Every Step Calculus

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Calculus Help

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Good morning Brandon,
My limits work exactly as the video, so I’ll send you again limits and you reload and try it.
Thanks for sending me problems 8, 9, 10
One thing to remember in calculus is that you can never integrate or differentiate multiplication or division. if you see a times sign * or division sign / you have to use different rules, for instance problem 10a.  To be correct there should be a times sign between the radical, in other words √(x)*(x^2+3).  When you see a times sign it’s a product rule and when you see a divide sign like in 10b its a quotient rule problem.  Noticing that then In my index8() menu you’d scroll down to either one of those  (quotient or product rule) and enter the function given.  Problem 9 you got correct but if you didn’t know you could have scrolled down to “equation of a line” in my menu you’d have the exact solution step by step.  Problem 8 “definition of a derivative” equals the same as the “difference quotient” which I’ve put into my menu now if you’ll notice.I’ve seen either one used in tests. I’m going to attach the exact answer to problem 8 and see if I can’t program that for you. Notice that if you were’nt asked to use the definition of a derivative you see a divide sign so you could get the answer via the quotient rule in my programs entering (1)/(x-3).  Always in your math future get used to using parenthesis around any function in a denominator.  The bonus problem notice it isn’t really clear (without the use of parenthesis) of whether the problem is √(x^2+3*x+1) – x  or √(x^2+3*x+1-x)  which would equate down to √( x^2+2*x+1 ) to me (I might be wrong) you could maybe get some credit for trying it by doing this √ ( ∞^2 + 2*∞ + 1) = ∞  or in the other case 
 √ ( ∞^2 + 3*∞ + 1) – ∞ = – ∞
I don’t know whether your professor used delta x  (∆x)  or “h” in explaining the difference quotient or definition of a derivative formula
Send me all of your problems for quizzes or tests or practice tests I’ll help you,  Thanks, Tom

Calculus Help

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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?

1.)  F ‘ (x) for f(x) = x^2 – 6x + 4
2.)  Find F ‘ (x) for f(x) = cubed root of x^2 + 1/x^2
I’m studying and practicing for a Calculus exam on Monday so I’m trying to learn and figure out your program today and tomorrow to assist me a little bit.
Thanks,
Nelson
F ‘ (x) for f(x) = x^2 – 6x + 4
I haven’t programmed this because it is too simple
I don’t think you need a program to solve this, and this is fundamental for your future calculus to learn, also easy
First you’ll notice that there are 3 items all seperated by minus and plus signs (x^2    -6x   and +4)
for a derivative, you take the exponent 2 bring it in front of the ‘x’ variable and multiply it times anything in front (in this case 1), then subtract one from the exponent to get x^(2-1) = x^1 = x   so   2x
Then put the minus sign down
and the derivative of -6x is just what’s infront of x or -6
the derivative of a number ( in this case 4) is zero so:
f ‘ = 2x-6
If the x^2 was 8x^2  then the exponent 2 would be multiplied by 8 to get 16 x
———————————————————–
Find F ‘ (x) for f(x) = cubed root of x^2 + 1/x^2
³√(x^2+1/x^2)
Anytime you see something like this you must get rid of the radical sign by the following:
(x^2+1/x^2)^(1/3)     (the 3 in front of the radical is really ^1/3)
If the problem was like this:
²√(x^2+1/x^2)  then it would be (x^2+1/x^2)^(1/2)    (the 2 in front of the radical is really ^1/2)
Anytime you see an exponent (1/3) in calculus out side a function enclosed in parenthesis  (x^2+1/x^2)
You think of the chain rule
You go to my menu  ( index8()  ) scroll down to chain rule and do the problem there
You have to press ALPHA first when you enter anything into my boxes for variables
Then:
(x^2+1/x^2)^(1/3)
Press ENTER
write everything you see down on your paper or test until the end answer
Pretty complicated problem and would not appear on a test, but maybe in homework, my programs are designed mainly for passing test questions.
Good luck, Tom

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

My Programming History

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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.

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?

Definition of Derivative, Calculus App, TI-89 Titanium

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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^2-x+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*h-1
@h=0
Answer
=6*x-1

Everystepcalculus.com, enjoy my program visit my website.

