For Sammi Jo

Question: A company knows that unit cost C(x) and unit revenue R(x) from the production and sale of x units are related by:


Find the rate of change of revenue per unit when the cost per unit is changing by $15 and the revenue is $2000.

Answer: So the above equation serves as the relation, this is the equation we’ll need to differentiate in order to obtain some related rates, which will allow us to solve the problem. We now know that:

    \[\frac{dC}{dx} = 15\]

and our goal is to solve for \frac{dR}{dx}. The question also states the revenue is $2000. First we must differentiate the equation relating C(x) and R(x):

    \begin{align*} \frac{C(x)}{dx} &= \frac{R^2(x)}{120000}+11439 \\ \frac{dC}{dx} &= \frac{d}{dx} \cdot \frac{1}{120000} \cdot R^2 + \frac{d}{dx} \cdot 11,439 \\ \frac{dC}{dx} &= \frac{1}{120000} \cdot \left( 2R \cdot \frac{dR}{dx} \right) + 0 \\ \frac{dC}{dx} &= \frac{dR}{dx}\cdot R \cdot \frac{1}{60000} \\ \frac{dC}{dx} &= \frac{dR}{dx} \cdot 2000 \cdot \frac{1}{60000} \\ \frac{dC}{dx} &= \frac{dR}{dx} \cdot \frac{1}{30} \\ 15 &= \frac{dR}{dx} \cdot \frac{1}{30} \\ \frac{dR}{dx} &= 450 \\ \end{align*}

The first line is simply the equation we were given from the question. The second line we obtain by beginning to differentiate the first line; I replaced the C(x) and R(x) notation with simply C and R to facilitate simplicity. \frac{1}{120000} is a scalar multiple so it can be pulled outside and is unaffected by differentiation. We then use the chain rule on R^2. Since 11439 is a constant, its derivative is simply 0. The next line is simplification. We then replace R with 2000 which was given in the question. Then we simplify further and solve for \frac{dR}{dx} and that’s that!

Alone With The Baby – Part II

My previous post was mostly about my anticipation of my night watching Cora alone. I mused about many possibilities. I, while not explicitly stated, believed my situation would fall into one of two possibilities. One, was that Cora would realize Mommy was not present and automatically default to me being the primary comforter. The other possibility, I figured more probable, was that Cora would be okay for the majority of the time but would break down at some key times, such as bed time and the middle of the night. Turns out I was wrong. Continue reading

Alone With The Baby – Part I

Well, today is the day. Currently I’m at work and I’m mentally preparing myself for my first night alone with Cora, my 6.5 month old daughter. Pathetic, right? I know, it is. Since she was born she has been breastfed, only at 4 months did we introduce formula and fruits and veggies. For the past 2.5 months Cora has been eating increasingly more formula and more fruits and veggies, however she is still exclusively breastfed at night. Her nocturnal breastfeeding habits are the sole source of my anxiety for tonight. Continue reading

Preparation for a 2,100 mile journey

Shortly after the birth of my daughter I quietly decided I wanted to apply to Montana State University’s graduate program in mathematics. My daughter, Cora, was born mid September about a week late. The past year had been as much of a surprise to me as it was for my girlfriend / de facto wife, Jenn. We had met freshmen year of college through a mutual friend. The day we had met was spent driving around with our mutual friends. Continue reading