Ch 1 1 Preliminaries The Real Numbers Visualized

Ch 1. 1: Preliminaries The Real Numbers Visualized on number line Set notation: A = {x : condition} Example A = {x : 0<x<5, x a whole number} = {1, 2, 3, 4} Reals

Interval Notation Open intervals (a, b) Closed intervals [a, b] Half open intervals (a, b], [a, b) Unbounded intervals; infinity notation Real numbers; interval notation

Proportionality Two quantities x and y are proportional if y = kx for some constant k Ex: The rate of change r of a population is often proportional to the population size p: r = kp

Proportionality Ex: 11(17) Experimental study plots are often squares of length 1 m. If 1 ft corresponds to 0. 305 m, express the area of a 1 m by 1 m plot in square feet Soln: Use proportionality. Let y be measured in feet, x in meters. Then y=kx k = y/x = (1 ft)/ (. 305 m) = 3. 28 Then y = 3. 28 x and (y ft) X (y ft) = (3. 28)(1) X (3. 28)(1) Ans: 3. 28 ft X 3. 28 ft

Lines Recall: x and y are proportional if y = kx for some constant k Suppose the change in y is proportional to the change in x: y 1 – y 0 = m(x 1 – x 0) This is the point-slope formula for a line

Equations of Lines Slope: m = (y 1 – y 0)/ (x 1 – x 0) Point-slope form y – y 0 = m(x – x 0) Slope-intercept form y = mx + b Standard form Ax + By + C = 0 Vertical Lines: x = a Horizontal lines: y = b

Equations of Lines Parallel Lines: m 1 = m 2 Perpendicular Lines: m 1 = -1/m 2

Equations of Lines Average CO 2 levels in atmospheres (Mauna Loa) Use data to find a model for CO 2 level Use the model to predict CO 2 levels in 1987 & 2005

Equations of Lines Example: Find the equation of the line that passes through (1, 2) and (5, -3). [Standard form] What is the slope of the line that is parallel to this line? Perpendicular? Example: Find the equation of the horizontal line that passes through (2, 3) Example: Find the equation of the vertical line that passes through (-4, 1)

Trigonometry: Angles There are two primary measures of angle Degrees: 360 deg in a circle Radians: 2 pi radians in a circle Conversion: y = radians, x = degrees y = kx

Trigonometry: Angles Example: Convert 30 deg into radians Example: Convert 60 deg into radians Example: Convert 45 deg into radians Example: Convert 1 rad into degrees Note: 1 rad is the angle for which the arc length is equal to the radius Graph common angles

Trigonometric Functions See Maple worksheet for more trig info.

Trigonometric Identities Other trig identities can be derived and used in problem solving.

Homework Read Ch 1. 1 10(7 -10, 15, 25 -29)