Hi Tom. How do I use this program to solve these two types of problems? What buttons do I push to locate the appropriate tools to use when solving these?
What is Calculus anyway?
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 massproduced 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
My Programming History
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 retook it the next semester and got an A. The semester after that was Calculus II, when we were required to purchase the TI92 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 TI92 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 GraphLink was able to download that – as notes – into my TI92. 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 TI89 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 TI92 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.
Critical numbers and critical points in graphing
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?
Definition of Derivative, Calculus App, TI89 Titanium
Raw Transcript
This is a video on the definition of the derivative also called a quotient, difference quotient. A very difficult program wouldn’t be difficult for you and me, if you have all day, all night and several days to study it and get the handle on it and really think about it). But this program may help in your calculus endeavors. Gonna get started, press 2nd Alpha (this shows that you can enter letters in the calculator). My code for getting into my program is index then you press alpha again to get number 8 and the (). Should be index8 (). Press enter and that takes you to the menu, where you can scroll down and select what you want. We’re working with definition of derivative here, select and press enter and put in formula or function. Press Alpha first then enter everything in the boxes on the program. We’re gonna do 3*x^2x+1 go to ok or change it if you want. Write down the formula that comes up. First thing to do when writing a problem is to write the formula down and then you start substituting x+h for every x in the function, which I’ve done here. 3*”x+h” or parentheses actually, squared, minus “x+h”+1 and then the original formula all over h.
F’ (x) =lim
h>0
=6*x+3*h1
@h=0
Answer
=6*x1
Everystepcalculus.com, enjoy my program visit my website.
Difference Quotient, Calculus App, TI89 Titanium
Difference Quotient, Calculus App, TI89 Titanium
Raw Transcript
Hello, this is Tom from everystepcalculus.com. I’m gonna do the difference quotient for you today or what others call the definition of a derivative.
We’re gonna get started. Turn calculator on, press 2nd Alpha and put index the alpha again, then 8, then () and press enter. Up comes my program menu, you can see the things I’ve programmed so far. There is a graphing section with the calculus and graphing, maximum and mins among others. But today we’ll be doing a definition of a derivative or difference quotient, which ever you know, either one will take you to the program. Press enter and you are in the actual program. Every time you see pause here, you can enter. We’re gonna put in an example, for instance, (you have to press alpha first when you put anything for these boxes) I like to use 7*^27*x. Now you’re gonna substitute for every x in the function, x+h
Should look like this: f(x) =7*x^27*x select ok or change it if you want. This is the formula, the f ‘ of x = the limit as h goes to 0 h represents the change of x.
Formula should be: f’ (x) = lim
h>0
= f (x+h) f (x)
h
h represents x
Jot down on your paper then put in your function. Notice I’ve substituted “x+h” for every function and the original function all divided by h.
F’ (x)
7*”x+h”^27*”x+h”
= (7*x^27*x)
h
Very complicated program, took me a long time, maybe a month to do this program. Worked a lot with research to do it, calculus books and such. So, you’re gonna write this on your paper, but you’ve taken the x squared of the function and worked it out.
F’ (x)
7*x^2+14*h*x+7h^2
““ (7*(x+h))
(7*x^27*x)
h
Keep writing these down and eventually you’ll get:
F’ (x) = lim
h>0
=14*x+7*h7
@ h=0
Answer
= 14*x7
And you’ve gotten a hundred percent on that problem and you look like you know what you’re doing and everything else.
Everystepcalculus.com you can go to a new problem, main menu or you can quit or do what you want.
