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

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

### The Fundamental Theorem of Calculus

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
- The Fundamental Theorem of Calculus 0:17
- Evaluating an Integral
- Lim as x → ∞
- Taking the Derivative
- Differentiation & Integration are Inverse Processes
- 1st Fundamental Theorem of Calculus 20:08
- 1st Fundamental Theorem of Calculus
- 2nd Fundamental Theorem of Calculus 22:30
- 2nd Fundamental Theorem of Calculus

### AP Calculus AB Online Prep Course

### Transcription: The Fundamental Theorem of Calculus

*Hello, welcome back to www.educator.com, and welcome back to AP Calculus.*0000

*Today, we are going to talk about the fundamental theorem of calculus, profoundly important.*0004

*This central theorem of calculus, probably one of the most important theorems in all human intellectual history.*0008

*Absolutely fantastic, let us jump right on in.*0015

*Let us start by evaluating an integral.*0020

*Let me go to black here.*0024

*Let us start by evaluating the integral from a, some constant, to x, which I'm going to leave variable.*0030

*The upper limit can be anything of the function t dt.*0046

*Here is what we are doing.*0058

*Let me draw this out real quickly.*0060

*This function f(t) is t, that is just this function right here, that is just a line.*0068

*We are saying pick some a, whatever that is, it does not matter where it is, here or there.*0088

*We are saying integrate to some point x.*0093

*We are leaving x open ended.*0101

*This can be, let us say 1 to 5, 6, 7, 8, we are just leaving it open ended.*0104

*t is the variable of integration here.*0111

*t is the variable of integration, in other words, it is the independent variable of this function.*0115

*This is the t axis, this is the f(t) axis.*0121

*It is the variable of integration, in other words, the independent variable of the function.*0126

*We simply leave x unspecified.*0152

*In other words, it can be anything.*0166

*Again, oftentimes, especially when you are just learning something, it is best to work formally.*0170

*In other words, do not worry if you have not wrapped your mind around what everything means, just work symbolically.*0178

*x does not mean anything, the interval symbolism does not mean anything.*0185

*a does not mean anything, t does not mean anything, d does not mean anything.*0188

*But you know how to manipulate these symbols, that is what is important.*0191

*Let us go ahead and evaluate this integral.*0195

*We know that the integral from a to x of f(t) dt is the limit as n goes to infinity of the sum *0199

*as i goes from 1 to n of f(t) sub i × Δt.*0227

*We know that, we are going to do the same thing.*0235

*This thing, our f(t) is t.*0238

*We are going to find Δt, we are going to find f(Δt).*0245

*We are going to multiply them, we are going to evaluate the sum.*0249

*And then, we are going to take the limit and that is going to give us the integral.*0252

*Let us start off with Δt here.*0258

*Δt, that is equal to x - a/ n.*0265

*I know that the t sub 0 is equal to a.*0276

*I know that t sub 1 is equal to a + Δx which is x - a/ n.*0280

*I know that t sub 2 = a + x – a/ Δn + Δx which is another x – a Δn.*0290

*It is going to be +2 × x - a/ n.*0298

*Therefore, I know that t sub i is going to equal a + i × x - a/ n.*0305

*I found my t sub i, I have my Δt.*0315

*I’m going to plug my t sub i a + i × x - a/ n into my f which is that.*0321

*Here, the f(t sub i) is equal to t sub i, because f(t) is equal to t.*0331

*Therefore, f(t sub i) × Δt is equal to a + i × x - a/ n × x - a/ n.*0343

*This is my f(t sub i), this is my t sub i.*0367

*When I do all of my multiplication, I get myself a × x - a/ n + i × x - a/ n × x - a/ n.*0371

*It is going to equal ax/ n - a²/ n + i × x² - 2xa + a²/ n².*0390

*It is going to equal ax/ n - a²/ n + ix²/ n² - 2i xa/ n² + i a²/ n².*0413

*This is my f(t) i Δt.*0439

*I’m going to evaluate the sum.*0454

*I’m going to evaluate this sum.*0458

*It is going to equal the sum of this, the summation symbol distributes over each of these.*0462

