30 the fundamental theorem of calculus homework

Fundamental Theorems of Calculus

In simple terms these are the fundamental theorems of calculus:

Derivatives and Integrals are the inverse (opposite) of each other.

When we know the indefinite integral:

We can then calculate a definite integral between a and b by the difference between the values of the indefinite integrals at b and a:

Let's explore the details:

First Fundamental Theorem of Calculus

For a continuous function f(x) on an interval [a, b] with the integral:

then the derivative of the integral F(x) gets us the original function f(x) back again:

This means that the derivative of the integral of f with respect to its upper limit is the function f itself.

Example: f(x) = 2x

The integral of 2x is x 2 , and using the second theorem (below):

Taking the derivative:

So the derivative of the integral of 2x got us 2x back again.

Example: Constant speed

A car travels at a constant speed of 50 km per hour for exactly one hour:

Note: "C" is however far the car had traveled already.

Second Fundamental Theorem of Calculus

When we have a continuous function f(x) on an interval [a, b] , and its indefinite integral is F(x), then:

In other words the definite integral of f(x) from a to b equals the difference in the values of F(x) at b and a

This makes calculating a definite integral easy if we can find its indefinite integral.

Imagine we're filling a tank with water, and the rate at which water flows into the tank is given by f(t) , where t is time in minutes.

If F(t) measures the total volume of water in the tank at any time t , then the amount of water added to the tank between times a and b is F(b) - F(a) .

Flood Warning: Edinburg & Brownsville Campuses Closed.

The Counties of Hidalgo and Willacy Counties have declared a flood emergency warning and asked that everyone stay off the roads. Campuses are closed.

The Center of Excellence in STEM Education College of Sciences

• UTRGV Calculus
• Definition of Function
• Domain and Range of a Function
• The Limit of a Function
• Defining the Derivative
• The Derivative Function
• Tangent Lines and Their Slopes
• Derivatives as Rates of Change
• Basic Differentiation Rules
• Derivatives of Trigonometric Functions
• Product Rule and Quotient Rule
• Derivatives of Logarithmic and Exponential Functions
• Logarithmic Differentiation
• Implicit Differentiation
• Related Rates
• Linear Approximations and Differentials
• Maximum and Minimum Values
• Mean Value Theorem
• Derivatives and Shapes of Graphs
• Indeterminate Forms and L'Hopitals Rule
• Optimization Problems
• Antiderivatives and Indefinite Integrals
• Approximating Area
• The Definite Integral
• The Fundamental Theorem of Calculus
• Average Value of a Function
• Net Change Theorem
• The Substitution Rule
• Integration By Parts
• Determining Volume by Slicing/Disk/Washer Method
• Arc Length of a Curve
• Physical Applications

Additional Resources

Openstax online textbook.

• Calculus ll

How to Read a Math Textbook

Additional instructional resources.

• Learning Desmos
• Calculus Applets in Geogebra
• Electronic flashcards for derivatives/integrals
• Flashcards on Trig and Calculus topics

Build Your Knowledge with UTRGV Training Services

UTRGV offers a variety of useful training sessions. Build your skills and knowledge. Register for training today!

Quick Links

• C-STEM Alerts
• Reserve the C-STEM
• Staff Directory
• H-E-B Planetarium
• C-STEM Mobile Lab
• College of Sciences
• College of Engineering & Computer Science
• Student Organizations

The Fundamental Theorem of Calculus- FTC5

• Describe the meaning of the Mean Value Theorem for Integrals.
• State the meaning of the Fundamental Theorem of Calculus, Part 1.
• Use the Fundamental Theorem of Calculus, Part 1, to evaluate derivatives of integrals.
• State the meaning of the Fundamental Theorem of Calculus, Part 2.
• Use the Fundamental Theorem of Calculus, Part 2, to evaluate definite integrals.
• Explain the relationship between differentiation and integration.

We can find the exact value of a definite integral without taking the limit of a Riemann sum or using a familiar area formula by finding the antiderivative of the integrand, and hence applying the Fundamental Theorem of Calculus.

The Fundamental Theorem of Calculus says that if f is a continuous function on [ a , b ] and F is an antiderivative of f , then

Hence, if we can find an antiderivative for the integrand f , evaluating the definite integral comes from simply computing the change in F on [ a , b ] .

