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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Interest Rates and Units
First Example (Interest Rates Without Units). The amount
A in a simple interest bank account at t years after a
deposit of an amount (principal) P is given by
where r is the annual interest rate. The
number t of years may be a whole number or a whole number
plus a fraction. The interest rate r is given here as a
percentage, e.g. 3.5%. The rule 100% = 1 implies that
1% = 0.01 = [1/100].
Now the slope of the above graph is
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DA
Dt |
= |
A2-A1
t2-t1 |
= |
(P+Prt2)-(P+Prt1)
t2-t1 |
= |
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The units of this slope m = Pr is units A over the
those of the quantity, more precisely the number t.
Recall t is the number of years. Now if the amount
of money A is measured in dollars
then the slope will have units [dollars /1].
To devise a second approach, let
be a measure of time (with units) and
let i = r/
yr = [(r)/yr ]. Then
|
A = P+Prt = P+P |
r
year
|
·(t year) = P+PiT |
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Second Example (Interest Rate With Units).
The amount A in a simple interest bank account at
time T since the deposit of the amount (principal) P is
also given by
where
i is the interest rate per year or per annum. The units of i are
a percentage per year, that is i = [1%/\yr ].
This yields a graph similar, very similar, to the previous one.
Now the slope
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DA
DT |
= |
A2-A1
T2-T1 |
= |
(P+PiT2)-(P+PiT1)
T2-T1 |
= |
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The units of this slope m = Pi is units of money over units of
time. If time is measured in terms in years and money in dollars,
then this slope will have units [\dollars /\year ] or
dollars per year.
Third Example (Interest Rate Without Units). With interest compounded annually, an
initial deposit P grows to amount A = P(1+i)n after n
years where i is the annual or yearly interest rate. The
interest rate i is usually given as a percentage. Compute the
final amount A in the case where an initial deposit of $100.00
compounds at 4% per year, for 3 years.
In the requested computation, i = 4% = 0.04, P = $100.00 and n = 3.
Therefore
A = P(1+i)n = $100.00(1+4%)3 = ¼.
REMARK. Daily,
weekly, monthly and annual interest rates are given by percentages or pure
numbers (the unit free approach). For example, a yearly or annual interest rate
of 5% is given by the number 5% = 5 ×[1/100] = [5/100]. Second, interest rates
per day, week, month or year refer to a percentage over a period of time. With
the latter, for example [5%/year] represents a 5% per year interest.
FOOTNOTE: Some conventions like these are needed for the consistent use of
units in computations. Without any such conventions, the use of units in
financial computations will depart from the practice in technology and
science.
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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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