*I have 1, 2, 3, 4, 5 summations, equals the sum as i goes from 1 to n ax/ n - the sum as i goes from 1 to n of a²/ n + *0468

*the sum as i goes from 1 to n of ix²/ n² - the sum as i goes from 1 to n of 2ix a/ n² +*0490

*the sum as i goes from 1 to n of i a²/ n².*0512

*We pull out the things that we can pull out.*0521

*I’m going to get ax/ n × the sum, I’m just going to leave off the i = 1 to n of 1.*0524

*We know that we are going from 1 to n - a²/ n × the sum of 1 + x² n².*0535

*I have to leave the i underneath, + x²/ n² × the sum of i - 2x a/ n²*0547

*× the sum of i + a²/ n² × the sum of i.*0564

*= ax/ n × n - a²/ n × n + x²/ n² × n × n + 1/ 2.*0582

*Because the summation from 1 to n of i is equal to this closed form expression, -2xa/ n × n × n + 1/ 2.*0604

*I’m just going to write it down here.*0619

*+ a²/ n² × n × n + 1/ 2 = ax - a² + x²/ n² +,*0620

*What is going on here, + a² + x²/ 2.*0662

*I’m going to multiply all this out, multiply all of these out.*0676

*+ x²/ 2n -, this is n².*0680

*I have to make sure that I got everything right here.*0699

*n, n, n², n², n², -x/ a - xa/ n + a²/ 2 + a²/ 2n.*0701

*We have our final summation that we have evaluated.*0736

*Now we take the limit as n goes to infinity of this expression.*0740

*I’m just going to go ahead and do it as is, instead of rewriting it.*0752

*First of all, let us go ahead cancel a few things.*0757

*ax and xa cancel, as n goes to infinity, this term goes to 0.*0761

*This term goes to 0, this term goes to 0.*0770

*I'm left with -a² + x²/ 2 + a²/ 2.*0774

*This and this, I'm left with x²/ 2 - a²/ 2.*0790

*There we go, the integral from a to x of t dt, I picked a specific function t dt is equal to x²/ 2 - a²/ 2.*0798

*No matter what x happens to be.*0817

*Notice, this integral gave us a function of x because we said x can be anything, the upper limit.*0821

*What I end up with is sum function of x.*0836

*Now I do this, now take the derivative of the thing that you just got.*0845

*Now take ddx of your x²/ 2 - a²/ 2.*0852

*The derivative of x²/ 2 is x and the derivative of this is 0.*0862

*There you go, here is what we did.*0870

*We have a function t, we integrate it to get this as a function of x.*0874

*We differentiated that and we ended up actually getting our x.*0880

*In some sense, the t and the x are the same.*0885

*You will see that in just a second.*0889

*I just wanted to throw that out there.*0891

*We found the integral, I just happen to take the derivative.*0893

*I will tell you why in just a second.*0897

*What we have is this, what we have is this, ddx of the integral from a to x of t dt is equal to x.*0903

*In general, the derivative with respect to x of the definite integral from a to x *0926

*of whatever function happens to be f(t) dt will always give you f(x).*0940

*In other words, this t is just a variable of integration.*0948

*If x is the upper limit of that integral, if I integrate the function *0954

*and then take the derivative of what I integrated, I end up just getting my x f(x) back.*0960

*In other words, this ddx operator and this integral operator, they cancel each other out leaving you just f(t), *0966

*but this upper limit goes into that t leaving you a function of x.*0974

*Differentiation and integration are inverse processes.*0979

*The logarithm of the exponential cancel, they cancel each other out.*0986

*The exponential of the logarithm, they cancel each other out.*0991

*The derivative of the integral, they cancel, leaving you just f(t).*0995

*The integral of the derivative, they cancel, leaving you just f(t).*0999

*They are inverse processes.*1004

*If the upper limit happens to be just the variable x, what you end up with is just your function of x.*1005