See the Desmos Demonstration on Evaluating Definite Integrals.

A slightly different perspective on the FTC allows us to restate it as the Total Change Theorem, which says that

for any continuously differentiable function f . This means that the definite integral of the instantaneous rate of change of a function f on an interval [ a , b ] is equal to the total change in the function f on [ a , b ] .

See the Desmos Demonstration.

Related Videos

Quick review - the fundamental theorem of calculus.

Fundamental Theorem of Calculus with power functions

Fundamental Theorem of Calculus with exponential functions

1201 W University Dr. Edinburg, TX 78539 Mathematics & General Classrooms EMAGC 2.412 (956) 665-STEM (7836)

Monday - Friday 9:00 AM - 5:00 PM CST

Find us on Social Media!

Like. follow. subscribe. add..

Watch CBS News

Teens come up with trigonometry proof for Pythagorean Theorem, a problem that stumped math world for centuries

By Bill Whitaker

Updated on: September 1, 2024 / 7:00 PM EDT / CBS News

This is an updated version of a story first published on May 5, 2024. The original video can be viewed  here . 

For many high school students returning to class this month, it may seem like geometry and trigonometry were created by the Greeks as a form of torture. 

So imagine our amazement when we heard two high school seniors had proved a mathematical puzzle that was thought to be impossible for two thousand years. 

We met Calcea Johnson and Ne'Kiya Jackson, at their all-girls Catholic high school in New Orleans. And, as we first reported this past spring, we expected to find two mathematical prodigies.  

Instead, we found at St. Mary's Academy , all students are told their possibilities are boundless.

Come Mardi Gras season, New Orleans is alive with colorful parades, replete with floats, and beads, and high school marching bands.

In a city where uniqueness is celebrated, St. Mary's stands out – with young African American women playing trombones and tubas, twirling batons and dancing - doing it all, which defines St. Mary's, students told us.

Junior Christina Blazio says the school instills in them they have the ability to accomplish anything. 

Christina Blazio: That is kinda a standard here. So we aim very high - like, our aim is excellence for all students. 

The private Catholic elementary and high school sits behind the Sisters of the Holy Family Convent in New Orleans East. The academy was started by an African American nun for young Black women just after the Civil War. The convent still supports the school with the help of alumni.

In December 2022, seniors Ne'Kiya Jackson and Calcea Johnson were working on a school-wide math contest that came with a cash prize.

Ne'Kiya Jackson: I was motivated because there was a monetary incentive.

Calcea Johnson: 'Cause I was like, "$500 is a lot of money. So I-- I would like to at least try."

Both were staring down the thorny bonus question.

Bill Whitaker: So tell me, what was this bonus question?

Calcea Johnson: It was to create a new proof of the Pythagorean Theorem. And it kind of gave you a few guidelines on how would you start a proof.

The seniors were familiar with the Pythagorean Theorem, a fundamental principle of geometry. You may remember it from high school: a² + b² = c². In plain English, when you know the length of two sides of a right triangle, you can figure out the length of the third.

Both had studied geometry and some trigonometry, and both told us math was not easy. What no one told  them  was there had been more than 300 documented proofs of the Pythagorean Theorem using algebra and geometry, but for 2,000 years a proof using trigonometry was thought to be impossible, … and that was the bonus question facing them.

Bill Whitaker: When you looked at the question did you think, "Boy, this is hard"?

Ne'Kiya Jackson: Yeah. 

Bill Whitaker: What motivated you to say, "Well, I'm going to try this"?

Calcea Johnson: I think I was like, "I started something. I need to finish it." 

Bill Whitaker: So you just kept on going.

Calcea Johnson: Yeah.

For two months that winter, they spent almost all their free time working on the proof.

CeCe Johnson: She was like, "Mom, this is a little bit too much."

CeCe and Cal Johnson are Calcea's parents.

CeCe Johnson:   So then I started looking at what she really was doing. And it was pages and pages and pages of, like, over 20 or 30 pages for this one problem.

Cal Johnson: Yeah, the garbage can was full of papers, which she would, you know, work out the problems and-- if that didn't work she would ball it up, throw it in the trash. 

Bill Whitaker: Did you look at the problem? 

Neliska Jackson is Ne'Kiya's mother.

