1: Analytic Geometry
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- Feb 8, 2025
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- 1.1: Prelude to Analytic Geometry
- In the (x,y) coordinate system we normally write the x -axis horizontally, with positive numbers to the right of the origin, and the y -axis vertically, with positive numbers above the origin. That is, unless stated otherwise, we take "rightward'' to be the positive xx -direction and "upward'' to be the positive y -direction. Often letters other than x and y are used, and often different scales are chosen in the horizontal and vertical directions.
- 1.2: Lines
- If we have two points A(x1,y1) and B(x2,y2), then we can draw one and only one line through both points. By the slope of this line we mean the ratio of Δy to Δx . The slope is often denoted mm : m=Δy/Δx=(y2−y1)/(x2−x1).
- 1.3: Distance Between Two Points; Circles
- The actual (positive) distance from one point to the other is the length of the hypotenuse of a right triangle with legs |Δx| and |Δy|. The Pythagorean theorem then says that the distance between the two points is the square root of the sum of the squares of the horizontal and vertical sides.
- 1.4: Functions
- A function y=f(x) is a rule for determining y when we're given a value of x. Functions can be defined in various ways: by an algebraic formula or several algebraic formulas, by a graph, or by an experimentally determined table of values. (In the latter case, the table gives a bunch of points in the plane, which we might then interpolate with a smooth curve, if that makes sense.)
- 1.5: Shifts and Dilations
- Many functions in applications are built up from simple functions by inserting constants in various places. It is important to understand the effect such constants have on the appearance of the graph.
- 1.E: Analytic Geometry (Exercises)
- These are homework exercises to accompany David Guichard's "General Calculus" Textmap.
