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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)

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Ch 1.1: Preliminaries

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  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)

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