For more information, please see full course syllabus of AP Calculus AB

For more information, please see full course syllabus of AP Calculus AB

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### Overview & Slopes of Curves

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Overview & Slopes of Curves 0:21
- Differential and Integral
- Fundamental Theorem of Calculus
- Differentiation or Taking the Derivative
- What Does the Derivative Mean and How do We Find it?
- Example: f'(x)
- Example: f(x) = sin (x)
- General Procedure for Finding the Derivative of f(x)

### AP Calculus AB Online Prep Course

### Transcription: Overview & Slopes of Curves

*Hello, welcome to www.educator.com, welcome to the first lesson of AP Calculus AB.*0000

*I thought what I would do is take about 5 or 10 minutes to give a nice overview of the course as a whole, *0007

*so that we have a sense of where we are going.*0012

*Then, we are going to launch right into the calculus proper.*0014

*Let us get started and welcome.*0016

*Calculus is basically going to be about two things, two processes.*0025

*We are going to spend the first half learning how to do something called differentiating a function.*0030

*We are going to spend the second half learning how to do something called integrating a function.*0033

*That is essentially it.*0037

*Each one of those tools is going to yield different applications.*0039

*That is all we are going to be doing.*0043

*Let me work in blue here.*0046

*Given a function f(x), we will be spending our entire time doing two things to this function.*0051

*The first thing is we are going to be differentiating it.*0087

*It also called taking the derivative.*0099

*Both of those terms, we will use interchangeably.*0106

*The second thing we are going to be doing to this function, *0110

*it is just going to occupy the second half of calculus, is we are going to be integrating it.*0114

*These are the two major rivers of calculus.*0123

*These are the two great rivers of calculus.*0128

*The differential and the integral, two basic set of tools that allow us to solve certain types of problems with the differential calculus,*0139

*certain other types of problems with the integral calculus.*0154

*Essentially, what is going on is this.*0158

*We have some function f(x).*0160

*One of the things that we can do to it is differentiate it.*0163

*When we differentiate a function, we are going to get another type of function.*0166

*We are going to symbolize that with f’(x).*0171

*The other thing that we can do to it is we can integrate this function.*0175

*And then, when we integrate a function, we are going to get yet another type of function*0179

*which we will symbolize normally with F(X).*0183

*This is a new function and we call this function the derivatives.*0187

*That is called the derivative because it is derived from the original function.*0195

*This one is also a new function.*0199

*We call this one the integral.*0206

*That is essentially what calculus comes down to.*0209

*It is finding ways to find derivatives of functions, applying it in certain cases, and then finding integrals of functions.*0212

*Applying the integral in certain cases.*0218

*Let us move on to the next page here.*0228

*These two tools namely differentiation and integration are very powerful.*0229

*It will allow us to solve the most extraordinary problems, that is what beautiful about the calculus.*0256

*So far in your mathematical studies from elementary school, all the way through high school,*0274

*there has been a pretty steady increase, in terms of the level of difficulty, the techniques that you develop.*0278

*Some or more problems that you can solve.*0284

*That is essentially what it is, it is just, you are making the class of problems that you can solve bigger and bigger.*0285

*Now the calculus, it is a huge jump.*0291

*It is not a little jump, it is not a stair step.*0294

*You are going to be introduced to these tools, differentiation and integration.*0296

*They are very powerful tools.*0300

*It is going to take you up here, in terms of mathematical sophistication and the types of problems that you can solve.*0301

*It is really quite extraordinary.*0308

*Everything that we enjoy in the modern world, I do mean everything, *0311

*It is worth writing down.*0331

*I do mean everything.*0332

*Let us capitalize this one.*0339

*I do mean everything is made possible with calculus.*0342

*It really is the single tool, these two things, differentiation and integration.*0358

*They have allowed us to absolutely enjoy everything that we enjoy, computers, cell phones, cars, you name it.*0364

*If there is anything in the modern world that you take for granted, that you enjoy, it is because of the calculus.*0372

*Once again, we are going to spend the first half learning and applying differential calculus.*0380

*And then, we are going to spend the second half, roughly, learning and applying the integral calculus.*0384

*It is going to turn out that these two independent techniques, the differentiation and the integration, *0391

*are in fact so deeply related, that we call this relationship the fundamental theorem of calculus.*0420

