1. Two Ideas, Vast Implications
Calculus is a subject of enormous importance and historical impact. It provides a dynamic view of the world and is an invaluable tool for measuring change. Calculus is applicable in many situations, from the trajectory of a baseball to changes in the Dow Jones average or elephant populations. Yet, at its core, calculus is the study of two ideas about motion and change.
1. Two Ideas, Vast Implications (info)
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13. Achilles, Tortoises, Limits, and Continuity
The integral's strategy of adding up little pieces solves a variety of problems, such as finding the volume of a pyramid or the total pressure on the face of a dam.
13. Achilles, Tortoises, Limits, and Continuity (info)
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2. Stop Sign Crime—The First Idea of Calculus—The Derivative
The example of a car moving down a straight road is a simple and effective way to study motion. An everyday scenario that involves running a stop sign and the use of a camera illustrates the first fundamental idea of calculus: the derivative.
2. Stop Sign Crime—The First Idea of Calculus—The Derivative (info)
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14. Calculators and Approximations
The Fundamental Theorem links the integral and the derivative. It shortcuts the integral's infinite process of summing and replaces it by a single subtraction.
14. Calculators and Approximations (info)
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3. Another Car, Another Crime—The Second Idea of Calculus—The Integral
You are kidnapped and driven away in a car. You can't see out the window, but you are able to shoot a videotape of the speedometer. The process by which you can use information about speed to compute the exact location of the car at the end of one hour is the second idea of calculus: the integral.
3. Another Car, Another Crime—The Second Idea of Calculus—The Integral (info)
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15. The Best of All Possible Worlds—Optimization
Calculus is useful in many branches of mathematics. The 18th-century French scientist Georges Louis Leclerc Compte de Buffon used calculus and breadsticks to perform an experiment in probability. His experiment showed how random events can ultimately lead to an exact number.
15. The Best of All Possible Worlds—Optimization (info)
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4. The Fundamental Theorem of Calculus
The moving car scenario illustrates the Fundamental Theorem of Calculus. This states that the derivative and the integral are two sides of the same coin. The insight of calculus, the Fundamental Theorem creates a method for finding a value that would otherwise be hard or impossible to get, even with a computer.
4. The Fundamental Theorem of Calculus (info)
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16. Economics and Architecture
Zeno's Arrow Paradox concerns itself with the fact that an arrow traveling to a target must cover half the total distance, then half the remaining distance, etc. How does it ever get there? The concept of limit solves the problem.
16. Economics and Architecture (info)
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5. Visualizing the Derivative—Slopes
Change is so fundamental to our vision of the world that we view it as the driving force in our understanding of physics, biology, economics—virtually anything. Graphs are a way to visualize the derivative's ability to analyze and quantify change.
5. Visualizing the Derivative—Slopes (info)
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17. Galileo, Newton, and Baseball
The real numbers in toto constitute a smooth, seamless continuum. Viewing the world as continuous in time and space allows us to make mathematical models that are helpful and predictive.
17. Galileo, Newton, and Baseball (info)
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6. Derivatives the Easy Way—Symbol Pushing
The derivative lets us understand how a change in one variable affects a dependent quantity. We have studied this relationship with respect to time. But the derivative can be abstracted to many other dependencies, such as that of the area of a circle on the length of its radius, or supply or demand on price.
6. Derivatives the Easy Way—Symbol Pushing (info)
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18. Getting off the Line—Motion in Space
Zeno's Arrow Paradox shows us that an infinite addition problem (1/2 + 1/4 + 1/8 + . . .) can result in a single number: 1. Similarly, it is possible to approximate values such as π or the square root of 2 by adding up the first few hundred terms of infinite sum. Calculators use this method when we push the "sin" or square root keys.
18. Getting off the Line—Motion in Space (info)
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7. Abstracting the Derivative—Circles and Belts
One of the most useful ways to consider derivatives is to view them algebraically. We can find the derivative of a function expressed algebraically by using a mechanical process, bypassing the infinite process of taking derivatives at each point.
7. Abstracting the Derivative—Circles and Belts (info)
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19. Mountain Slopes and Tangent Planes
We have seen how to analyze change and dependency according to one varying quantity. But many processes and things in nature vary according to several features. The steepness of a mountain slope is one example. To describe these real-world situations, we must use planes instead of lines to capture the philosophy of the derivative.
19. Mountain Slopes and Tangent Planes (info)
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8. Circles, Pyramids, Cones, and Spheres
The description of moving objects is one of the most direct applications of calculus. Analyzing the trajectories and speeds of projectiles has an illustrious history. This includes Galileo's famous experiments in Pisa and Newton's theories that allow us to compute the path and speed of projectiles, from baseballs to planets.
