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Chapter 16
Velocity Approximation
Again, saying precisely how to compute a quantity
defines it.
This chapter explains the approximation and then the
computational definition of speed and velocity at an
instant of time.
Distance Versus Time
The graph of Harry Snail's position on a
straight (or curved) road, his
distance to the origin versus time, follows. In the
following diagram a skier is drawn, using some poetic license.
The ski midpoint is assumed to lie on the tangent to the
curve. The slope of the line segment (chord) joining
(t1,d1) to (t2,d2) is assumed or expected to
approach the slope of the tangent line for times t2
close to time t1.
· The average slope mavg between times t1 and
t2 is given and defined by the slope of the segment
joining (t1,d1) to (t2,d2). In particular,
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mavg = |
d2-d1
t2-t1
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= |
f(t2)-f(t1)
t2-t1
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When t1 £ t £ t2, the slope of the above position
versus time curve at the point (t,f(t)) may be
approximated by mavg.
· For times t in between t1 and t2, the line
segment joining (t1,d1) to (t2,d2) provides an
approximation to the graph of the function. That is, in or near the
interval t1 to t2, his position
where the symbol » means approximately equal.
The error in this approximation depends on the
behavior of f(t), and once f(t) is given, also on the
values of t1 and t2, or the distance between them.
For some functions, the approximation error may become
smaller if the interval t2 to t1 is made smaller.
The error in this approximation is zero, that is, it
vanishes, at the two times t = t2 and t = t1.
· The average speed or velocity of travel between t1 and t2
is given (defined by)
The velocity at the time or instant t1 is obtained if
the second time t2 in this calculation is allowed to
approach t1. The concept of instantaneous velocity will be
treated next. A definition will make the foregoing idea
more precise.
What is Velocity or Speed?
In the following graph d2 = f(t2). In it, t2 = t1+Dt with t1 fixed,
that is non-moving, while the difference Dt = t2-t1 -> 0. The
arrow -> , as before, indicates goes to or approaches.
For smaller and smaller values of Dt, the slope of
the line segment through (t1,d1) and (t2,d2) should approach that of a
tangent line touching the curve at the point (t1,d1) = (t1,f(t1)).
This expectation provides the motivation for the following
definition of the slope m to the curve at the fixed, non-moving
point (t1,d1) = (t1,f(t1)).
The slope m = mtangent of the tangent line
through (t1,d1) = (t1,f(t1)) is defined by
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mtangent = f¢(t1) = |
lim
Dt -> 0
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f(t2)-f(t1)
t2-t1
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= |
lim
Dt -> 0
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Dd
Dt
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The units of mtangent here are those
of [distance/time], that
is, distance over time. The slope mtangent gives or
should give the
limiting value of average velocity over the time interval
t1 and t2 = t1+Dt as Dt goes closer and
closer to zero. This physical interpretation provides
motivation for the following definition.
Definition: [Velocity at an Instant] The velocity v at the
instant or time t1
is given by
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v = mtangent = f¢(t1) = |
lim
Dt -> 0
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f(t2)-f(t1)
t2-t1
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= |
lim
Dt -> 0
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Dd
Dt
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and the speed at instant t1 is given by s = |v|.
Instantaneous velocity v is another name for the
velocity v = mtangent = f¢(t) at an instant t1.
Likewise, instantaneous speed refers to the
velocity magnitude or absolute value s = |v| at an instant
t1.
Linear or Tangent Line Approximation
For times t near t1, the distance
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d = f(t) » d1+mtangent·(t-t1) = d1+v·(t-t1) |
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The error |f(t) -[d1+mtangent·(t-t1)]| in this linear approximation vanishes when t = t1. This
approximation is best if the velocity between t
and t1 is almost constant. Otherwise, the approximation
error could be large.
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Why Slopes
and
More Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7
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Read slowly, Volumes 2 & 3 may ease or avoid
calculus difficulties. Take the risk.
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Chapters and Appendices
Content Guide
Foreword
2nd Content Guide
1. Introduction
Geometric Calculus Preview (1983)
2. Algebraic 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.
What is a Variable?
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent
or Independent
Variable, a Matter of Choice
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For Parents & Teachers: Speaking
Skills, Reading
& Writing,
Preparing for Science, ends,
values and methods for work and study, parent- friendly mathematics booklets for ages 4-14.
-
Math
Education
Essays (opinions,
possibilities, references)
- POMME, a two
level program for future skill development in
schools and colleges worldwide. Address content &
motivation gaps with ends, values & methods for skill
development to say which way to go, how and why. -
Present Day Curriculum:
(A) Secondary
I Mathematics
consolidate fractions and measurement, skills and
sense consolidation,
(B)
Secondary II Mathematics
year of algebra and proportionality
(C) See too the following:
- Arithmetic
& Number Theory Practices (horribly put, but
useful)
- Algebra and
Logic SubProgram
(well put, extremely useful)
For Avid Readers in School & Out -
Online Books
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
Tour their forewords.