*You will often see it as FTC.*0461

*Differentiation and integration, they are reasonably independent techniques.*0464

*There is no reason to believe that one is actually related to the other.*0469

*They solve different set of problems.*0472

*Yet, there is a very deep relationship that exists between them.*0474

*This relationship is what we call the fundamental theorem of calculus.*0477

*What is really interesting about the fundamental theorem of calculus,*0481

*something that many of you who go on to higher mathematics will discover, *0483

*is that this relationship that exists between differentiation integration is actually true, not just in one dimension.*0487

*In other words the real line, which is what we have essentially been dealing with ever since elementary mathematics.*0495

*We stayed on with the real numbers.*0499

*It is true in any number of dimensions, dimensions 2, 3, 4, 5, 6, 15, 34, or 147.*0502

*It is true in any number of dimensions.*0508

*It is a very profound relationship.*0509

*We will actually be getting to that, when we are getting to integration.*0517

*We will study the fundamental theorem of calculus.*0521

*Now the nice thing is, I can already tell you now what this relationship is.*0523

*You have an idea of where is that we are going.*0526

*I can tell you now what this relationship is.*0532

*Later, we will explore this relationship.*0535

*I can tell you now what this relationship is.*0540

*Again, there is nothing here, this is just overview.*0547

*Just giving you an idea of what it is that we are in for, so that you are not going into this blindly.*0549

*So you just have some sense of why we are doing what we are doing.*0554

*I can tell you now what this relationship is.*0557

*The relationship is, each of these processes namely differentiation and integration, *0564

*each of these processes, dif and int is the inverse of the other.*0573

*In other words, if I start with some function f(x), we said that I can differentiate it.*0592

*You know what I’m going to do actually here, I will go ahead and write dif down here.*0619

*I said that I could differentiate it to come up with a new function, the derivative.*0627

*If I want to go back, I integrate this function.*0632

*If I integrate, it will take me back to my original function.*0638

*Or if I integrate, it will give me some other function which we call the integral.*0642

*If I want to go back to the original, I just differentiate.*0648

*That is the relationship between the two.*0653

*Two entirely independent techniques, for the most part, because they solve entirely different set of problems.*0654

*Yet, the relationship is one is the inverse of the other.*0661

*The same way that if you will take a number and if you take the logarithm of that number, you end up with some other number.*0663

*If you want to go back, you exponentiate this thing and it takes you back.*0670

*The logarithm and the exponential are inverse processes.*0675

*The sin and the inverse sin are inverse processes.*0681

*Cos, inverse cos, are inverse processes.*0685

*Differentiation or integration are inverse processes.*0688

*This is extraordinarily deep, extraordinarily beautiful.*0691

*Again, there is no reason to the world to believe that they are connected and yet they are.*0694

*Let us write that last part.*0701

*There is no obvious reason why they should be connected, at least cosmetically.*0704

*There is no reason to just sort of look and say that this is related to that.*0711

*There is no obvious reason why differentiation and integration should be related, but they are.*0715

*This relation has consequences that go further and deeper than you can imagine.*0746

*One of the beautiful things about mathematics is that, you will have different people *0779

*or perhaps the same person will investigate different areas of mathematics, to solve a certain type of problem.*0783

*When you take a look at this set of mathematics that you develop for this problem *0792

*and this set of mathematics that you develop for this problem, *0796

*when you realize that there is actually a connection between those two mathematics, they come together, you unify that.*0799

*That relationship that exists between the various areas of mathematics takes you to a deeper level, *0806

*a deeper understanding of reality, a deeper understanding of how the physical world works.*0813

*This is what we strive for, we strive for unification.*0819

*This is what makes it beautiful, things that should not be related, at least, as far as our intuition is concerned,*0823

*they end up being not only related but very deeply related.*0829

*The consequences of those relationships are profound.*0833

*Anyway, this is really beautiful stuff and it begins right here, with your first course in calculus.*0838

*Once again, welcome, and let us get started.*0845

*We are going to spend the first half talking about differentiation.*0856

*We are going to put integration on the shelf, for the time being.*0859

*We will come back to it later.*0861

*We are going to begin with taking derivatives, differentiation.*0861

*We begin with differentiation.*0865

*Actually, let me write something here.*0880

*Differentiation or also called taking the derivative.*0883

*I will just write it over here, taking the derivative.*0890

*In other words, starting with some f(x), performing the differential operation on it,*0896

