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# آموزش مفاهیم کلیدی حسابداری

**Publisher:**TTC

Calculus has had a notorious reputation for being difficult to understand, but the 24 lectures of Change and Motion: Calculus Made Clear are crafted to make the key concepts and triumphs of this field accessible to non-mathematicians. This course teaches you how to grasp the power and beauty of calculus without the technical background traditionally required in calculus courses. Follow award-winning Professor Michael Starbird as he takes you through derivatives and integrals-the two concepts that serve as the foundation for all of calculus. As you investigate the field's intellectual development, your appreciation of its inner workings and your skill in seeing how it can solve a variety of problems will deepen.

24 lectures

31 minutes each

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.x

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.x

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.x

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.x

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.x

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.x

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.x

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.x

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.x

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.x

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.x

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.x

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.x

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.x

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.x

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.x

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.x

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.x

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.x

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.x

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.x

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.x

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.x

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.x

One of the greatest achievements of the human mind is calculus. It justly deserves a place in the pantheon of our accomplishments with Shakespeare's plays, Beethoven's symphonies, and Einstein's theory of relativity. In fact, most of the differences in the way we experience life now and the way we experienced it at the beginning of the 17th century emerged because of technical advances that rely on calculus. Calculus is a beautiful idea exposing the rational workings of the world; it is part of our intellectual heritage. The True Genius of Calculus Is Simple Calculus, separately invented by Newton and Leibniz, is one of the most fruitful strategies for analyzing our world ever devised. Calculus has made it possible to build bridges that span miles of river, travel to the moon, and predict patterns of population change. Yet for all its computational power, calculus is the exploration of just two ideas—the derivative and the integral—both of which arise from a commonsense analysis of motion. All a 1,300-page calculus textbook holds, Professor Michael Starbird asserts, are those two basic ideas and 1,298 pages of examples, variations, and applications. Many of us exclude ourselves from the profound insights of calculus because we didn't continue in mathematics. This great achievement remains a closed door. But Professor Starbird can open that door and make calculus accessible to all. Why You Didn't Get It the First Time Professor Starbird is committed to correcting the bewildering way that the beauty of calculus was hidden from many of us in school. He firmly believes that calculus does not require a complicated vocabulary or notation to understand it. Indeed, the purpose of these lectures is to explain clearly the concepts of calculus and to help you see that "calculus is a crowning intellectual achievement of humanity that all intelligent people can appreciate, enjoy, and understand." He adds: "The deep concepts of calculus can be understood without the technical background traditionally required in calculus courses. Indeed, frequently the technicalities in calculus courses completely submerge the striking, salient insights that compose the true significance of the subject. "In this course, the concepts and insights at the heart of calculus take center stage. The central ideas are absolutely meaningful and understandable to all intelligent people—regardless of the level or age of their previous mathematical experience. Historical events and everyday action form the foundation for this excursion through calculus." Two Simple Ideas After the introduction, the course begins with a discussion of a car driving down a road. As Professor Starbird discusses speed and position, the two foundational concepts of calculus arise naturally, and their relationship to each other becomes clear and convincing. Professor Starbird presents and explores the fundamental ideas, then shows how they can be understood and applied in many settings. Expanding the Insight Calculus originated in our desire to understand motion, which is change in position over time. Professor Starbird then explains how calculus has created powerful insight into everything that changes over time. Thus, the fundamental insight of calculus unites the way we see economics, astronomy, population growth, engineering, and even baseball. Calculus is the mathematical structure that lies at the core of a world of seemingly unrelated issues. As you follow the intellectual development of calculus, your appreciation of its inner workings will deepen, and your skill in seeing how calculus can solve problems will increase. You will examine the relationships between algebra, geometry, trigonometry, and calculus. You will graduate from considering the linear motion of a car on a straight road to motion on a two-dimensional plane or even the motion of a flying object in three-dimensional space. Designed for Nonmathematicians Every step is in English rather than "mathese." Formulas are important, certainly, but the course takes the approach that every equation is in fact also a sentence that can be understood, and solved, in English. This course is crafted to make the key concepts and triumphs of calculus accessible to nonmathematicians. It requires only a basic acquaintance with beginning high-school level algebra and geometry. This series is not designed as a college calculus course; rather, it will help you see calculus around you in the everyday world.

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