Calculus Prep or Help: See Volumes 2 & 3,
and this bigger
Calculus
Guide.
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Senior
High School &
Calculus Students
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Free Live Lesson
- Operations with Decimals - Comparison, Subtraction and Long Division
- Click here
to attend.
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For Senior
High School Mathematics & Calculus
Intro to Solving
Linear Equations
- a different paths for junior and even senior high
school students.
5
wordy Logic
Chapters
4 curious Algebra
Chapters
Words before & besides symbols. A Key Algebra
forward & backwards Chapter
First Calculus
Preview (1st intro)
Four Calculus
Chapters
(2nd intro)
Intro to Complex
Numbers (long)
Intro to Mathematical
Induction (romantic & wordy at first)
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Many More Topics
1. Decimal
Arithmetic Reference!
2. Integers
- Intro to Signed No.s
3. Fractions
- fully explained.
4. Fractions
with Units
5. Number
Theory,
6. Solving
Linear Equations
7 Formulas
Use Forward & backwards -
8. Proportionality,
Back- & For-wards.
9. Logic
Chapters:
10. Euclidean-Geometry
11. Slopes
& Equations of Straight Lines. (Take
I. See take II below)
12. Why
Study Slopes.
13. Maps,
Plans, Similarity & Trig,
(Take II included here)
14. Quadratics:
Starter lessons
15. Polynomials:
Starter lessons
16 Why
Factor Polynomials:
17 Functions
- Forwards & Backwards.
18. Exponents,
Radicals & logs.
19. Complex
Numbers before trig (new advance/ starter lesson)
20. DC
Electric
Circuits Etc
21. Real
Analysis
22. The
Olde Complex No, Trig
& Vector Section.
23. More
Calculus Stuff
- written after Volumes 2 and 3.
More For Instructors
-
Education
Essays
(opinions,
possibilities, references)
- POMME, a two
level program for instruction K1-14
- Math & Logic How-TOs
1. Arithmetic
2. Algebra
3. More Algebra
4. Beginner Geometry
5. More Geometry
6. Calculus
7. Show Work or Logic
These may be too dense for students.
Level I Material: New Stuff
Time and Date Matters
Level I Arithmetic.
Money Matters
Measurement Matters
Matters of Chance (Risk Control)
Logic
Chapters
(leave what's not clear in Level I to Level II)
Using/Making Maps and Plans.
(A variant of
Maps,
Plans, Similarity & Trig, to
appear here).
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Skill Development Tips
For All
Standards: (A) Take
care to avoid the domino effect of errors & approximations; (B) Do and
record steps in an manner that allows skill mastery to be seen or
corrected. Anything represent substandard work.
Key Numerical Methods
- To multiply signed numbers, prefix the product of their signs to the product
of their lengths or unsigned parts. The product is negative if the no of
negative sign in it is odd.
- To add signed numbers with like signs, prefix the common sign to the sum of the
lengths.
- To add signed numbers with opposite signs, prefix the sign of the longest to
the difference: length of longest minus length of shortest.
- Should we study roots and powers of real numbers with formulas involving exponential and log.
- How does adding and multiplying points in the plane and rotating the midpoint
of a line segment lead to mastery of complex numbers and the thought-based
development of their properties, all before trig?
- New Axioms for High School Mathematics: In accounting, totals of assets
and debts may be calculated by dividing the assets and debts into
non-overlapping (disjoint) groups and then adding subtotals. In general, sums
(and products) of counts and numbers, positive and negative numbers
included, can be obtained by adding subtotals (and multiplying
subproducts, respectively). These practices may be cast as axioms in
secondary mathematics. Then operations on polynomials are easily implied
justified by these "axioms" and the geometric introduction of column
methods for expanding a products of two sums. While set theory in pure
mathematics may imply the above axioms in university mathematics programs
instruction, an earlier and more accessible explanation based on easily accepted
and understood geometric and counting practices derivation of the above
axioms is possible at the high school for students heading for college programs
in science.
In Volume 2: Prep for Calculus
- What is the difference between saying A if B and saying A if and
only if B. Being aware of the difference will sharpen ye wits.
- What is a chain of reason?
-Are your arithmetic skills OK?
-Have words been missing in the introduction of algebra?
- Can ye talk about numbers & quantities varying apart from or before the
use of letters & functions?
- Do ye know about the forward & backward use of formulas?
-Contrapositive: is that backward use of A if B?
-What is a variable x? Answer before speaking of function f(x) = x.
-What a twist! There are no rules of algebra for subtraction and division. But
if you replace them by addition of -x and multiplication by 1/x, rules of
algebra (properties of arithmetic) can be used.
In Volume 3: Calculus Slowly?
-Why are slopes studied and polynomials factored in high school?
- Volume 3 suggests how to ease or delay algebra shock in
calculus *& beyond. In Calculus, derivatives and integrals introduced and defined by limits, but calculated
without when possible by using differentiation rules forwards and backwards. The
second site calculus section may help in differential calculus.
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