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Calculus 4E - Ch 1-8
Calculus 4E - Chapters 1 - 8 REVIEW
| Question | Answer |
|---|---|
| THALES of Miletus, the first mathematician, set the agenda for western philosophy with his attempt to EXPLAIN | WHY the world around us changes while maintaining so much order. |
| The Pythagorean motto | "All is number." |
| The underlying theses of the Pythagorean motto (and thus the Scientific Revolution) were that: | 1) The universe is ordered according to perfect mathematical laws. 2) Divine reason is the orderer. 3) Human reason can discern the divine mathematical pattern. |
| Wigner said, "The miracle of the APPROPRIATENESS of the languages of mathematics for the formulation of the laws of | PHYSICS is a wonderful gift which we cannot understand or deserve." |
| Wigner meant that we cannot understand how math | "works." |
| Plato believed that the world was mathematical because | God made it that way. |
| The Western world gets the notion of the LIBERAL ARTS from | Plato's REPUBLIC. |
| Plato's CHALLENGE was | to mathematically describe that apparently chaotic motion of the heavens. |
| Zeno's outrageous suggestion was that __________ simply CANNOT occur. | motion |
| Zeno discussed the endless ability to divide distances in half as his support for the idea that | motion could not occur. |
| Mathematical lines & physical distance in space are different because mathematical lines are ___________________ objects, not _______________ ones, whereas physical distance is physical. | mathematical; physical |
| Number lines unite the realms of ______________ and ____________. | numbers; points |
| The paradox of SPEED is that INSTANTANEOUS speed | does NOT seem to make any sense. NO object moves at all during one instantaneous MOMENT in time. |
| A ZERO in the denominator of a fraction is _______________ or meaningless (forbidden). | undefined |
| Aristotle said that a dropped ball fell because of its ___________________. | nature. |
| Gravity is an external cause while Aristotle's theory regarding falling objects made the cause ___________________. | internal |
| Greek word meaning nature | physis |
| Aristotle said that the "principle of change" is | an object's nature. |
| "Two magnitudes balance at distances RECIPROCALLY proportional to the magnitudes." | Archimedes' Law of the Lever |
| Formula for average velocity that "shows" the paradox of speed | Vave = Δx/Δt = 0/0 |
| Draw a diagram for Archimedes' Law of the Lever | (see pg. 66 of the book) w1/w2=d2/d1 |
| The Scientific Revolution overthrew the scientific regime of ____________________. | Aristotle |
| The Scientific Revolution was characterized by the resurgence of the | Platonic/Pythagorean tradition, which looked on nature in geometric terms, arguing that the cosmos was constructed according to the principles of mathematical order. |
| The 3 main results of the Scientific Revolution: | 1. The overthrow of Aristotle; 2. The reintroduction of the Platonic-Pythagorean project; 3. The mathematization of motion |
| What is a "bonus" result of the Scientific Revolution? | the development of the mechanical philosophy, which views the universe like a machine, as opposed to like a living organism |
| Galileo had a _______________________ or ______________________ view of cosmology. | Copernican; heliocentric |
| The Mathematical Principles of Natural Philosophy, first published in 1687 | Newton's Principia; contained the set of laws that Newton had derived that unified physics |
| the mathematics of change | calculus |
| Calculus was invented to solve 4 problems: | 1.describing continually changing velocity & acceleration; 2. finding the slope of a curve; 3. finding the tangent to a curve; 4. finding the area under a curve |
| To represent a physical property with numbers is to _______________ it. | quantify |
| A GRAPH is a picture of a ______________; it SHOWS how variables behave together. | function |
| a mathematical system that combines ALGEBRAIC formulas with PICTURES in Cartesian coordinates; algebraic representation of shapes & the geometric representation of algebraic formulas | analytic geometry |
| The 3 main concepts of calculus are: | 1) the limit of a function; 2) the derivative of a function; 3) the integral of a function |
| The LIMIT of a function is the value you APPROACH but NEVER ______________. | reach |
| The method of substituting a limit into a function can be MISLEADING because it LOOKS like | we are allowing our input value to EQUAL the forbidden value when we can really only APPROACH the forbidden value. |
| The metaphor of STANDING vs. SEEING helps explain why limits are exact because you do NOT need to | STAND on a gap to know WHERE it is. You can see EXACTLY where it is, even without standing directly on it. |
Created by:
MrsHough
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