*and ending up with a new f’(x), a new functions that is going to give us other information.*0904

*Either by the situation, or it is going to give us information about the original function, whatever it is.*0910

*But it is a new function that we have derived.*0915

*Two questions, the obvious questions.*0920

*Two questions, what does the derivative mean and how do we find it?*0926

*Given this, how do we find the derivative and once we have the derivative, what does the derivative mean?*0933

*What does it give us, what does it tell us, what problems does it solve?*0939

*Two questions, what does the derivative mean?*0943

*Two, how do we find it?*0958

*How do we find it given some f(x)?*0961

*How do we derive and get?*0967

*The answers are as follows.*0971

*The answers, the answer to number 1, what does a derivative mean?*0974

*The derivative is the slope of a function curve.*0978

*A function is just some curve that you draw on the xy plane at a given point on the curve.*0999

*We will explain what that means more, in just a second.*1011

*That is pretty much it.*1015

*A derivative is a slope of a curve at a particular point.*1017

*The strange thing is you have been doing derivatives for many years now.*1024

*If you have the function y = 3x + 4, you know that is the equation of the line.*1027

*The curve itself is a straight line but we call it a curve.*1037

*In general, it is a line in space.*1040

*What is the slope of this line, it is 3.*1043

*The function is 3x + 4, what is the derivative of that function?*1046

*The derivative is 3 because no matter where you are on that line, the slope is 3.*1049

*That is what you have been doing, you have been finding derivatives.*1056

*Now in calculus, we are not just going to find the slopes of straight lines.*1058

*We are going to find the slopes of curves.*1062

*What is that slope, the slope there, slope there.*1066

*The slope is going to change as you move along the curve, that all we are saying.*1069

*We are just giving you a fancy name and calling it the derivative, that is all.*1074

*Number 2, let us go back to blue here.*1080

*The answer to the question, how do you find it?*1083

*Here is how you find, f’(x), in order to find f’(x), here is what you do.*1087

*Limit as h goes to 0, you form f(x) + h, given whatever f is, you subtract from it the original f(x).*1094

*You divide it by h and then you subject it to this process called taking the limit as h goes to 0.*1106

*We will be discussing what this means, how does one do this.*1114

*It is actually quite simple, just algebraically tedious.*1119

*How do we find it? We find it like this.*1124

*What does it mean, it means it is a slope.*1126

*We are going to start with number 1.*1131

*As far as number 2 is concerned, the how, I’m going to leave that for a future lesson.*1133

*For the next couple of lessons, I’m going to be talking about what the derivative is, slope of curves, *1139

*getting ourselves comfortable with the idea of a slope of a curve, as opposed to just the slope of a straight line.*1145

*And then, once we have a reasonably good sense, once we feel comfortable with that, *1150

*we will talk about how to find this so called f’(x), the derivative.*1154

*Let us go over here, I will stick with blue.*1163

*Let us start with number 1, in other words, the meaning, what does it mean?*1170

*The derivative f’(x) is a function which gives us the slope of the curve or graph.*1194

*You know what, maybe I will just call it a graph, which gives us the slope of the graph of the original function at various values of x.*1232

*Here is what this means.*1260

*Let us take a look at let us say the sine functions.*1262

*Our function f(x), original function is sin(x), also y = sin(x).*1271

*We are going to be working in the xy coordinate system.*1280

*We are never going to be moving out of that.*1282

*Whenever you see f(x), you can just replace it with y, it is the same thing.*1284

*We know what the sine function looks like, it looks like this.*1289

*This is 0, this is going to be our π, this is going to be 2 π.*1302

*Over here at π/2, it is going to hit a value of 1.*1306

*Over here at 3π/ 2, it is going to hit a value of -1, standard sine function like that.*1310

*The slope of the curve is, basically, what you are doing is you are finding the line that touches the curve at a given point.*1322

*What we call the tangent line.*1335

*The slope of that line is the slope of the curve, at that particular value of x.*1338

*At π/2, think of it as just some tangent line that is following the curve along.*1343

*You have that slope.*1351

*At this point over here, the slope is that.*1353

*At this point over here, the slope is that.*1356

*At this point over here, the slope is that.*1359

*This point over here, the slope is that.*1362

*Here, the slope is that.*1365

*You can see that the slope changes depending on where you are.*1366

*That is what f’(x) gives.*1372

*F’(x), you have the original function, you do something to it, which we will talk about later, how to find the derivative of it.*1374

