The Calculus Diaries - Jennifer Ouellette [77]
Normally we would take a derivative with respect to all possible values of f, but in this case we would take the derivative only with respect to those values for f that are allowed—namely, those that can be changed without exceeding our boundary conditions of total calories and cost.
What about the integral? It plays a role in the Thermodynamics Diet too, specifically with regard to how many calories we burn. It all comes down to the burn rate. We can take an integral of our rate of burning calories with respect to time and get the total number of calories burned—the calorie meter on an exercise machine at the gym is secretly doing this calculation. But as we’ve seen, that burn rate is affected by numerous variables: metabolism, level of exertion, muscle mass, and so forth, all of which complicate the equation. So most machines are incorrectly calibrated. The best those machines can manage is a ballpark figure.
THIS MORTAL CURVE
Not only is it possible to use math and calculus to optimize our diet and exercise regimen and maintain a healthy weight; we can also use it to determine the probability that we will die in any given year, thanks to the work of an obscure British actuary named Benjamin Gompertz. Gompertz hailed from a family of wealthy merchants who emigrated to England from Holland. Because he was Jewish, he was denied admission to university and thus was largely self-educated. He acquired his mathematical knowledge by reading Newton’s works, among others, thereby becoming proficient at calculus.
One day, when he was just eighteen, Gompertz stopped in at a secondhand bookstore, and struck up a friendship with the bookseller, John Griffiths, who was a mathematician. Initially Gompertz wished to be tutored, but Griffiths quickly realized the young man’s knowledge already outstripped his own. Instead, he introduced him to the Spitalfields Mathematical Society (later to become the London Mathematical Society), of which he was then president. Gompertz joined the Society, despite the fact that the minimum age was technically twenty-one, and found himself with more than enough math tutors at his disposal, enabling him to advance rapidly in his knowledge. (The society had a rule whereby, if a member asked another for help or information, the second member was required to provide that assistance or else be fined a penny.)
He married the daughter of another wealthy Jewish family with strong ties to the stock exchange, and that connection enabled him to join the exchange himself. He eventually became an actuary and head clerk for his brother-in-law’s nascent insurance company, where his mathematical skills proved very useful. Apparently he had a great capacity for “sustained complex computation” in compiling detailed actuarial tables. In particular, Gompertz found he could apply the principles of calculus to human mortality to determine the cost of life insurance. “It is possible that death may be the consequence of two generally co-existing causes,” he wrote around 1825. “The one, chance, without previous disposition to death or deterioration; the other, a deterioration, or an increased inability to withstand destruction.”
In other words, assuming one doesn’t meet with a fatal accident, such as being hit by a bus, it is possible to use calculus to model the probability of the likelihood that one would die in any given year—a probability that increases with age. Gompertz tested his hypothesis by comparing the proportion of people in