Science:Math Exam Resources/Courses/MATH103/April 2015/Question 02 (e)
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Question 02 (e) 

The concentration of a certain hormone in the blood changes periodically over 24h. The graph of the production rate p(t) and removal rate r(t) of the hormone are shown below. In the figure, clearly mark the times (i) at which the total concentration of the hormone is highest, and (ii) at which the hormone concentration is increasing at the fastest rate. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Find the relation between the total concentration of the hormone in hours and the two given functions and . 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Let be the total concentration of the hormone in hours. Since and represent the production and the removal rate of the concentration of the hormone, respectively, the rate of change for the total concentration of the hormone can be written as This implies that . Using this, we can find the time at which achieves its maximum as in the graph shown below.
. Now, we claim that is as in the graph. As we explained above, only in the interval , the concentration is increasing. Since is the greatest at , at we have the fastest increasing rate. 