*In other words, this symbol and this integral symbol go away, all you have to do is put this into here.*1011

*Whatever f happens to be, just write it as a function of x, that is what is actually happening.*1017

*Now if I define a function as a definite integral with the upper limit being x,*1027

*if I define a function by means of an integral which I can do, it is not a problem.*1033

*By integral such as, if I set the g(x) is equal to the integral from a to x of f(t) dt,*1048

*it is a strange way of actually defining the function.*1063

*Not strange in the sense that you have never seen it before.*1065

*But all this is saying that, if I’m given some random function f(t), if I actually form the integral of it,*1068

*leaving the upper limit of integration as a variable x, what I end up with is a function of x g(x).*1075

*I'm saying that g(x) is this.*1082

*Instead of giving you the f(x) explicitly, I'm giving it to you in terms of another function f(t).*1085

*I'm saying f take f(t), integrate that function, evaluate that sum, leave x variable that gives me a function of x.*1092

*As you just saw, the integral of t dt was x²/ 2 – a²/ 2, that is a function of x.*1100

*It is strange in a sense that you have never seen it, but it is perfectly valid to define a function this way.*1109

*If I define a function by means of an integral such as this, if you ever see this,*1115

*then, g’(x), the derivative of this g is nothing more than f(x).*1124

*In other words, just get rid off the symbol and just put x in for t.*1132

*Differentiation and integration are inverse processes, very profound, very deep.*1142

*There is no reason in the world to believe that this should be true.*1159

*The two branches of calculus actually developed independently.*1163

*Without any sort of relationship between the two, we ended up discovering that there is a relationship between the two.*1169

*That is a really profound moment in intellectual history.*1176

*Two people working on separate things, or the same person working on two separate things, and somehow they come together.*1183

*The relationship is, these things are inverse processes of each other.*1188

*If I have a function, I could integrate it.*1192

*If I differentiate that, I get my function back.*1194

*If I have a function, I can differentiate it.*1198

*If I integrate what I get, I get my function back, that is amazing.*1200

*Let us go ahead and write the first fundamental theorem of calculus.*1206

*The first fundamental theorem of calculus, abbreviated as FTC.*1213

*If f(x) is continuous, if f is continuous on the closed interval ab, *1232

*and we define g as g(x) equals the definite integral from a to x of f(t) dt,*1253

*then g(x) itself, 1, is continuous, 2, it is continuous on ab.*1273

*It is differentiable on ab, and 3, g’(x) is nothing more than f(x) in essence.*1294

*Start with f, integrate to get some other function.*1318

*Take that function, if you differentiate, it brings you back to f, very simple.*1333

*The second fundamental theorem of calculus which will allow us to evaluate definite integrals directly.*1343

*It is actually one theorem, you can call it first part, second part.*1357

*I just tend to think of it as first fundamental theorem, second fundamental theorem.*1359

*They are the same thing really.*1364

*Second fundamental theorem of calculus.*1365

*If f(x) is continuous on the closed interval ab, then the integral from a to b of f(x) dx *1375

*= f(b) – f(a), where f is any antiderivative of f.*1396

*F’ is equal to f.*1423

*If I have any integral from a to b of f(x) dx, I find the antiderivative.*1426

*The antiderivative, I stick in b, and then I stick in a, and I subtract the f(a) from the f(b).*1434

*That actually gives me the value of that, that is the second fundamental theorem of calculus.*1443

*Thank you so much for joining us here at www.educator.com.*1450

*We will see you next time for example problems on the fundamental theorem of calculus.*1452

*Take care, bye.*1457

2 answers

Last reply by: Peter Ke

Sat Jul 23, 2016 10:58 AM

Post by Peter Ke on July 17, 2016

At 14:28 shouldn't the derivative of this ---> http://prntscr.com/bu4942

is x - a? How is the derivative of a^2 / 2 be 0? I thought it was just "a" because the 2 cancels out and the exponent is just a 1.