Zack Bajin                                    Calculus Intro - The Ultimate “AHA”!                                       Fall 2020

www.gbcprof.com

                                                “Knowledge is Power”, Sir Francis Bacon – See also: Newton, Bacon, Descartes

Some useful words and symbolic expressions, here and in general:

     Smidgen – very small chunk of something (ice cream, material)

     Bit – small quantity of something (tiny portion of anything, smallest piece of information)

     Quantum – minimum amount allowed or required (money, energy)

     Zilch – next to nothing, not any (practically zero of anything)

     Δ − delta (capital D in Greek alphabet), tiny difference between two values (e.g.: Δx = x2 – x1)

     y(x) – “y is a function of x” or "y at x" for short, meaning y depends on what one picks for x at will (Euler’s notation).

               e.g.: y(x) = 3x2 – 5x + 7      According to Descartes (“I think therefore I am”) , a graph can be drawn

                                                                       on a flat surface having y as the vertical axis and x as the horizontal axis

                                                                           (N.B.: he invented xy coordinates lying in bed observing flies on the ceiling)     

     y(x + Δx) - "y at x + Δx" - same meaning as above.

Isaac Newton, the 18th century founder of classical science and engineering from London, England, had a problem:

     When an object’s speed is constant, it’s a line in a distance vs. time graph, or more generally Descartes‘ (Cartesian) xy system.

     Speed is the slope of that line (change of distance in an amount of time) and is easy to calculate:

     On the graph, it’s a ratio of vertical rise to horizontal run between two points: (y2-y1)/(x2-x1), Δy/Δx for short.

     But when the speed is varying the graph of distance vs. time is not a line any more, it’s a curve.

     The speed (slope in the graph) varies from point to point on the curve, no matter how close the two points are.

     The slope, rise over run, in one point only is 0/0! How can it be calculated? Newton took almost 20 years to think of a way.

Leibnitz, a mathematician from Leipzig in Germany, found a way to calculate the slope of a tangent on a curve.

     NB: Newton and Leibnitz both claimed to have found it and Newton accused Leibnitz of plagiarism!

     This is how Leibnitz did it:

     Select two points as close as possible on a given curve and draw a line between them. The resulting slope of the line

     (Δy/Δx or rise over run) is between the slopes of the tangents through each of the two points.

     If one shrinks the smidgen of Δx between the two points to zilch, the distance Δy between the two points shrinks to zilch as well

     and the connecting line and the two tangents converge to one tangent with a specific slope through one single point.

     Instead of writing its slope as 0/0, or y’(fluxion) as Newton did, Leibnitz writes dy/dx (zilch of y divided by zilch of x)

     This precise tangent slope through one point on a curve Leibnitz called the DERIVATIVE (We ended up using mostly his

     terminology and notation). By carefully applying conventional algebraic steps, one tediously arrives at a simple formula like this:

Using above example: y(x) = 3x2 – 5x + 7 (with the table of values one can see that each point sits on a parabola)

Another point, Δy away: y+Δy = y(x+Δx) = 3(x+Δx)2 – 5(x+Δx) + 7 (insert x+Δx into the original equation everywhere there was an x)

Slope of the line between them: Δy/Δx = [(y+Δy) - y]/Δx = [y(x+Δx) - y(x)] /Δx = {[3(x+Δx)2 – 5(x+Δx) + 7] – [3x2 – 5x + 7 ]} /Δx

     (Note that today many high school books use h instead of Δx, making lim h = dx missing the ultimate AHA!)

     PHEW, the setup is done!

But now the algebraic workout starts:

     Δy/Δx={[3(x2 + 2xΔx + Δx2) – 5x – 5Δx + 7 – 3x2 + 5x – 7}/Δx={6xΔx + 3Δx2 – 5Δx}/Δx     Δx can now cancel.

     Δy/Δx = 6x + 3Δx – 5   GOTCHA! Turn the smidgen of Δx to zilch ( lim Δx = 0, better: dx). Then: dy/dx = 6x– 5

Therefore:

     Formula for any term in a polynomial: If y(x) = a xn then dy/dx = a•n xn – 1 Easy to memorize!

     Formula (shortcut) for a trig function: If y(x) = cos x then dy/dx = – sin x is found the same way.

     Just as easy to memorize, but it’s only one of far too many! (See the proofs in Derivatives of Transcendental Functions)

     FYI: The above method, called the Delta Process, is the only way a tangent slope is found on any curve, including

     trigonometric, logarithmic, and other transcendental (non-algebraic) functions.

Unfortunately, this amazing AHA experience is missed by too many Calculus “experts” burdening themselves and those they end up teaching with memorized formulas, starting with the polynomial formula. This makes calculus a subject impossible to grasp. Many who learned it this way end up questioning its usefulness before realizing how powerful calculus really is.

Result: Newton had to invent calculus to explain his discovery of mechanics basics but now many authors of new physics books are trying to avoid calculus all together, working mostly with averages and approximations avoiding derivatives.

A major lack of essential knowledge and loss of empowerment!