Neliska Jackson: Personally I did not. 'Cause most of the time I don't understand what she's doing (laughter).

Michelle Blouin Williams: What if we did this, what if I write this? Does this help? ax² plus ….

Their math teacher, Michelle Blouin Williams, initiated the math contest.

Bill Whitaker: And did you think anyone would solve it?

Michelle Blouin Williams: Well, I wasn't necessarily looking for a solve. So, no, I didn't—

Bill Whitaker: What were you looking for?

Michelle Blouin Williams: I was just looking for some ingenuity, you know—

Calcea and Ne'Kiya delivered on that! They tried to explain their groundbreaking work to 60 Minutes. Calcea's proof is appropriately titled the Waffle Cone.

Calcea Johnson: So to start the proof, we start with just a regular right triangle where the angle in the corner is 90°. And the two angles are alpha and beta.

Bill Whitaker: Uh-huh

Calcea Johnson: So then what we do next is we draw a second congruent, which means they're equal in size. But then we start creating similar but smaller right triangles going in a pattern like this. And then it continues for infinity. And eventually it creates this larger waffle cone shape.

Calcea Johnson: Am I going a little too—

Bill Whitaker: You've been beyond me since the beginning. (laughter) 

Bill Whitaker: So how did you figure out the proof?

Ne'Kiya Jackson: Okay. So you have a right triangle, 90° angle, alpha and beta.

Bill Whitaker: Then what did you do?

Ne'Kiya Jackson: Okay, I have a right triangle inside of the circle. And I have a perpendicular bisector at OP to divide the triangle to make that small right triangle. And that's basically what I used for the proof. That's the proof.

Bill Whitaker: That's what I call amazing.

Ne'Kiya Jackson: Well, thank you.

There had been one other documented proof of the theorem using trigonometry by mathematician Jason Zimba in 2009 – one in 2,000 years. Now it seems Ne'Kiya and Calcea have joined perhaps the most exclusive club in mathematics. 

Bill Whitaker: So you both independently came up with proof that only used trigonometry.

Ne'Kiya Jackson: Yes.

Bill Whitaker: So are you math geniuses?

Calcea Johnson: I think that's a stretch. 

Bill Whitaker: If not genius, you're really smart at math.

Ne'Kiya Jackson: Not at all. (laugh) 

To document Calcea and Ne'Kiya's work, math teachers at St. Mary's submitted their proofs to an American Mathematical Society conference in Atlanta in March 2023.

Ne'Kiya Jackson: Well, our teacher approached us and was like, "Hey, you might be able to actually present this," I was like, "Are you joking?" But she wasn't. So we went. I got up there. We presented and it went well, and it blew up.

Bill Whitaker: It blew up.

Calcea Johnson: Yeah. 

Ne'Kiya Jackson: It blew up.

Bill Whitaker: Yeah. What was the blowup like?

Calcea Johnson: Insane, unexpected, crazy, honestly.

It took millenia to prove, but just a minute for word of their accomplishment to go around the world. They got a write-up in South Korea and a shout-out from former first lady Michelle Obama, a commendation from the governor and keys to the city of New Orleans. 

Bill Whitaker: Why do you think so many people found what you did to be so impressive?

Ne'Kiya Jackson: Probably because we're African American, one. And we're also women. So I think-- oh, and our age. Of course our ages probably played a big part.

Bill Whitaker: So you think people were surprised that young African American women, could do such a thing?

Calcea Johnson: Yeah, definitely.

Ne'Kiya Jackson: I'd like to actually be celebrated for what it is. Like, it's a great mathematical achievement.

Achievement, that's a word you hear often around St. Mary's academy. Calcea and Ne'Kiya follow a long line of barrier-breaking graduates. 

The late queen of Creole cooking, Leah Chase , was an alum. so was the first African-American female New Orleans police chief, Michelle Woodfork …

And judge for the Fifth Circuit Court of Appeals, Dana Douglas. Math teacher Michelle Blouin Williams told us Calcea and Ne'Kiya are typical St. Mary's students.  

Bill Whitaker: They're not unicorns.

Michelle Blouin Williams: Oh, no no. If they are unicorns, then every single lady that has matriculated through this school is a beautiful, Black unicorn.

Pamela Rogers: You're good?

