Why Slopes
and
More Math
Volume 3
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| Vol 2, Three
Skills for Algebra covers many topics in algebra and
logic that students starting calculus should have mastered or will
have to master. Also includes arithmetic review problems to catch
common mistakes. A fourth skill gives a unifying theme
for high school maths. |
Content Guide
Foreword
Chapter Descriptions
1. Introduction
Calculus Appetizer (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
Appendices:
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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Chapter 12.
More On Units and Slopes
First Derivatives
Suppose or let y = f(x). Then the units of the slope
m = f¢(x)
are given by the ratio of the units of y over those of
x. That is,
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units of m = units of f¢(x) = |
units of y
units of x |
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The slope is given by a real number if the units of x and
y are equal. The case where the units of y are
distance and the units of x are time gives units of
velocity in the form
Many different ratios of units are
possible and allowable in the computation of slopes.
Second Derivatives
Now let z = g(x) = f¢(x) be the slope function for
y = f(x). The units of the slope M = g¢(x) = f¢(x)
are given by the ratio of the units of z over those of
x. That is,
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units of z = f¢(x)
units of x
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(units of z)· |
1
units of x
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units of y
units of x |
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1
units of x |
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Therefore the units of
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f¢(x) = |
units of y
[units of x]2 |
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The case where the units of the first quantity y is meters and
the units of the second quantity x is also meters gives units
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units of y
[units of x]2 |
= |
1
meter |
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This case appeared in the
previous chapter.
The case where the units of the first quantity y is
meters and the units of the second quantity x is
seconds gives units
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units ofy
[units of x]2 |
= |
meter
sec2 |
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This last situation occurs in the discussion of
acceleration, that is, the rate of change of speed or velocity
respect to time. In one such discussion below, the
role of the quantity x will be played by a time t and
the role of the quantity y will be played by a distance
d = f(t). Again, various combinations of units can appear.
Units or Dimensions In physics
courses, in place of talking about units of distance, time
and mass, there is talk about dimensions of distance D,
time T and mass M. With this specialized use of the
word dimension, the foregoing notes on units can be
rewritten as follows.
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dimensions ofy
dimensions of x
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dimensions ofy
[dimensions of x]2 |
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Finally in some physics books, the square brackets [y] is
shorthand for the phrase units of y, or more precisely
the dimension of y. (There is a subtle difference between
units and dimensions which will not be discussed here.) The last
statements can be rewritten one more time as
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www.whyslopes.com
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Parents: Help
your Child/Teen Learn covers Speaking
Skills, Reading
& Writing,
Preparing for Science &
Having Patience, etc
Math How-TOs
1. Arithmetic
2. Algebra
3. More
Algebra 4. Geometry
5 More
Geometry 6. Calculus
>> densely written
>> use as skill checklists
Online
Volumes (orders)
1, Elements of
Reason. 1996
1A. Pattern
Based Reason 1995
1B. Math
Curriculum Notes 1996
2. Three
Skills for Algebra 1995
3 .Why.Slopes.&.More.Math.1995
Skill
& Concept
Review or Development
1. Decimal
Arith - Video Based ]
2 Fractions
3. Fractions
with Units
3. Solving
Linear Equations -
making alg easier
4. Formulas
forwards & Backwards - unifying theme for Algebra
5. Proportionality,
Back- & For-wards - theme at work.
6. Logic
- Math Free, good for precision in work & studies
7. Euclidean-Geometry
(leanly)
8. Slopes
and Lines
9. Why
Study Slopes - a context
10. Quadratics
11 Polynomials
12 Factored
Polys - a context
13 Functions
- For-& Back -wards
14 Number
Theory, Richly
15. Exponents,
Radicals & logs.
16 Calculus
- Examples & Advice
17. Real
Analysis
18
Electric
Circuits Etc (So So)
19 Maps,
Similarity & Trig, (alt view)
20 Complex
numbers
21
Logic with Symbols+truth tables
22 Consistent
Story Telling
23. Even
More Logic
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