M1410101 - (Section 1.1: Rectangular Coordinates) 1.01...

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(Section 1.1: Rectangular Coordinates) 1.01 CHAPTER 1: FUNCTIONS AND THEIR GRAPHS SECTION 1.1: RECTANGULAR COORDINATES We’re used to drawing two-dimensional graphs in the Cartesian (or rectangular ) plane . Historical Note: The Cartesian plane is named after René Descartes (1596-1650), the great French philosopher and founder of modern analytic geometry. He is known for his famous Latin quote, “Cogito ergo sum (I think, therefore I am).” Descartes devised the Cartesian plane while he was in a hospital bed watching a fly buzzing around a corner of his room. See p.124 . The horizontal x -axis and the vertical y -axis are the coordinate axes , which are real number lines. They intersect at the origin, O . Points in the xy -plane are described by an ordered pair of the form x -coordinate, y -coordinate ( ) . For example, the origin is described by the ordered pair 0,0 ( ) , and the red point below is described by 2,3 ( ) . In some problems, the x -axis is replaced by the t -axis (with t typically representing time) or some other axis, and the y -axis may also be replaced.
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(Section 1.1: Rectangular Coordinates) 1.02 In Chapters 6 and 10 , we will study an alternate two-dimensional coordinate system called the polar coordinate system. In Calculus: In Multivariable Calculus ( Calculus III: Math 252 at Mesa ), you will study three three-dimensional coordinate systems: Cartesian, cylindrical, and spherical. Distance Formula The distance between points x 1 , y 1 ( ) and x 2 , y 2 ( ) in the Cartesian plane is given by: d = x 2 x 1 ( ) 2 + y 2 y 1 ( ) 2 or, equivalently, x 1 x 2 ( ) 2 + y 1 y 2 ( ) 2 This is proven using the Pythagorean Theorem. See p.4 and Notes on Sections 4.2- 4.4, Part A . The midpoint of the line segment in the figure above is given by: x 1 + x 2 2 , y 1 + y 2 2 . Observe that the x -coordinate is the average of the x -coordinates of the endpoints of the line segment, and the y -coordinate is the average of the y -coordinates.
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(Section 1.1: Rectangular Coordinates) 1.03 Think About It: The figure below is a square! Its diagonals are perpendicular! (How?)
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(Section 1.2: Graphs of Equations; Circles and Relatives) 1.04 SECTION 1.2: GRAPHS OF EQUATIONS; CIRCLES AND RELATIVES The Basic Principle of Graphing The graph of an equation consists of all points (corresponding to solutions) whose coordinates satisfy the equation. This principle applies to any coordinate system: Cartesian, polar, cylindrical, spherical, …. We will typically do our graphing in the Cartesian xy -plane. Warning: This may seem like a simple idea, but students often forget to apply it!
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This note was uploaded on 09/08/2011 for the course MATH 141 taught by Professor Staff during the Fall '11 term at Mesa CC.

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M1410101 - (Section 1.1: Rectangular Coordinates) 1.01...

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