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You are here: Home / Archives for Calculus 1

Limits Calculus

August 30, 2012 by Tommy Leave a Comment

 

Limits in Calculus on the TI-89: Raw Transcript

This is a video on limits as it applies to calculus, generally calculus one, and let’s
get started. You have to press second alpha to get to my menu. You can scroll down from the menu many things you want, all in alphabetical order, we’re going to go down to limits and press enter. Here’s the choices in the limit program you have with regards to limits. You can compute the limit, see the definition of a limit formula, complete a table of limits, and prove that a limit is equal to the computed limit. You can also find delta given epsilon. We’re going to do the first choice now finding delta given epsilon. You have to press alpha before you enter anything in the boxes that come up in my programs. We’re going to try two times x minus five equals one as x approaches three and the epsilon they give you, They will give you these things on the test problem they give you, so epsilon equals point zero one. So I show you what you’ve entered. The limit of two times x minus five equals one, as x approaches 3, while epsilon equals point zero one. The next screen you have the choice of changing it or saying that it is ok. I say it’s ok so I’ll press one for the choice. Here’s the first formula, absolute f of x minus L is less than epsilon. I took the one over here and added it to the minus five to get minus six equals zero. That’s an absolute value sign not a one, and that all is less than point zero one. You’ll keep writing this stuff down on your paper exactly
as it shows. This is the way it done on youtube videos, or at least what I could find, and
delta equals point zero zero fine. Now we can go back to choose main menu or new problem, we want to do a new problem so I’ll choose one again and then go to number two and compute the limit. You can put anything in you want within reason. Alpha again, anytime you are doing division you have to put parenthesis in — in any problem — so we want to make sure we get the parenthesis in there. Let’s go parentheses x minus two closed parentheses, divided by, parenthesis x squared minus x minus two, closed parenthesis. As x goes to alpha two. Here’s the problem you’ve entered and you want to compute the limit. You say it’s ok. Now at x equals two, f of x equals zero and therefore doesn’t exist, so you use the following to find the limit. You add point zero one to every x in the formula so you are a very little away from the limit given and then compute it. The limit is point three, I give you other ways of showing that, because some tests require other answer forms.
Let’s do another problem, let’s do a table of limits — number four — let’s clear
that out and do another one. Alpha closed parenthesis x minus closed parenthesis divided
by x squared minus four, as x approaches two, and here’s the table you’d mark down on
your paper. Let’s do another problem, this is always fun. Prove that the limit is L — the limit
— this is done in always calculus one and throws every student off. Youre panicked because
they are talking about epsilon and delta and the definition of it, which sounds quite complicated, and of course it is useless for the rest of calculus in your life. Let’s put a problem in here. Alpha three times x plus five equals thirty-five as x approaches ten, You have to press alpha again to enter the ten. and here’s what your problem is. Here’s the
formula again. F of x minus L is less than epsilon. You write this stuff down exactly
as shown on your paper. You factor it and then x minus ten is less than epsilon divided
by three. Epsilon becomes delta, and here’s the proof. If zero is less that absolute x
minus c — c is a constant — and is less that delta, then zero is less that absolute
x minus 10 is less than epsilon over three and therefore the original function is less
than epsilon. Put this on your paper and get one hundred percent on that problem. EveryStepCalculus.com check it out and check out the blogs also.

Filed Under: Calculus 1, Limits Tagged With: Calculus Limits, Calculus Videos, Limits Calculus, Limits in 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

Filed Under: Calculus 1, Professors Tagged With: Calculus App, definition of a derivative, derivatives, first day of calculus, point slope form, step by step calculus, TI-89 Titanium, what is a derivative