*When you find the derivative, it is going to be another function which we symbolize with f’(x).*1382

*The different values of x, it gives us some number, when you actually solve that function.*1386

*That number is the slope of the line that touches the graph, at that particular xy value.*1393

*As we can see for various values of x, the slope at that point, the slope at the point xy,*1401

*if this is x, this is the point xy, it is different.*1439

*We can see it geometrically, if that is the case.*1447

*It is a straight line, for tangent line it is just going to be touching it at a certain point, the slope is going to change.*1449

*Once again, given f(x), we differentiate it and it gives us f’(x).*1457

*This thing, this tells us what the numerical value of the slope is.*1465

*The derivative itself is a function of x, we do not know.*1488

*We have to put in different values of x, to see what the slope is.*1491

*It tells us what the numerical value of the slope is for different values of x.*1494

*Let us go over here.*1522

*Let me draw a little bit of a curve, I will draw it like this.*1524

*I will take a point on that curve.*1529

*Again, we are looking for the slope of the line that just touches that curve at one point.*1531

*This is our f(x).*1539

*This is called our tangent line.*1544

*This point here is going to be xy, or x, if you prefer f(x), however you want to list it.*1552

*Once again, the line that touches a curve at a single point is called the tangent line.*1564

*It is kind of redundant, the tangent line of the curve at that point, I just said that.*1606

*We will just say, it is called the tangent line.*1611

*I think it is perfectly clear what we are talking about.*1613

*It is called the tangent line.*1616

*The slope of the tangent, it is the specific numerical value of the derivative of the function at that point.*1619

*It is the derivative at that point.*1644

*Again, the point itself is a point xy in the plane.*1666

*When we find f’(x), let us say we have some f(x), we find f’(x), that is the derivative, it is the x value of the point.*1674

*It is the x value of the point that we put into f’(x).*1697

*Notice, it is a function of x not a function of y.*1706

*If we want to find the particular numerical value of the derivative, if we have f’(x), it is a function of x.*1709

*We are going to put the x value in there and it is going to spit out some number.*1716

*That number is the slope of that line.*1720

*It is the x value that you put in.*1723

*When we find f’(x), it is the x value of the point that we put into f(x), in order to get a numerical value for the slope.*1724

*Let us see what we have got, let us do some examples here.*1754

*Some examples, I think what we will do is we will let f(x), let us go ahead and take the same function y = sin(x).*1759

*f(x) = sin(x) or y = sin(x).*1772

*I’m going to go ahead and tell you what the derivative is here.*1777

*Again, we will talk about how we got this later on.*1780

*It will turn out that f’(x) actually equals cos(x).*1783

*The original function is sin(x), its derivative is going to turn out to be cos(x).*1798

*When I put in different values of x into cos(x), *1804

*that will give you the numerical value of the slope of the tangent line touching the sine curve, at that point, like this.*1807

*I think I will work in red for this one, that will be nice, a little change of pace here.*1818

*Of course, we have that there.*1828

*Let us go ahead and draw our sine curve again.*1830

*We said we have 0, we have π, we have 2 π.*1834

*This is 1 and this is -1.*1838

*Now let us take the point 0.*1841

*When x = 0, let us find the y value.*1845

*We know y is 0 but let us actually find it.*1849

*f(0) is equal to sin(0) which = 0.*1856

*Yes, our point is going to be 0,0.*1863

*Again, sometimes we do not have a graph to work with, which is the reason I went through it analytically here.*1866

*It is very clear that this is going to be 0,0.*1872

*It is going to be very clear that this point up here is going to be π/2, 1, 3π/ 2, -1, π 0, 2π 0.*1876

*We can see it graphically but we would not always have a graph.*1884

*f’(0) of f’ is cos(x), that is the cos(0).*1889

*What is the cos(0)?*1897

*The cos(0) = 1.*1898

*What that means is that the slope of the tangent line through 0,0 has a slope of 1.*1901

*What is that tangent line?*1926

*Let me extend this out a little further so it goes down that way.*1930

*My tangent line is the line that touches the graph at that point, that is my tangent line.*1936

*The slope of that line is given 1, because the derivative of sin x is cos x.*1944

