www.whyslopes.com
Volume 3, Why Slopes and More Math
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Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
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Caution: Site advice is approximately
correct, for some circumstances, not all. . That leaves room for thought and
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Online Volume 2, Three Skills
for Algebra, Chapters 1 to 25 - skip 18., verbalizes and explains key skills
and concepts, those needed in calculus, again to make the hard easier. A visual
understanding of complex numbers
may serve as back ground info for partial fraction decomposition.
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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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For help in calculus, explore
Volumes
2. Three Skills
for Algebra
and 3. Why
Slopes & More Math, and Calculus
Introduction site area. See how to learn or teach key skills and
concepts, some not all.
Foreword, One Calculus preview and Online Chapters:
(V) signals video (RealPlayer Format) to
watch
Area Intro
Foreword
Chapter Descriptions
1. Introduction
2. Calculus Starter Lesson
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
Content Guide
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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