Difference Quotient, Calculus App, TI-89 Titanium

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Difference Quotient, Calculus App, TI-89 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*^2-7*x. Now you’re gonna substitute for every x in the function, x+h
Should look like this: f(x) =7*x^2-7*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”^2-7*”x+h”
= -(7*x^2-7*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^2-7*x)
h

Keep writing these down and eventually you’ll get:
F’ (x) = lim
h->0
=14*x+7*h-7
@ h=0
Answer
= 14*x-7
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

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Hi I’m Tom the calculus and physics guy that programs the TI-89 Titanium calculator uh… today we’re going to do the curl on the titanium and show you step by step how to do the curl enjoy my programs at EveryStepCalculus.com the first thing you enter is index eight close parentheses in my programs the home screen and uh…up will come the menu and you can see where you can scroll down for many many other things which then again takes you to other many many things so what were going to do the curl today I’m going to press number 8 or you can press the number of the alphabet before it and you notice we have instructions curl definition or do curl problem uh… whatever your choice is there were going to do the problem quickly give you the choice of the P Q R or the M N P variables
I’m going to use the P Q R system and here it is in the cartesian form I J K or the vector form with the two arrows on the sides and the P Q R inside of the arrows that’s called the vector form and the first function we’re going to put in for P and press alpha first uh… y times z squared press it twice 2nd function is alpha you have to press alpha first as I just said uh… x times y and the next example is alpha y squared times z it shows you what you’ve entered in case you want to change it or you made a mistake that’s good also shows you the cartesian system in case you need to write that down you can okay or change it this will say its okay here’s the formula for the curl the matrix system uh… you’re doing cross product of the functions those little indications before the letters are partial signs and that’s a partial of r with respect to the partial of y minus partial q with prospective partial of z etcetera and here’s the actual calculations and there’s the curl in vector form here’s the curl in the cartesian form next thing it gives you what you want to go conservative or you wanna find the divergence or you want to compute the curl at points or you want to compute the divergence at points or new variables or what pretty neat huh? EveryStepCalculus.com

Video Example: Critical Points for Graphing Program

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Raw Transcript

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

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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=243-666
=-423
Y = 222x+-423

Video Example: Equation of a line program

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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 TI-89 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 sixty-four, 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

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This video is on natural log and the derivatives of that, or integrals depend on the choice, these are my fabulous programs that I’ve programmed under the titanium calculator that you can download from my website and do all these like you were an expert. I’m going to clear the entry line here by pressing clear and then I’m going to clear the screen by pressing F1 and 8, we need to enter index 8 in the home screen and I’m going to show you how to do that. Press second alpha, highlights that you are in the numbers or letter “1index* and then alpha 8 and close parenthesis which tell the calculator it’s a program and out comes my menu. You can see how many things are to do, you can scroll down this or whatever…area of a parallelogram, there is he chain rule, there cross product, all kinds of complicated stuff that you can do easily on this one, definite integral that’s always fun, it’s all alphabetical and then I give you a sheet if you buy the programs where you can see it on an 8 by 10 an outline of everything that’s on the calculator so it’s really easy to learn quickly and go to them. Anyway what are we doing? Oh we’re doing the natural log, there’s a natural log, in another video I did log to base A, they call it base a. So we are going to do the natural log right now and differentiate rather than integrate so I want to do those kind of problems and that’s what we’re into. I gave you the choice when you look at this menu because it’s one of the difficult things in calculus, certain for me was how to choose which problem to do but what function is necessary to do the problem; is it a chain rule problem, is it a product rule, is a quotient rule, is it U-substitution, what is it? Well and you pick out your problem and your test or homework and you say “oh gee, it look kind of like this number 3 here” you know that could be log of 5X cubed plus 6 or whatever in here and if it looks similar to that, we can press the number before that which I’ll do now and it tells you to that’s it’s U-substitution and now we are going to enter the function, so we’re going to press second and then log, here is the natural log function there, we are to press alpha and then we are going to enter our 5xX cubed plus 6 and so we are going to close the small screen so we’ll have to check it you know but it gives you the option of changing it if you did something wrong so you don’t worry about that, for instance I say it ok so we’re going to go on with it and you can see that here is the original function used in the inside your 5xX36 and U prime is which is the derivative of 15X squared. You
write this stuff on your paper to get 100% on this problem. Notice the U-substitution is U prime over U. so here is the U prime and here is the U and dividing it by the other heres 15 x squared plus that. Pretty neat huh. Ok everystepcalculus.com, check it out

