Back To Course

Math 104: Calculus16 chapters | 135 lessons | 11 flashcard sets

Are you a student or a teacher?

Try Study.com, risk-free

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Try it risk-freeWhat teachers are saying about Study.com

Already registered? Login here for access

Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Sarah Wright*

The fundamental theorem of calculus is one of the most important equations in math. In this lesson we start to explore what the ubiquitous FTOC means as we careen down the road at 30 mph.

The **fundamental theorem of calculus** says that if *f(x)* is continuous between *a* and *b*, the integral from *x*=*a* to *x*=*b* of *f(x)dx* is equal to *F(b)* - *F(a)*, where the derivative of *F* with respect to *x* is equal to *f(x)*. The big *F* is what's called an anti-derivative of little *f*. This is one of the most key points in all of mathematics, and it's called the fundamental theorem of calculus. But what the heck does 'fundamental' mean?

Let's look at a graph of velocity as a function of time - so this is *f(t)* - between some point in time *a*, and some point in time *b*. Let's say that *f(t)* is constant between *a* and *b*, just to make things simple. I know that, according to the fundamental theorem of calculus, the integral from *a* to *b* of' *f(t)dt* - so that's the area under this curve - is equal to *F(b)* - *F(a)*, where *F`(t)*=*f(t)*. That's the fundamental theorem of calculus. But let's look at it.

Let's say *f(t)*=30 miles per hour (mph). Let's first find an *F* that you can take the derivative of to get 30. So here I want to find what's called an anti-derivative of *f*. Let's say that *F*=30*t*. If I take the derivative of 30*t*, I get *d/dt*(30*t*), which is equal to 30*d/dt*(*t*), which is just 30. So right now, I know that *F`(t)*=*f* and the derivative of 30*t* is equal to 30, so the derivative of this is equal to this.

Okay, so now I've got *f(t)*, which is 30, and my anti-derivative here is 30*t*, so let's plug those in. Let's say I've got my velocity here, 30, and I'm integrating it between *a* and *b*. According to the fundamental theorem, this is equal to *F(b)*, so that's *F*, which is 30*t*, evaluated at *t*=*b*, so that equals 30*b* - *F(a)*, which is 30*a*. So according to the fundamental theorem of calculus, the integral from *a* to *b* of 30*dt*=30*b* - 30*a*. I can simplify this right-hand side to equal 30(*b* - *a*). Well let's take a look at our graph. Our graph is just a straight line, so the integral in this case is just this rectangle here. Well, the area of a rectangle is the height times the width. My height is 30, because *f(t)*=30. My width is *b* - *a*. So sure enough, that's what this right-hand side equals: it equals my height times my width, which is exactly the area. So this kind of works, but what does it mean?

Let's take a look at what this anti-derivative is. This anti-derivative, *F`(t)*, is really saying *dF/dt*. I can write *F`t*=*f(t)* is the same as *dF/dt*=*f(t)*, right? So what's on the left-hand side? Remember that a derivative is just a slope, so this is the slope of the tangent of this function, *F*. On a really, really small scale, I could write this as *delta F*/*delta t*. So this is like finding a slope on a graph. The difference in the height, *F*, divided by the difference in *t*. So if I write *dF/dt* as *delta F*/*delta t* on the left, and I keep my *f(t)* on the right-hand side, well, I could multiply both sides by *delta t*, so that's my change in time here, and I end up with the change in this anti-derivative, *F*=*f(t)delta t*. But wait a second: *f(t)delta t* - that's like a Riemann rectangle! That's just my height times my width, this area right here. If I start adding all of these up, I'm going to end up with an integral for little teeny tiny *delta t*'s. So it kind of makes sense; if I add them all up, I end up with the left-hand side of my fundamental theorem. I end up with the area under the curve - this is a Riemann sum. My *delta F* is my change in *F* - that's all this right-hand side is. It's the change in *F* between one point and another point.

Okay, so seeing it is kind of difficult at first. But what does this really mean, from a practical standpoint? It means that if you have some funky function, and you want to find the area under this curve, all you need to know is a function that this is the derivative of. You just need to find the anti-derivative of this function. Once you do that, finding the area is easy. So no more Riemann sums, no more infinite limits. All you have to do is find an anti-derivative. This - believe me - will make your life very, very happy.