*Let us find what the slope is at π/2.*1952

*When x is equal to π/2, the f value, the y value, the f(π/2) which is equal to sin(π/2), that is equal to 1.*1959

*Therefore, I know that the point that I’m talking about is π/2 and 1.*1972

*I think I will do a little purple.*1979

*I know my tangent line is there, it is the line that touches the graph at that point.*1985

*That is the tangent line.*1991

*What is the slope of that tangent?*1992

*Graphically, just by looking at it geometrically, I know that the slope is 0.*1994

*We would not always have a graph, let us do it analytically.*1999

*Analytically, I know that the derivative is cos(x).*2002

*f’ at π/2, remember, we put in the x value, is equal to cos(π/2) that is equal to 0.*2008

*As you can see geometrically, analytically here, the slope of the tangent line through π/2, one has a slope of 0.*2021

*Let us do one more.*2050

*Let us take the point 7π/ 6, not quite so easy this time.*2054

*We will take x = 7π/ 6.*2059

*f(7π/ 6), f is sin, sin (7π/ 6), we are just trying to find the y value first, to find out where the point is.*2064

*It is going to be -1/2.*2077

*Our point is 7π/ 6 is our x value, -1/2 is our y value.*2086

*You are looking at it on the graph.*2096

*7π/ 6 is somewhere like right over here.*2097

*This point right here, that point is our point 7π/ 6, -1/2.*2103

*If this is going to be 1, that is probably going to be that way.*2115

*I have not drawn it that great but you get the idea.*2119

*Let us try this again, shall we, all this crazy writing.*2127

*This point over here, on the graph it is 7π/ 6 and -1/2, that is the coordinate of it.*2132

*What about f’, the derivative?*2145

*f’(7π/ 6) = cos(7π/ 6) because the derivative of sin x is cos x.*2148

*The cos(7π/ 6) is –√3/2.*2160

*The slope of the tangent line, that tangent line, *2175

*the slope of the tangent line through the point 7π/ 6, 1/2 is –√3/2.*2181

*Geometrically, we can see that it is going to have to be a negative slope because it is going from top left to bottom right.*2202

*Numerically, analytically, we have to use the formula for the derivative, to find its actual numerical value.*2207

*The slope is a derivative.*2215

*When we say find the derivative of a function, we are saying do whatever you need to do to find the derivative of the function, *2217

*which is going to be another function of x.*2223

*And then, put in the x value of whatever point on the curve you want, that will give you the slope of the tangent line.*2225

*That is the derivative.*2232

*The derivative of 7π/ 6 – 1/2 of sin x is equal to -√3/2.*2234

*The derivative of the function is cos(x).*2241

*The derivative of the function, the numerical value.*2245

*Let us stick with red here.*2255

*It would be very nice to have a general procedure for finding the derivative of f(x).*2258

*We have a general procedure.*2291

*That general procedure says, let me write it a little bit more clearly here.*2293

*And then later on, we will be a little bit more messy.*2302

*The limit as h approaches 0 of f(x) + h - f(x)/ h, this is our general procedure.*2306

*It is a procedure that we are going to address in a later lesson, not right now.*2316

*I’m going to save the procedure for how to find the how, I'm going to save for another lesson.*2322

*For right now, I want to concentrate on the y.*2327

*What does it mean, we want to get a feeling for this.*2331

*I will start discussing this procedure in a future lesson.*2339

*The first thing we are going to do is, when we do this, first, we will discuss what this part means, what that means.*2361

*The second thing we will do, then, we address the whole thing.*2382

*If you saying to yourself, why does he keep writing this thing over and over again?*2397

*There is a reason for it, there is a pedagogical reason for it.*2400

*This is a very important thing.*2403

*I’m writing it over and over again so that by the time you actually do see it, it will be a sort of like you have seen it before.*2406

*That is the reason I'm doing it.*2412

*It is not because I’m obsessive compulsive, over h.*2414

*When we actually discuss this in a future lesson, the how, I’m going to discuss what limits are first, how to find limits.*2419

*And then, we will go ahead and address how to take the limit of this particular quotient, which will give us the derivative.*2426

*For the next few lessons, we will continue with slopes of curves and what derivatives mean.*2436

*Once again, we want to become familiar with this idea of the slope of a curve.*2475

