, call it $q_{*}(k_2)$. Sketch a plot of $q_{*}(k_2)$ and explain why you drew it that way. Hint: don't forget that $k_1<k_2$!