Recall equation (1)
Obtain the first derivative dW/dt by differentiating the equation on both sides, i.e.
In relation to the graph W vs t given by (1), dW/dt at a tvalue will give the slope of the tangent at that tvalue. What will be the slope of the tangent at the point where a graph has a minimum? Of course, it is zero (if you are not sure about this, review the section on slope). Therefore, in (2), if we make
one obtains
Now, by solving (3), we can find the tvalue
At this juncture, let us pause for a minute. Do we know for sure that this tvalue corresponds to the minimum value of W? No, we do not! This tvalue may very well correspond to the maximum value for W, because the slope of the tangent at the point where a graph has a maximum is also zero!! For the case study presented in this lesson, show that the tvalue in (4) corresponds to a minimum. So, the minimum weight will occur 6.5 days after the baby is born and that minimum weight will be
This means that the baby will start to grow after 6 and a 1/2 days from birth. Question: Could we have found this result without using the concept "derivative"? The answer is yes. See how with the derivativefree and algebraic approaches.