Enjoy my programs
Curl Program
Raw Transcript
Graphing: Maximum & Minimum Program
Video Example: Critical Points for Graphing Program
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So, this video is on critical points and critical numbers, with regard to graphing by hand in calculus. Calculus one, or all through calculus I guess. Ahh Let’s get started. You press second alpha on your calculator titanium. See this box here which turns black which shows you that you can enter letters. And the code for my menu of course is I n d e x, and then you have to press alpha to get back to numbers and parenthesis, and you’re into my menu. My menu has many many things in it. Right now we’re talking about critical points, so – were gonna go – uhm – to critical points. Pull up the program, and I tell you to mark on your paper right away, on your test paper or whatever, ya know, a graph with these, ahh all labeled in numbers so you can mark down whatever comes up in my program, and then you can, ya know, connect the dots and have your graph completed – by hand. So we’re going to enter a function, a from a test, we’re going to have to – for any of these boxes that come up in my programs you have to add alpha – you have to press alpha first – so we press alpha, and we’re going to put in x cubed minus six times x squared, plus nine times x and plus two, and then press enter twice – and I show you what you’ve entered, in case you want, made a mistake, you can go back and do it, you press enter again, you can, ok or change it – I’m saying its ok – my programs also, you can press the number before these what the choice is, you can scroll down, or and then press enter, but you can also just press the number and it will go right to there. So were at critical points, we want critical points in the menu, instead of scrolling we’re going to just press three on the calculator – and – and I – discuss a critical points and numbers in a blog in my web site, so check that out. Most tests, come up with, ask you for critical points, and there’s a difference between critical numbers, critical points – and I discuss that – here I put a little bit of information about it. Critical point is really an xy point on the graph, and critical numbers are – are on the x axis, just what the x value is – however – professors and tests I’ve seen – uhm – ah – you know use both and – and it’s not correct, their different. Ahh, So – in critical numbers, your gonna – you’re going to set the uhm, first derivative to zero, and then solve for x – so we factor the first derivative, here, and we come up with the critical numbers – x equals three or x equals one – that’s what you’d put down – You put everything on your test, just like this. You can’t find the first derivative unless you put the function down – then you find the first derivative – and go from there – uhm – I do the – I find critical numbers and critical points on my programs – so – we add – we take three – one of the critical numbers of three – and plug it into the first function – three cubed, times, minus – six times three squared, plus nine times three, and you write this on your paper and you come with two – y equals two – so the first critical point is three and two – second one you plug in one for critical number into the primary function, and come up with six – so the second critical point is one and six – and then, it takes you back – you can find more parameters here, press one, you can get more parameters, and go whatever you want – local maximum min – Intervals of increase decrease – inflection point – what ever you need to complete the graph
Equation of Tangent Line to a Curve Program
Raw Transcript
This video is gonna be on the equation of a tangent to the x point on a curve. This is the line when you have found the first derivative and you find the slope. This is the equation that is in the line at the point where you find the slope. This program I pretty neat, check it out on my website. Turn calculator on and clear screen to get to the menu of the program by pressing F1 8. Now add index 8 () to get to the menu of the program. Press 2nd Alpha, enter index, press alpha again to get back numbers and parentheses add 8 (). Here we are, select the equation of a tangent line, scroll down and press enter. It shows you what programs are added here. Enter the function 8*x to the 3rd power + 6*x+9
You can change or press ok if you want. Now you want a point, maybe at point 3, press Alpha 3 (it shows x=3). If youmade a mistake you can change it or say ok.
Function: F (x) = 8*x^3+6*x+9
F (x) = slope
= 24*x^2+6
The derivative o 6x is 6 f’ (x) =24*x^2+6
@ x= 3
F’ (3) = 24*”3”^2+6
=222
You have to go back and put 3 in the function to find y
Y = f (x) = 5s^2+6x+9
@ x = 3
Y = 8*’3”^3+6*”3”+9
=243
So (x,y) = (3,243)
Y=243 (222 is the slope)
243 = mx+b
243= 222(3) +b
B=243666
=423
Y = 222x+423
Video Example: Equation of a line program
Raw Transcript
Ah, this video is going to be on the equation of a line, with regards to my fabulous programs that I’ve programmed on the TI89 calculator, and ah, these are pretty simple calculations, but it’s easy to forget how to do it. It’s nice to have a program to be able to do it. So, I’m going to clear the calculator here and we’re going to put in; 2nd Alpha, 2nd has to appear here, and then the alpha has to, to become darkened to go for the letters. And I have to put that in the entry line – I n d e x – and you will to if you buy the programs, ah alpha, it goes back to the number system, and then put the closed parentheses in, and press enter and we’re into the ah menu. You’ll notice there are many, many things on this menu. Definite integral, ahh derivatives, you know in algebra all of derivatives, ah transforming ah problems. Curl, product rule, quotient rule, whatever, but we’re going to do equation of a tangent line, you can press alpha e, or you can scroll down like this, and ahh, and press enter equation of a tangent line. Ah we’re going to enter the function, five times x squared, plus six times x, plus one let’s say – notice I forgot to press alpha, first, which is easy to do on this calculator, and so I’m gonna go back with two, I’m going to change it, and I’ll press alpha, five time x squared, plus six times x, plus one, and that’s better, five x squared plus six x plus one that’s the function – now say it’s ok – so we’re gonna, I generally press the one before the choice – and then we’re going to enter the x y component, you know, x, the values for x and y, so let’s do that now, let’s do it. You have to the parenthesis in first – you have to go alpha again – and put, let’s go three comma sixtyfour, something – close out the parenthesis – and let’s see – now we got three, x y z equals three sixty four, that’s cool – press one – and we have the original function – we find the derivative of it, the slope – which equals ten x plus six – at three – our x choice that we had, is the derivative, derivative at three is ten, and then add the three to the x and there’s the – thirty six is the slope which is m – press enter again – shows you the formula – you write this stuff on your paper as you go – and at y equals sixty four – which was our choice – sixty four equals m x plus b, so sixty four equals thirty six that was the slope – times 3 – which was x – plus b, sixty four minus one o eight minus forty four – here’s the formula – y equals 36 s+ minus 44 – for that point pretty neat, huh? everystepcalculus.com – check it out
Natural Log Program
Raw Transcript
Quotient Rule Step by Step
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Product Rule Step by Step
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so i’m going to demonstrate the uh… product rule on the titanium written you know with regard to my program start program a calculator uh… and to we need to get to the home screen here and you type index that i’m gonna clear this out and reapply it so they can show you how to do that you go second alpha get that little dark mark in there to show you you’re gonna put letters in the calculator i_n_d_e x and then alpha again to switch the numbers in parentheses and your into my programs there’s a menu of many many things depending on what i want to put in there but you know product rule like this one chain rule, quotient rule, quotient um difference quotient, limits trig integrals, derivatives log of base a or you know natural log derivatives, derivatives of thosebut anyways, we’re going to go team also goes straight up and then you can scroll down to go here’s velocity and stuff that you would need in calculus, um a lot of it were going to do the quotient rule in the next video but anyways were going to do the product rule now you can see that I’ve highlighted that, while that’s loading the formula for the product rule is of course h of x because you’re doing two functions f of x and g of x h prime of x is this uh… formula here prime of x g event epa vexed times g prime objectives the part of it for the and then were going to enter in the parentheses like this country an example so we have to go alpha and then enter the first parentheses and then you can enter whatever let’s do 5 times x squared plus six closed parentheses without the prince of sleaze which shows you what you’d entered the program you write that down in your paper dash if you think you want it if it’s okay if you like if you made a mistake and go back and change it whatever so saying it’s ok so each prime although it had to be good at wildness times the either function plus the other function times riveted the other functions of that so you write that on your paper ten x etcetera etcetera we come to the three h primal inferences that explicitly plus the derivatives at this when you add it up or multiply it up at fortyfive expert ninety extra eighteen and that’s the product rule opt regarding my programs check them out of my website at least of calculus dot com
Chain Rule Program Step by Step
Raw Transcript
Equation of Tangent Line on TI89
Equation of Tangent Line to a Curve
Raw Transcript
This video is gonna be on the equation of a tangent to the x point on a curve. This is the line when you have found the first derivative and you find the slope. This is the equation that is in the line at the point where you find the slope. This program I pretty neat, check it out on my website.
Turn calculator on and clear screen to get to the menu of the program by pressing F1 8. Now add index 8 () to get to the menu of the program. Press 2nd Alpha, enter index, press alpha again to get back numbers and parentheses add 8 (). Here we are, select the equation of a tangent line, scroll down and press enter. It shows you what programs are added here.
Enter the function 8*x to the 3rd power + 6*x+9
You can change or press ok if you want. Now you want a point, maybe at point 3, press Alpha 3 (it shows x=3). If you made a mistake you can change it or say ok.
Function: F (x) = 8*x^3+6*x+9
F (x) = slope
= 24*x^2+6
The derivative o 6x is 6 f’ (x) =24*x^2+6
@ x= 3
F’ (3) = 24*”3”^2+6
=222
You have to go back and put 3 in the function to find y
Y = f (x) = 5s^2+6x+9
@ x = 3
Y = 8*’3”^3+6*”3”+9
=243
So (x,y) = (3,243)
Y=243 (222 is the slope)
243 = mx+b
243= 222(3) +b
B=243666
=423
Y = 222x+423
How Professors teach compared to my programs
Here is what you get as an answer usually when you ask a question on Calculus or even Physics in my experience and evidently the asker was satisfied. The person answering is a professor at a college. This girl asked on line for help on what the derivative of 4cos(5x2) was. This is a chain rule problem. How would you like it answered?

Best answer as selected by question asker.
For a function f(x) = g(h(x)), express h(x) as y.
Then f(x) = g(y), f’(x) = [d {g(y)}/ dy]*(dy/dx).
Here we have to find the derivative of f(x)= 4 cos (5x2).
Let y=5x2, this gives f(x)= 4 cos y
f’(x)= [d (4 cos y)/dy]*[d(5x2)/dx]
We also know that the derivative of cos x= sin x.
=> [d (4 cos y)/dy]= 4 sin y
[d(5x2)/dx]= 5
Therefore f’(x)= [d (4 cos y)/dy]*[d(5x2)/dx]
= (4 sin y)*5
=4*sin (5x2)*5
=20 sin (5x2)
Therefore the derivative of 4 cos (5x2) is 20 sin (5x2)
Calculus
I had a person question me over the difference quotient. He couldn’t load 4x3 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 4x1 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.
Point Slope Form: The relation to calculus
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^(21)
= 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 TI89 calculator: (Click Links below)
What is a derivative?
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 reteaching 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!!