Chapter 2. Slopes and Ski Trails
- [Play
Video] 80 seconds: Slope Sign Interpretation for Linear
Functions. (Appeared earlier in Why Slopes Appetizer)
Slopes of Line Segments
A tour of calculus begins. Recall how the slope to a straight line is computed.
The slope m of a straight line segment between two points (x1,y1)
and (x2,y2) may be calculated as follows.

| slope m = |
Dy
Dx |
= |
y2-y1
x2-x1 |
= |
rise
run |
|
|
The point-slope formula for a line
implies
The latter says that the change Dy in y
is proportional to the change in Dx in x.
For a straight line segment, the slope m is a constant of proportionality
between Dy = y-y1
and Dx = x-x1.
Remark. A quantity Q2
is said to be proportional to a quantity Q1 when and
only when there is a constant k such that Q2 = k·Q1.
If a quantity Q2 is proportional to a quantity Q1
then the graph of Q2 versus Q1 is a
straight line through the origin whose slope m = k is the
constant of proportionality.
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Why Slopes
and
More Math
understanding & explaining
Reason and Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7
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Content Guide Foreword Chapter Descriptions 1. Introduction Preview -why slopes in 1983 2. Second Preview Begins 2 Skier in Motion (V) 2 The Skier (V) 2. Position Dependent (V) 3 Slope & Extrema (V) 4 Single Factor Analysis (V) 4 Two Factor (V) 4 More Factors (V) 4 With Divisors (V) 5 Maxima & Minima Tests 6 Jumps & Discontinuities 8 Review (optional) 9 On Calculus Studies 11 Slope of Slope 13 Acceleration 14 Limits & Error Control (V) 14 Limit of a Fn. 14. Limited Error Control 14 Signif. Digits 14 Cauchy Limits 14 Sequence Limits 14 Decimal Arith. 15 What is Slope (V) 15 Slope Calculation (V) 15 Slope, a Limit 15 Tangent Lines 15 Linear Approx., 15 Limits via Algebra (V) 15 Recap. PS.Chain Rule for Polys PS Chain Rule- General (V) - PS More Chain Rule (V) PS - Sign Analysis (V) 16 What is Velocity 17 What is Area 18 Integration 18 Area Calculation 18 Fn DefN, 6 Ways 19 Logs & Powers 19 Natural Log. 19 Exponential Fn. 20 What's Next 21 Add Vectors 22 Complex #'s 23 Complex #'s 23 Trig Identity 23 Proofs of. 24 Complex Logs etc
Units in Calculations:
7 Velocity 7 Varying Velocity Example 7. Velocity Calculation 7 Changing Units 7 Same Velocity Motions 10 Slopes without Units. 10 Units & Slopes 10 Units in Cost vs. Quantity 10 How Units Appear 10 Unit Elimination 10 Partial Elimination 10 Interest & Units 12 More on Units
Enriched material: The Appendices of Volume
3 are located in the Real
Analysis Area.
Pigeon Hole Principle
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side
Theorem
Integration
& Lipschitz
Continuity
These appendices continue the
decimal viewpoint of limits, error
control and continuity begun
in Chapter 14. The One Sided
Range Theorem is a postscript,
not in printed version.
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