My Programming History

August 9, 2012 by Tommy Leave a Comment

About 19 years ago (1993)  I started college at San Diego State at the age of 50.  Electrical Engineering Major.  Calculus one was miserable for me and I actually flunked it, then re-took it the next semester and got an A. The semester after that was Calculus II, when we were required to purchase the TI-92 calculator, which had just come out from Texas Instruments.  Cost was around $185 I think.  I was pissed about buying that because of the cost and I thought my old HP calculator would work just fine.   Turns out that the TI-92 purchase certainly helped my college career in all my classes and has allowed me to sell my programs for all these years. First I found out that it had a word processor in it and so I started to scan my homework, study problems and whatever into my computer and via Graph-Link was able to download that – as notes – into my TI-92.  I could find topics via word search and it helped me somewhat for tests.  Anything like that however is like an open book test where one has to find the problem, read it, then add your variables and try to get the problem correct.  To me a very slow process and in many instances to slow to even finish all the problems in a test.  Then one day – in desperation for a better system – I happened to read and discover,  in the TI manual, the subject of programming the calculator.  I discovered the fabulous programming capabilities of the TI calculators. Wow what a system for me or anyone. I had an edge over anyone in class from then on, and even better for me, was the ability to never forget a problem. To desperately avoid the waste of time system in college,  of cramming – testing – and forgetting – (CTF) which is the main system of college even today.  I can still do all those problems;  Calculus, Physics, Electronics, Lasers, Optics, even Geology problems. Even if you took fabulous complete notes in classes and college you still couldn’t add the variables and complete a problem, after a while, without out again studying. When you are young like most college students, you don’t know you are wasting time, and don’t care for that matter, but when you attend at the age of 50 its a different story, CTF and wasting time is, and was, not acceptable. I got so good at programming the TI calculators, that  I wrote a manual on programming and used to sell that. However after the Titanium came out that ended.  I would have never found programming, to any helpful extent with TI Connect and the TI-89 calculators, Titanium included. The programming system is still in those calculators but extremely impractical.  Wouldn’t have happened.  I still think that the greatest calculator ever from Texas Instruments was, and is, the TI-92 Plus calculator. Better than the Voyage 200, the NSpire Cas or the Titanium.  The Nspire Cas Cx is pathetic, with no practical programming capabilities to my knowledge.  Anyway enjoy my programs, there is nothing like them.

Filed Under: Calculus 1, Professors

Calculus

March 16, 2012 by Tommy 5 Comments

I had a person question me over the difference quotient.  He couldn’t load 4x-3 into my programs.  I had not programmed that “straight line” into my programs I had only programmed something with x^2 (curve).  He gave me a you tube video feed that 4x-1 was a valid function.  I guess it is after watching the feed.  So I programmed it for him and now you.  I’m a guy who thinks practical about things.  I programmed my calculator because I was smart and allowed to use the calculator for tests.

 

Filed Under: Calculus 1 Tagged With: calculus, derivatives, difference quotient

Point Slope Form: The relation to calculus

March 14, 2012 by Tommy 1 Comment

The equation of a line to a point on a curve (point slope form) includes the slope and the position of that line on that curve function.  It’s better than the derivative because the derivative only tells us the slope. Again in Algebra the professor forgot to tell us the importance of that and the relationship to the derivative. Didn’t make it interesting enough to sink in and how it relates to the real world.

You have a function.  Has to have x^2 in it to be a curve from my understanding,  Example y or f(x) = 3x^2

Graph that and you have some form of curve in this case a “valley” parabola, (my own word), -3x^2 and you have a “mountain” parabola (again my own word).

Pick any point “(x,y)” Example: (3,12)

Point = (3,12)

x = 3

y = 12

 Find the derivative:   f(x) = 3x^2

f’(x) = (2)(3)x^(2-1)

= 6x^(1)

= 6x

Compute the derivative at the point “x”

f’(3) = 6(3)

= 18 = m = slope

Point slope form = y = mx + b

y   = 12 so:

 12 = mx + b

m  = 18

12 = 18x + b

x    = 3  so:

12  = 18(3) + b

= 54 + b

b    = 12 – 54

= – 42

y    = mx + b

= 18x + -42

 If you graph this equation along with the original function you’ll see the tangent line to that point on the curve

The slope =  18/1   (rise over run)

The angle of that tangent line = tan^(-1)(18/1) = 86.8 degrees

(make sure your calculator mode is in APPROXIMATE and DEGREES)

Fabulous and exciting, right?  lol Tom

p.s. You’ll love my programs

Have a test or quiz on point slope form? Here is a video example using the programs on the TI-89 calculator: (Click Links below)

Point Slope Form Calculator

Point Slope Form Given Two Points

Filed Under: Calculus 1, Derivatives, Point Slope Form, Professors Tagged With: algebra, calculus, derivatives, point slope form

First day of calculus I

March 14, 2012 by Tommy 2 Comments

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???

Filed Under: Calculus 1, Derivatives, Point Slope Form, Professors Tagged With: definition of a derivative, derivatives, first day of calculus, what is a derivative

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