Quotient Rule Step by Step

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Raw Transcript

So this is a video on the quotient rule as it applies to my fabulous programs
that I’ve programmed on the Titanium or any calulator that uses the ti system and that can do an integral or derivative where my programs can work uhm, there’s so many now that they’re almost overloading the calculator but uhm I’m going to show you this quotient rule works I’m going to turn it on here and get out of this program here and get to the home screen F one eight clears the screen and then we can clear the home entry line by the clear button here you have to put index eight closed parenthesis on here, we do that by going second, here’s second alpha, it makes this dark, therefore you have the letters you can put in letters i n d e x and then alpha again to put in the numbers- number 8 and the closed parenthesis and we’re into my program you’ll notice that this is, you can scroll up with the scroll keys to go up as far as you want, there’s all kinds of things in here product rule quotient rule, difference quotient, limits, trig integrals and derivatives, not everything in a calculus book is in here, but just what I’ve done on tests, and I wanted to pass test, that’s what I did using the calculator uhm, so I’m going to go down to all alphabetical, so, here’s the product rule I showed you in another video, quadratic formula here’s the quotient rule right there, we’re gonna wait for it to load here, ah there it is the quotient rule and you are taking one function and dividing it by another function.and the formula is: g of x times f prime of x minus f of x time g prime of x all divided by g of x squared, that would be fun to do on a test wouldn’t it, from memory, I don’t think so, but maybe anyways much easier with this let’s put something in here we have to put the parenthesis first then put a divide sign in between it as I show you up here, we have to press alpha, and there’s the, we have to put in x squared plus three x divided by uhm let’s say two times x plus five and we have to press it twice, let’s see if we’ve done that, x squared plus three times x over two x plus five, very good and you always have a chance to change my programs or change what you’ve entered in case you’ve made a mistake, that’s always — and so here we go we got the g of x plus f prime of x and g prime of x times f of x, over g of x squared and we work it down here, we work and keep doing the calculations, which you write on your paper, don’t even think about it just write it. write down the next thing, write sloppy. nerds and mathematicians love a to write sloppy and all over the page, you do the same unless you already do that, that’s fine and we have, ahh the answer is two x squared plus ten x + fifteen, all over two x + five squared all step by step, working it step by step ah pretty neat huh? check it out at my web site every step calculus dot com

Product Rule Step by Step

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Raw Transcript

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 forty-five 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

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Raw Transcript

In this video I’m going to do the chain rule, I’m sure you know how my fabulous program works on the titanium calculator. We have to enter index 8 into the calculators home screen, you can do that by going second alpha, you can see second alpha highlighted here so you can put the letters in “index” and then you have to press alpha 8 to get to numbers in parenthesis and then is my menu that comes up. Menu is quite long all the way down here,all alphabetical and you can choose what you want; product rule, quotient rule, limits whatever you might need, not everything that you need but most of the stuff is on tests, I took these off of my tests and I programmed them. We got to do the chain rule so we can either scroll down to it or you can press the number in front of it, I’m going to press 5 and go to the number and we are going to put two systems U and g of x, I’m going to use the g of x system. And we have to press alpha and put in function, let’s try it 5Xx2+6 close parenthesis to the 8 power, shows you what you entered 5X squared to 6 to the 8th power, find the derivative of that, always give you a chance to change the *2:08* mistake, doesn’t look right or something, now we’re into the problem, here is the fundament for the chain rule *2:16*, you write this on your paper or homework. f[g<x>] is U over 8 and f1[g<x>] is the first derivative of x which is shown here and g(x) is the *2:36* portion 5X2+6 and the fist derivative of that is 10 to the X, and we put it together *2:46*, we have to substitute 5X2 for U, so we do that and here is *3:02* and then we can go back and do another problem or whatever. Pretty neat huh, I’ll tell you everystepcalculus.com and check it out, my website.

Equation of Tangent Line on TI-89

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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=243-666
=-423
Y = 222x+-423

How Professors teach compared to my programs

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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(5x-2) was.  This is a chain rule problem.  How would you like it answered?

www.everystepcalculus.com

Tom’s Programs for the Ti Calculators Answer!

 

Chain Rule:

 

h(x)   = f [g(x)]

h’(x)  = f ‘ [g(x)]*g’(x)

 

= 4cos(5x-2)

= (4)-sin(5x-2)*d/dx(5x-2)

= (4)-sin(5x-2)*(5)

= (5)(4)-sin(5x-2)

= -(20)sin(5x-2)

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 (5x-2).

Let y=5x-2, this gives f(x)= 4 cos y

f’(x)= [d (4 cos y)/dy]*[d(5x-2)/dx]

We also know that the derivative of cos x= -sin x.

=>   [d (4 cos y)/dy]= -4 sin y

[d(5x-2)/dx]= 5

Therefore f’(x)= [d (4 cos y)/dy]*[d(5x-2)/dx]

= (-4 sin y)*5

=-4*sin (5x-2)*5

=-20 sin (5x-2)

Therefore the derivative of 4 cos (5x-2) is -20 sin (5x-2)

 

Calculus

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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.

 

Point Slope Form: The relation to calculus

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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