A Mind Unleashed

December 25, 2022January 10, 2023 / Michael Xue / Leave a comment

In 1665, following an outbreak of the bubonic plague in England, Cambridge University closed its doors, forcing Issac Newton, then a college student in his 20s, to go home.

Away from university life, and unbounded by curriculum constraints and tests, Newton thrived. The year-plus he spent in isolation was later referred to as his annus mirabilis, the “year of wonders.”

First, he continued what he had begun at Cambridge: “forging the sword” in mathematical problem solving; Within a year, he gave birth to differential and integral calculus.

Next, he acquired a few glass prisms and made a hole in his window shutter so only a small beam could come through. What he saw after placing a prism in the sunbeam sprung his theories of optics.

And yes, there was an apple tree in the garden! One fateful day in 1666, while contemplating celestial body movements under that tree, Newton was bonked by a falling apple. It dawned on him that the force pulling the apple to the ground might be the same force that holds celestial bodies in orbit. The epiphany led him to discover the law of universal gravitation.

Newton returned to Cambridge in 1667 after the plague had ended. Within six months, he was made a fellow of Trinity College; two years later, the prestigious Lucasian Chair of Mathematics.

To William: on dy/dx and FTC

April 1, 2017March 2, 2023 / Michael Xue / Leave a comment

You have asked good questions.

Given $x = x(t), y=y(t)$ and $x=f(y).$ i.e.,

$x(t) = f(y)$ where $y = y(t).$

By Chain Rule,

$x'(t) = f'(y)\cdot y'(t).$

here,

$x'(t), y'(t)$ – derivative of function $x(t), y(t)$ with respect to $t$ respectively.

$f'(y)$ – derivative of function $f(y)$ with respective to $y.$

Suppose $y'(t) \ne 0,$ then

$\frac{x'(t)}{y'(t)} = f'(y).$

Written in Leibniz’s notation, $\frac{\frac{dx}{dt}}{\frac{dy}{dt}} = \frac{df}{dy}.$

Even though Leibniz’s notation is convenient at times, it often gives the WRONG idea that the derivative of a function is the ratio of two mysterious quantities ( $dy, dx$ ) that are not zero and yet, smaller than any positive numbers.

So saying “split the derivative” is correct (from looking at the Chain Rule) since derivative implies a process of taking the limit:

$f'(t) = \lim\limits_{\delta t \to 0} \frac{\delta f}{\delta t}= \lim\limits_{\delta t \to 0}\frac{f(t +\delta t) - f(t)}{\delta t},$

not simply $\frac{\delta f}{\delta t}.$

To see how antiderivative and definite integral are related. one must dive into the proof of the Fundamental Theorem of Calculus (For example, see “A Sprint to FTC“). After that, you can decide for yourself how the concepts of rate of change and “area under the curve” are logically linked.