*We want to be able to handle a few things, in basic brute force way.*2481

*We want to know what is going on, how this idea of the slope, *2487

*how we are going to relate it to what we have done with slope before.*2492

*We want to get comfortable with it, before we start actually introducing calculus ideas.*2495

*That is what is going to occupy us, for the next probably three lessons.*2500

*We are going to spend a couple of lessons discussing what these things mean and*2505

*we are going to do a lesson on some example problems.*2508

*We will begin by discussing this idea of a limit of a function.*2511

*What does this limit as h approaches 0 mean.*2516

*With that, I will go ahead and stop this first lesson there.*2519

*Thank you again for joining us, I hope this turns out to be a wonderful experience for you.*2522

*Thank you and see you next time, take care.*2527

1 answer

Last reply by: Professor Hovasapian

Fri Mar 25, 2016 10:44 PM

Post by Eric Liu on March 18 at 02:35:33 PM

Hello Mr. Hovasapian,

I love your lectures, is there a time table for when AP Calc BC will come out?

Thanks!

Eric Liu

2 answers

Last reply by: Professor Hovasapian

Wed Jan 27, 2016 4:05 PM

Post by Harold Snook on January 13 at 05:07:35 PM

Mr. Hovasapian,

The equation which I found on the Internet a couple of weeks ago has disappeared and the paper I wrote it down on, along with symbol designation, has also disappeared. While searching for the old one, I found another,

V=h[r^2 arc cos(r-d/r)-(r-d)sq rt (2dr-d^2). d=depth of fuel, r=radius, h=height of tank when standing on end.

Of course, I am not sure either is accurate.

I am not nearly as concerned about finding a formula that will work as I am about how to come up with an equation for solving the problem. I don't even know how to start.

It has me stumped.

Thanks,

Harold

2 answers

Last reply by: nathan lau

Mon Jan 11, 2016 4:55 PM

Post by nathan lau on January 9 at 06:52:02 PM

and one more question, when you take the derivative of a function, is the answer not directly related to the slope, but rather the x value of the point that we plug into the f'(x) to get the actual slope? That sort of confuses me, because i thought the derivative of a function was just the slope of tangent line at that point. And if the problem you were dealing with was all symbolic, than how would you end up getting a numerical value for the slope by plugging the newly found value for x into f'(x)? For example, if the original function was f(x)=x^3+x^2+2, than f'(x) would be 3x^2+2x+0. Than to find the numerical value for the slope, i believe you would have to plug the derivative back into f'(x), that would make the function needed to find the numerical valve for the slope f'(3x^2+2x+0), but wouldn't that give you the 2nd derivative, or is the 2nd derivative just written that way to make things easier to look at? sorry, i know this is really wordy and i may be looking at things wrong, i just want to get my understanding straight. thanx :)

1 answer

Last reply by: Professor Hovasapian

Mon Jan 11, 2016 1:33 AM

Post by nathan lau on January 9 at 05:36:07 PM

hey, so i have already learned all of semester 1 of AP calculus ab. i have a final on monday, do you know how long this course is up to the second half(integration)? i just want to know if it is even possible to listen to all the lectures in time. i know everything pretty well, but i get allot of anxiety while taking tests, especially because i have some gaps in my knowledge. i just need to stultify all my knowledge and the rules to feel fully confident to do well, thanx! :)

2 answers

Last reply by: Professor Hovasapian

Thu Jan 7, 2016 11:55 PM

Post by Harold Snook on December 31, 2015

Mr. Hovasapian,

I am an "older" student who enjoys learning. I find your lectures very well done, challenging and enjoyable.

This is not a question about this lecture. I write it here because I cannot find a better place to include it on Educator and I was told by Katie that you are very knowledgeable in mathematics.

I am attempting to find how a formula is derived. Recently, I needed a formula to calculate the volume of a cylindrical fuel tank, which is laying on its side, by measuring the height of the fuel when sticking a ruler through the inlet. I found an algebraic formula on the Internet but cannot find how it was derived.

A=pi*a^2/2-a^2*arcsin (1-h/a)-(a-h)*sqrt(h(2a-h)).

Would you be able to show me the derivation? Or is there another equally good formula with an easier derivation? I realize this would probably be lengthy, so if this is not practical, do you know of a web site or book that would give me the derivation?

Thanks,

Harold Snook