Pamela Rogers, St. Mary's president and interim principal, told us the students hear that message from the moment they walk in the door.

Pamela Rogers: We believe all students can succeed, all students can learn. It does not matter the environment that you live in. 

Bill Whitaker: So when word went out that two of your students had solved this almost impossible math problem, were they universally applauded?

Pamela Rogers: In this community, they were greatly applauded. Across the country, there were many naysayers.

Bill Whitaker: What were they saying?

Pamela Rogers: They were saying, "Oh, they could not have done it. African Americans don't have the brains to do it." Of course, we sheltered our girls from that. But we absolutely did not expect it to come in the volume that it came.  

Bill Whitaker: And after such a wonderful achievement.

Pamela Rogers: People-- have a vision of who can be successful. And-- to some people, it is not always an African American female. And to us, it's always an African American female.

Gloria Ladson-Billings: What we know is when teachers lay out some expectations that say, "You can do this," kids will work as hard as they can to do it.

Gloria Ladson-Billings, professor emeritus at the University of Wisconsin, has studied how best to teach African American students. She told us an encouraging teacher can change a life.

Bill Whitaker: And what's the difference, say, between having a teacher like that and a whole school dedicated to the excellence of these students?

Gloria Ladson-Billings: So a whole school is almost like being in Heaven. 

Bill Whitaker: What do you mean by that?

Gloria Ladson-Billings: Many of our young people have their ceilings lowered, that somewhere around fourth or fifth grade, their thoughts are, "I'm not going to be anything special." What I think is probably happening at St. Mary's is young women come in as, perhaps, ninth graders and are told, "Here's what we expect to happen. And here's how we're going to help you get there."

At St. Mary's, half the students get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. Here, there's no test to get in, but expectations are high and rules are strict: no cellphones, modest skirts, hair must be its natural color.

Students Rayah Siddiq, Summer Forde, Carissa Washington, Tatum Williams and Christina Blazio told us they appreciate the rules and rigor.

Rayah Siddiq: Especially the standards that they set for us. They're very high. And I don't think that's ever going to change.

Bill Whitaker: So is there a heart, a philosophy, an essence to St. Mary's?

Summer Forde: The sisterhood—

Carissa Washington: Sisterhood.

Tatum Williams: Sisterhood.

Bill Whitaker: The sisterhood?

Voices: Yes.

Bill Whitaker: And you don't mean the nuns. You mean-- (laughter)

Christina Blazio: I mean, yeah. The community—

Bill Whitaker: So when you're here, there's just no question that you're going to go on to college.

Rayah Siddiq: College is all they talk about. (laughter) 

Pamela Rogers: … and Arizona State University (Cheering)

Principal Rogers announces to her 615 students the colleges where every senior has been accepted.

Bill Whitaker: So for 17 years, you've had a 100% graduation rate—

Pamela Rogers: Yes.

Bill Whitaker: --and a 100% college acceptance rate?

Pamela Rogers: That's correct.

Last year when Ne'Kiya and Calcea graduated, all their classmates went to college and got scholarships. Ne'Kiya got a full ride to the pharmacy school at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University.

Bill Whitaker: So wait a minute. Neither one of you is going to pursue a career in math?

Both: No. (laugh)

Calcea Johnson: I may take up a minor in math. But I don't want that to be my job job.

Ne'Kiya Jackson: Yeah. People might expect too much out of me if (laugh) I become a mathematician. (laugh)

But math is not completely in their rear-view mirrors. This spring they submitted their high school proofs for final peer review and publication … and are still working on further proofs of the Pythagorean Theorem. Since their first two …

Calcea Johnson: We found five. And then we found a general format that could potentially produce at least five additional proofs.

Bill Whitaker: And you're not math geniuses?

Bill Whitaker: I'm not buying it. (laughs)

Produced by Sara Kuzmarov. Associate producer, Mariah B. Campbell. Edited by Daniel J. Glucksman.

Bill Whitaker is an award-winning journalist and 60 Minutes correspondent who has covered major news stories, domestically and across the globe, for more than four decades with CBS News.

More from CBS News

What's the mortgage interest rate forecast for September 2024?

4 Las Vegas teens plead guilty in classmate's deadly beating

11 killed as bus crashes into students, parents outside school in China

Some Social Security recipients won't get September checks. Here's why.