1 / 14

Ch 1.1: Preliminaries

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)

bbingham
Download Presentation

Ch 1.1: Preliminaries

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. 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

  2. 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

  3. 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

  4. 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 = k x • k = y/x = (1 ft)/ (.305 m) = 3.28 • Then y = 3.28x and • (y ft) X (y ft) = (3.28)(1) X (3.28)(1) • Ans: 3.28 ft X 3.28 ft

  5. 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: • y1 – y0 = m(x1 – x0) • This is the point-slope formula for a line

  6. Equations of Lines • Slope: m = (y1 – y0)/ (x1 – x0) • Point-slope form • y – y0 = m(x – x0) • Slope-intercept form • y = mx + b • Standard form • Ax + By + C = 0 • Vertical Lines: x = a • Horizontal lines: y = b

  7. Equations of Lines • Parallel Lines: m1 = m2 • Perpendicular Lines: m1 = -1/m2

  8. Equations of Lines • Average CO2 levels in atmospheres (Mauna Loa) • Use data to find a model for CO2 level Use the model to predict CO2 levels in 1987 & 2005

  9. 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)

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

  11. 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

  12. Trigonometric Functions See Maple worksheet for more trig info.

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

  14. Homework • Read Ch 1.1 • 10(7-10,15,25-29)

More Related