So let's review the **fundamental theorem of calculus**. If some *f(x)* is continuous between point *a* and point *b*, then I can write the integral from *a* to *b* of *f(x)dx* as being equal to the anti-derivative at *b* minus the anti-derivative at *a*, where the anti-derivative is a function such that when you take the derivative of it, you end up with *f(x)* back.

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Login here for access

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
10 in chapter 12 of the course:

Back To Course

Math 104: Calculus16 chapters | 135 lessons | 11 flashcard sets

- Go to Continuity

- Go to Series

- Go to Limits

- Summation Notation and Mathematical Series 6:01
- How to Use Riemann Sums for Functions and Graphs 7:25
- How to Identify and Draw Left, Right and Middle Riemann Sums 11:25
- What is the Trapezoid Rule? 10:19
- How to Find the Limits of Riemann Sums 8:04
- Definite Integrals: Definition 6:49
- How to Use Riemann Sums to Calculate Integrals 7:21
- Linear Properties of Definite Integrals 7:38
- Average Value Theorem 5:17
- The Fundamental Theorem of Calculus 7:52
- How to Find the Arc Length of a Function 7:11
- Go to Area Under the Curve and Integrals

- Computer Science 335: Mobile Forensics
- Electricity, Physics & Engineering Lesson Plans
- Teaching Economics Lesson Plans
- U.S. Politics & Civics Lesson Plans
- US History - Civil War: Lesson Plans & Resources
- HESI Admission Assessment Exam: Factors & Multiples
- HESI Admission Assessment Exam: Probability, Ratios & Proportions
- HESI Admission Assessment Exam: 3D Shapes
- HESI Admission Assessment Exam: Punctuation
- HESI Admission Assessment Exam: Linear Equations, Inequalities & Functions
- CPCE Prep Product Comparison
- CCXP Prep Product Comparison
- CNE Prep Product Comparison
- IAAP CAP Prep Product Comparison
- TACHS Prep Product Comparison
- Top 50 Blended Learning High Schools
- EPPP Prep Product Comparison

- History of Sparta
- Realistic vs Optimistic Thinking
- How Language Reflects Culture & Affects Meaning
- Logical Thinking & Reasoning Questions: Lesson for Kids
- Exceptions to the Octet Rule in Chemistry
- Database Hacking: Attack Types & Defenses
- Pride and Prejudice Discussion Questions
- Quiz & Worksheet - Frontalis Muscle
- Quiz & Worksheet - Dolphin Mating & Reproduction
- Octopus Diet: Quiz & Worksheet for Kids
- Quiz & Worksheet - Fezziwig in A Christmas Carol
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies
- Rubrics | Rubric Definition and Types
- 6th Grade Math Worksheets & Printables

- ASSET Numerical Skills Test: Practice & Study Guide
- Remedial 12th Grade English
- Discover Health Occupations Readiness Test: Practice & Study Guide
- Virginia SOL - World History & Geography to 1500: Test Prep & Practice
- CSET Science Subtest II Physics (220): Test Prep & Study Guide
- US Citizenship - MTEL Political Science/Political Philosophy
- Major Religions - ORELA Middle Grades Social Science
- Quiz & Worksheet - Virtue Ethics & Utilitarianism
- Quiz & Worksheet - Hoover's Rugged Individualism
- Quiz & Worksheet - Criticisms of Humanistic Psychology
- Quiz & Worksheet - General McClellan's Role in the Civil War
- Quiz & Worksheet - Adhesion of Water

- Herbert Hoover vs Al Smith: The Election of 1928
- The Gulf Stream: Properties & Discovery
- How to Choose a Dissertation Topic
- What is the Center for Deployment Psychology?
- Tennessee Science Standards for 3rd Grade
- AP English Book List Example
- Plant Cell Project Ideas
- 8th Grade Colorado Science Standards
- Caps for Sale Lesson Plan
- National Bullying Prevention Month
- Ancient History Documentaries
- Soil Activities for Kids

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject