Some Applications of Derivatives — Part I

Sensitivity to Change.

When a small change in x produces a large change in the value of a function f(x), we say that the function is relatively sensitive to changes in x. The derivative f^{'}(x) is a measure of the senstivity to change at x.

Example:

The Austrian monk Gregor Johann Mendel (1822-1884), working with garden peas and other plants, provided the first scientific explanation of hybridization. His careful records show that if p (a number between 0 and 1) is the frequency of the gene for smooth skin in peas (dominant) and (1-p) is the frequency of the gene for wrinkled skin in peas, then the proportion of smooth-skinned peas in the population at large is

y = 2p(1-p)+p^{2}=2p-p^{2}

The graph of y versus p is an inverted parabola. If you plot and see, it suggests that the value of y is more sensitive to a change in p, when p is small than when p is large. Instead, this is borne out by the derivative graph which shows that (\frac{dy}{dp}) is close to 2, when p is  near zero, and close to zero, when p is near 1.

More later,

Nalin Pithwa

PS: why peas wrinkle

British geneticists had discovered that the wrinkling trail comes from an extra piece of DNA that prevents the  gene that directs starch synthesis from functioning properly. With the plant’s starch conversion impaired, sucrose and water build up in the young seeds. As the seeds mature, they lose much of this water, and the shrinkage leaves them wrinkled.

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