Provide and are suitably restricted, there is an Extended Binomial Theorem. Namely,

where

Although Issac Newton is generally credited with the Extended Binomial Theorem, he only derived the germane formula for any rational exponent (i.e., ).

We offer a complete proof as follows:

Let

Differentiate (1) with respect to yields

We have

That is,

From

we see that is a solution of initial-value problem

By we also have

Express (3) as

and then differentiate it with respect to

gives us

i.e.,

Since

we see that is also a solution of initial-value problem (2).

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.

A guided missile is launched to destroy a fighter jet (Fig. 1).

Fig. 1

We introduce a coordinate axes such that at the missile is at origin and the jet at . The jet flies parallel to the x-axis with constant speed . The missile has locked onto the jet so it is always pointing at the jet as it moves. Its speed is

Find the time and position the missile strikes its target.

“The missile has locked onto the jet so it is always pointing at the jet as it moves” means that the tangent to the missile’s path at any point will pass through the position of the jet. The equation of the tangent is

In addition,

Notice are functions of time

Differentiate (1) with respecte to

By (1), substituting for ,

Let we express (2) as

Solving for gives

Submitting (4) into (3),

Let

(5) becomes

We solve this non-linear differential equation as follows: