Science:Math Exam Resources/Courses/MATH104/December 2010/Question 01 (d)

MATH104 December 2010

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[hide] Question 01 (d)
A drug that stimulates reproduction is introduced into a colony of bacteria. After t minutes, the number of bacteria is given approximately by $\displaystyle {}N(t)=1000+30t^{2}-t^{3},\quad {}t\geq {0}$ When is the rate of growth $\displaystyle {}N'(t)$ decreasing?

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

[show] Hint 1
First, note that this problem is not asking for the interval(s) where $N(t)$ is decreasing, but rather where its derivative $N'(t)$ is decreasing.

[show] Hint 2
When a differentiable function is decreasing, what can you say about its first derivative?

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[show] Solution
As long as $N''(t)$ is negative, $N'(t)$ is decreasing. Differentiating $N(t),$ $N'(t)=60t-3t^{2}$ Differentiating again, $N''(t)=60-6t.$ This is negative whenever $6t>60,$ i.e. $t>10.$ We can conclude that $N'(t)$ is decreasing for all $t>10.$ (Furthermore, $N'(t)$ is continuous, we also know that it is increasing on the closed interval, $[10,\infty )$ . On an exam, either answer, $(10,\infty )$ or $[10,\infty )$ , would be acceptable.)