step by step calculus Archives | Every Step Calculus

What is Calculus anyway?

August 13, 2012 by Tommy Leave a Comment

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

August 1, 2012 by Tommy 2 Comments

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?

What is a derivative?

March 14, 2012 by Tommy 2 Comments

Ask anybody “what is a derivative?” and you’ll quickly find out that nobody can tell you exactly what it is in no uncertain terms without any question. If fact, most people you ask who have taken calculus can’t even come close. Think I’m wrong, try it and you’ll find out what I did, it took me eight years after college to find out and even then not exactly. That’s pathetic and unacceptable in my opinion. Of course I was programming the TI Calculators in calculus at the time so I had some interest to even ask the question. I mean, who cares right? No one out of college will ever calculate a derivative or integral again in any job outside of re-teaching it as a professor, so I/we understand that.

I had some interest because of programming step by step calculus into the calculators and while teaching tennis to this guy named Mike, I find out – he at one time worked at NASA. At the time of me teaching him tennis he had left NASA was a professional black jack player. Went all over the world making money at black jack in the casinos. I was also fooling around with on line poker at the time and asked him how come black jack and not poker? He said poker was too much gambling and black jack is relatively sure. He said he used differential equations to help him count cards and change the odds in his favor. Anyway I asked him what was a derivative? He said immediately that it was the slope of a tangent line to a curve. I said “but when I graph the derivative there is a line – but no tangent line and the slope is off.” He said the graph of the tangent line is meaningless, “of no value”. He said when you compute the derivative of a function at an “x” value you come up with a number and that is the slope of the tangent line at that point on the function. I thought even that was fabulous and an eye opener, but we had finished picking up tennis balls and so I let it go and started to again teach him tennis.

After that moment, I kept thinking and thinking and thinking of what he said and then it dawned on me. The number you get when computing the derivative at the chosen point “x”, no matter how deep the original function is the numerator of a fraction with the denominator = to 1. That was rise over run. If the numerator is 12, that is 12/1. You go 12 spaces up the “y” axis and one space over on the “x” axis. Draw a line from that point through the origin of a graph (0,0) and that is the exact angle or slope of the line of the derivative. For that line to me tangent, it must touch the curve at only one minute point, so that line has to be transposed to do that, however it will still be the exact slope. To transpose that line you have to compute the equation of that line to the desired point (point slope form) and then graph that function and you have the perfect picture of what a derivative is. That’s fabulous. (Equation of a line at a point on a curve is in my calc1 programs). Now at the next party you know all there is of “what’s a derivative” and can look like a nurd, I mean nerd. lol, Tom

p.s. (The exact angle of that slope by the way is tan^(-1)(12,1), Fabulous!!