8. Circles, Pyramids, Cones, and Spheres (info)
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20. Several Variables—Volumes Galore
After developing the ideas of calculus for cars moving in a straight line, we have gained enough expertise to apply the same reasoning to anything moving in space—from mosquitoes to planets.
20. Several Variables—Volumes Galore (info)
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9. Archimedes and the Tractrix
Optimization problems—for example, maximizing the area that can be enclosed by a certain amount of fencing—often bring students to tears. But they illustrate questions of enormous importance in the real world. The strategy for solving these problems involves an intriguing application of derivatives.
9. Archimedes and the Tractrix (info)
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21. The Fundamental Theorem Extended
Calculus plays a central role in describing much of physics. It is integral to the description of planetary motion, mechanics, fluid dynamics, waves, thermodynamics, electricity, optics, and more. It can describe the physics of sound, but can't explain why we enjoy Bach.
21. The Fundamental Theorem Extended (info)
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10. The Integral and the Fundamental Theorem
Formulas for areas and volumes can be deduced by dividing such objects as cones and spheres into thin pieces. Ancient examples of this method were precursors to the modern idea of the integral.
10. The Integral and the Fundamental Theorem (info)
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22. Fields of Arrows—Differential Equations
Many money matters are prime examples of rates of change. The difference between getting rich and going broke is often determined by our ability to predict future trends. The perspective and methods of calculus are helpful tools in attempts to decide such questions as what production levels of a good will maximize profit.
22. Fields of Arrows—Differential Equations (info)
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11. Abstracting the Integral—Pyramids and Dams
Archimedes devised an ingenious method that foreshadowed the idea of the integral in that it involved slicing a sphere into thin sections. Integrals provide effective techniques for computing volumes of solids and areas of surfaces. The image of an onion is useful in investigating how a solid ball can be viewed as layers of surfaces.
11. Abstracting the Integral—Pyramids and Dams (info)
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23. Owls, Rats, Waves, and Guitars
Whether looking at people or pachyderms, the models for predicting future populations all involve the rates of population change. Calculus is well suited to this task. However, the discrete version of the Verhulst Model is an example of chaotic behavior—an application for which calculus may not be appropriate.
23. Owls, Rats, Waves, and Guitars (info)
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12. Buffon’s Needle or π from Breadsticks
The integral involves breaking intervals of change into small pieces and then adding them up. We use Leibniz's notation for the integral because the long S shape reminds us that the definition of the integral involves sums.
12. Buffon’s Needle or π from Breadsticks (info)
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24. Calculus Everywhere
There are limits to the realms of applicability of calculus, but it would be difficult to exaggerate its importance and influence in our lives. When considered in all of its aspects, calculus truly has been—and will continue to be—one of the most effective and influential strategies for analyzing our world that has ever been devised.
24. Calculus Everywhere (info)
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1. The Joy of Math—The Big Picture
Professor Benjamin introduces the ABCs of math appreciation: The field can be loved for its applications, its beauty and structure, and its certainty. Most of all, mathematics is a source of endless delight through creative play with numbers.
1. The Joy of Math—The Big Picture (info)
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13. The Joy of Trigonometry
Trigonometry deals with the sides and angles of triangles. This lecture defines sine, cosine, and tangent, along with their reciprocals, the cosecant, secant, and cotangent. Extending these definitions to the unit circle allows a handy measure of angle: the radian.
13. The Joy of Trigonometry (info)
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2. The Joy of Numbers
How do you add all the numbers from 1 to 100—instantly? What makes a square number square and a triangular number triangular? Why do the rules of arithmetic really work, and how do you calculate in bases other than 10?
2. The Joy of Numbers (info)
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14. The Joy of the Imaginary Number i
Could the apparently nonsensical number the square root of –1 be of any use? Very much so, as this lecture shows. Such imaginary and complex numbers play an indispensable role in physics and other fields, and are easier to understand than they appear.
14. The Joy of the Imaginary Number i (info)
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3. The Joy of Primes
A number is prime if it is evenly divisible by only itself and one: for example, 2, 3, 5, 7, 11. Professor Benjamin proves that there are an infinite number of primes and shows how they are the building blocks of our number system.
3. The Joy of Primes (info)
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15. The Joy of the Number e
Another indispensable number to learn is e = 2.71828 ... Defined as the base of the natural logarithm, e plays a central role in calculus, and it arises naturally in many spheres of mathematics, including calculations of compound interest.
15. The Joy of the Number e (info)
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4. The Joy of Counting
Combinatorics is the study of counting questions such as: How many outfits are possible if you own 8 shirts, 5 pairs of pants, and 10 ties? A trickier question: How many ways are there to arrange 10 books on a shelf? Combinatorics can also be used to analyze numbering systems, such as ZIP Codes or license plates, as well as games of chance.
