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What is Calculus anyway?

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To me this stuff is never taught in Calculus Class but should be taught and reviewed over and over again.  As we struggle with the concept of Calculus – and why we’re required to study it in college to the extent they teach it (me as an electrical engineering student at the age of 50, 3 semesters) – and ponder over the seemingly insane extent to finding derivatives and integrals that the classes get into – the same question appears for most of us – what the hell is calculus, what’s a derivative, what’s an integral, and so what?

Sir Isaac Newton (1642 – 1727) (lived 85 years) born in England, never married (no wonder), had no children that we know about, is credited with discovering Calculus along with –  Gottfried Wilhelm Leibniz (1646  – 1716) (lived 70 years) Born in Germany who also came up with the way we notate calculus today such as the integral sign ( ∫ ), and dy/dx.

 

Lagrange invented the f ’(x) notation (derivative of   f (x)  )

Leibniz invented the “y = f (x)” notation and the definition of a derivative as:

 

f (x + ∆x) – f (x)

lim       ——————–

∆x→0      (x + ∆x) – x

 

Notice how close the above is to the “definition of a derivative” or “difference quotient” that I have programmed and exampled on YouTube for you:

 

f (x + h) – f (x)

lim       ——————–

h→0               h

 

Newton worked on solutions in analytical geometry of drawing tangents to curves (differentiation) and defining areas bounded by curves (integration), so:  If somebody asks you what is calculus you say:

 

Calculus is the study of tangent lines to curves (differentiation) and areas under curves (integration); to me It’s that simple.

 

However my gripe is that it should be condensed and taught only for one semester – unless you’re a math major – There has to be much more important things to teach an engineering student in that field – in those extra two semesters – than tangent lines and areas. It’s hard to believe for me that after programming and studying the quotient rule, product rule, integration by parts, transcendental derivatives and integrals and all else that comes with finding a derivative, for all these years – that when you solve for x in that derivative you come up with a number and that number is the slope of a line.  Not a tangent line yet oh no!! – you have to go to my program of “ tangent line to a curve” to get the line to be placed on that tangent point on the functions curve.  The number you get after solving for x in the derivative lets say 15, you go 15 notches up on the y axis and 1 over on the x axis, draw a line down through (0,0) and that’s the slope of that line and what you found. After all that!!

 

Another thing is that calculus with regards to derivatives only works with functions.  The actual function is the trick, and that’s found by experimentation to be able to come up with data points (x,y) or (x,y,z) to be able to graph it.  When a professor says that the first derivative is also velocity, which is true, – making you think that calculus discovered it –  the thing Is – velocity has already been found at that point or any point on the curve by the genius who designed the function in the first place. Incidentally that 15 number above would be 15 meters per second at that computed point on the curve with regards to velocity, slope is just a number.

 

Like John Goodman says in the Big Lebowski, “am I wrong?”

 

One more thing before I let you go – from Wikipedia

 

Leibniz became one of the most prolific inventors in the field of mechanical calculators. While working on adding automatic multiplication and division toPascal’s calculator, he was the first to describe apinwheel calculator in 1685[4] and invented theLeibniz wheel, used in the arithmometer, the first mass-produced mechanical calculator.

 

Is it any wonder why I programmed the calculator for my own use and now yours – I’m in good company – with the following from wikipedia regarding first mechanical calculators.

 

“The desire to economize time and mental effort in arithmetical computations, and to eliminate human liability to error, is probably as old as the science of arithmetic itself. This desire has led to the design and construction of a variety of aids to calculation, beginning with groups of small objects, such as pebbles, first used loosely, later as counters on ruled boards, and later still as beads mounted on wires fixed in a frame, as in the abacus. This instrument was probably invented by the Semitic races and later adopted in India, whence it spread westward throughout Europe and eastward to China and Japan.
After the development of the abacus, no further advances were made until John Napier devised his numbering rods, or Napier’s Bones, in 1617. Various forms of the Bones appeared, some approaching the beginning of mechanical computation, but it was not until 1642 that Blaise Pascal gave us the first mechanical calculating machine in the sense that the term is used today.”

 

Enjoy my programs,

 

Tom

everystepcalculus

Critical numbers and critical points in graphing

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It seems that most calculus tests I receive to check my programs with, and with regards to graphing a function by hand, they always have you find: ” critical points”.  Then the answer is always just what “x” equals.  You factor the first derivative f ‘(x), find the value of x or x’s and mark it down on your test problem to get it correct x=2 or x=5 or whatever.  If I was your professor (you wish) you’d have gotten only partial credit because you found the “critical numbers” and not the “critical points”.  However, critical points are actually the value of (x,y).  You find the values of x, from the first derivative, plug those values into the original function f(x) to find the value of y, and you have the critical point or points (x,y).  When tests ask for critical numbers the professor actually means critical numbers.  There is a difference!! In my program I find both for you critical numbers and critical points (step by step of course) and leave it up to you as to what or how your professor teaches this, in most cases teaching it incorrectly, what else is new?

First day of calculus I

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In my first day of calculus I – the chalk was flying.  That professor started with the fundamental theorem of calculus, probably said “if it exists” a hundred times, and never let up from then on. It got worse from then on and never better. I, with others – I’m sure – sat there in disbelief and in a fog.  I was 50 years old that day. I remember 3 semesters later I turn to the guy next to me and ask, “What the hell is a derivative?”  He whispers,  “I think it’s an angle of a line or something”.

Try it yourself go up to any one of your friends, ask them first if they’ve taken calculus and then ask them what a derivative is, and see what the say or don’t say.  That’s the way all of my professors taught in my experience of college, I never enjoyed one class.  If it was me – and my inability to learn in a class – that kid would have told me what a derivative was without any question as well as others in the class, however nobody knew (I asked several others after that kid) they didn’t know either. We were all three semesters into calculus and didn’t know what a derivative was. What the hell is that?  Isn’t knowing what a derivative is, in no uncertain terms, more important than the fundamental theorem of calculus???