When Do Newborn Babies Start to Grow?
(A case study on the use of first derivatives)
V.S."Mano" Manoranjan
Department of Pure and Applied Mathematics
Washington State University, Pullman, Washington
mano@alpha.math.wsu.edu
Joshua Yeidel
IT Learning Systems Group
Washington State University, Pullman, Washington
yeidel@wsu.edu
Randy Lagier
IT Learning Systems Group
Washington State University, Pullman, Washington
lagier@wsu.edu
TABLE OF CONTENTS
Abstract: brief overview of the learning module
Learners: intended audience and prerequisite competencies
Objectives: learning aims
Intended Use: on how the learning module can be used
Learning Efforts: expected effort
Lesson: the text of the learning module
Comments: comments by the learners
After birth, an infant is known to lose weight for a few days before starting to grow. We make use of this information in order to develop a mathematical module that illustrates the concept of "minimum". At the same time, this module will allow one to learn the use of calculus in real situations. When monitoring the health of a newborn baby, it is very important for a medical practitioner to know, roughly, when a baby should stop losing weight. For, if a baby continues to lose weight beyond the day determined by available medical data, that may be worrisome and measures should be taken to ensure the well-being of the baby.
First, we will suppose that the functional relationship between the average weight of a baby and the number of days from birth is known. (How does one obtain such a relationship?) Then, we will estimate the day on which the average weight of the baby will be the lowest (minimum) using a trial and error approach. Will this be a good estimate? We will demonstrate how to employ the idea of "derivative"(what is a derivative of a function?) from calculus to systematically find the exact time (after
birth) on which the minimum weight occurs. Also, we will determine that minimum weight. The newborn baby will start to grow beyond the day on which the minimum weight occurs.
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High School students, college freshman interested in business, engineering, mathematics or sciences, or any individual who wants to know and learn about the uses of simple mathematical concepts in real life problems.
Prerequisite Competencies
There are two expected learning outcomes:
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2. In-class use by a non-math instructor as a review module in a class where calculus is pre-requisite.
3. Out of class use by a student as a module for self-paced study (this could include distance learning too).
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The module is designed to be completed with approximately thirty minutes of student effort.
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It is a known fact that after birth, an infant normally loses weight and then starts growing. Let us suppose that based on data collected at maternity hospitals, the average weight W (in lbs.) of infants over the first 14 days (two weeks) following birth is given by the functional relationship
W = 0.03t2 - 0.39t + 7.3 -------(1)
where time t is measured in days. Note that t can take any value between 0 and 14. Now, when will a newborn baby start to grow? Obviously, the baby will start to grow after its weight has reached the minimum value. So, we need to determine the day (i.e. the value of t) on which (W) will be the minimum.
Trial and Error Approach
The date at which the minimum weight occurs can be determined using a trial and error approach. Use the included exercise to explore this approach.
How good is your estimate? Let us find out by determining the exact t-value where the minimum occurs in a systematic fashion.
Systematic Approach
We consider equation (1) and obtain the first derivative dW/dt by differentiating the equation on both sides.
i.e. dW/dt = 0.03*2t - 0.39 -------(2)
In relation to the graph W vs t given by (1), dW/dt at a t-value will give the slope of the tangent at that t-value. 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
dW/dt = 0
one obtains
0.03*2t - 0.39 = 0 -------(3)
Now, by solving (3), we can find the t-value
t = 0.39/(2*0.03) = 6.5 -------(4)
At this juncture, let us pause for a minute. Do we know for sure that this t-value corresponds to the minimum value of W? No, we do not! This t-value 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 t-value 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
W = 0.03*(6.5)2 - 0.39*(6.5) + 7.3
Determine this value using the calculator below (push the button).
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 and in what follows we will see how to do this.
Derivative-free Approach
Consider equation (1)
W = 0.03t2 - 0.39t + 7.3
i.e.
W = 0.03(t2 - 13t + 730/3) -------(5)
Now, let us complete the squares on the right hand side of equation (5).
Then,
W = 0.03(t2 -13t + 132/22 - 132/22 + 730/3)
or
W = 0.03(t2 - 13t + 132/22) + 0.03(730/3 - 132/22)
i.e.
W = 0.03(t-13/2)2 + 0.03(730/3 - 132/22).
If we look at the right hand side of the above equation, the first term 0.03(t-13/2)2 is either zero or positive for any t value and the second term is positive (check this) all the time. This means that W will have a minimum when the first term is zero (are you clear on this?)
i.e. W will have a minimum when
0.03(t - 13/2)2 = 0
or when
t = 13/2 = 6.5
and this is exactly the same answer we got earlier in equation (4)!
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