4. The Joy of Counting (info)
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16. The Joy of Infinity
What is the meaning of infinity? Are some infinite sets "more" infinite than others? Could there possibly be an infinite number of levels of infinity? This lecture explores some of the strange ideas associated with mathematical infinity.
16. The Joy of Infinity (info)
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5. The Joy of Fibonacci Numbers
The Fibonacci numbers follow the simple pattern 1, 1, 2, 3, 5, 8, etc., in which each number is the sum of the two preceding numbers. Fibonacci numbers have many beautiful and unexpected properties, and show up in nature, art, and poetry.
5. The Joy of Fibonacci Numbers (info)
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17. The Joy of Infinite Series
Starting with the analysis of the proposition 0.999999999 ... = 1, this lecture explores what it means to add up an infinite series of numbers. Some infinite series converge on a definite value, while others grow arbitrarily large.
17. The Joy of Infinite Series (info)
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6. The Joy of Algebra
Arguably the most important area of mathematics, algebra introduces the powerful idea of using an abstract variable to represent an unknown quantity. This lecture demonstrates algebra's golden rule: Do unto one side of an equation as you do unto the other.
6. The Joy of Algebra (info)
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18. The Joy of Differential Calculus
Calculus is the mathematics of change, and answers questions such as: How fast is a function growing? This lecture introduces the concepts of limits and derivatives, which allow the slope of a curve to be measured at any point.
18. The Joy of Differential Calculus (info)
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7. The Joy of Higher Algebra
This lecture shows how to solve quadratic (second-degree) equations from the technique of completing the square and the quadratic formula. The quadratic formula reveals the connection between Fibonacci numbers and the golden ratio.
7. The Joy of Higher Algebra (info)
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19. The Joy of Approximating with Calculus
Exploiting the idea of the derivative, we can approximate just about any function using simple polynomials. This lecture also shows why a formula sometimes known as "God's equation" (involving e, i, p, 1, and 0) is true, and how to calculate square roots in your head.
19. The Joy of Approximating with Calculus (info)
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8. The Joy of Algebra Made Visual
Algebra can be used to solve geometrical problems, such as finding where two lines cross. The technique is useful in real-life problems, for example, in choosing a telephone plan. Graphs help us better understand everything from lines to equations with negative or fractional exponents.
8. The Joy of Algebra Made Visual (info)
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20. The Joy of Integral Calculus
Geometry and trigonometry are used to determine the areas of simple figures such as triangles and circles. But how are more complex shapes measured? Calculus comes to the rescue with a technique called integration, which adds the simple areas of many tiny quantities.
20. The Joy of Integral Calculus (info)
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9. The Joy of 9
Adding the digits of a multiple of 9 always gives a multiple of 9. For example: 9 x 4 = 36, and 3 + 6 = 9. In modular arithmetic, this property allows checking answers by "casting out nines." A related trick: mentally computing the day of the week for any date in history.
9. The Joy of 9 (info)
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21. The Joy of Pascal's Triangle
A geometric arrangement of binomial coefficients called Pascal's triangle is a treasure trove of beautiful number patterns. It even provides an answer to the song "The Twelve Days of Christmas": Exactly how many gifts did my true love give to me?
21. The Joy of Pascal's Triangle (info)
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10. The Joy of Proofs
Professor Benjamin begins his discussion of mathematical proofs with intuitive cases like "even plus even is even" and "odd times odd is odd." He builds to more complex proofs by existence and induction, and ends with a checkerboard challenge.
10. The Joy of Proofs (info)
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22. The Joy of Probability
Mathematics can draw detailed inferences about random events. This lecture covers major concepts in probability, such as the law of large numbers, the central limit theorem, and how to measure variance.
22. The Joy of Probability (info)
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11. The Joy of Geometry
Geometry is based on a handful of definitions and axioms involving points, lines, and angles. These lead to important conclusions about the properties of polygons. This lecture uses geometric reasoning to derive the Pythagorean theorem and other interesting results.
11. The Joy of Geometry (info)
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23. The Joy of Mathematical Games
This lecture applies the law of total probability and other concepts from the course to predict the long-term losses to be expected from playing games such as roulette and craps and understand what is known as the "Gambler's Ruin Problem."
23. The Joy of Mathematical Games (info)
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12. The Joy of Pi
Pi is the ratio of the circumference of a circle to its diameter. It starts 3.14 and continues in an infinite nonrepeating sequence. Professor Benjamin shows how to learn the first hundred digits of this celebrated number, making it look as easy as pie.
12. The Joy of Pi (info)
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24. The Joy of Mathematical Magic
Closing the course with a magician's flair, Professor Benjamin shows a trick for producing anyone's phone number, how to create a magic square based on your birthday, how to play "mathematical survivor," a technique for computing cube roots in your head, and a card trick to ponder.
24. The Joy of Mathematical Magic (info)
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