Chapter 16. Velocity Approximation
Volume 3, Why Slopes and More Math.
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.
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The average slope $m_{avg}$ between times $t_1$ and $t_2$ is given
and defined by the slope of the segment joining $(t_1,d_1)$ to
$(t_2,d_2)$. In particular, \begin{eqnarray*}
m_{avg}&=&\frac{d_2-d_1}{t_2-t_1} \\ &=&\frac{f(t_2)-f(t_1)}{t_2-t_1}
\end{eqnarray*} When $t_1 \le t \le t_2$, the slope of the above position
versus time curve at the point $(t,f(t))$ may be approximated by
$m_{avg}$.
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For times $t$ in between $t_1$ and $t_2$, the line segment joining
$(t_1,d_1)$ to $(t_2,d_2)$ provides an approximation to the graph of
the function. That is, in or near the interval $t_1$ to $t_2$, his
position \[ d=f(t)\approx d_1+m_{avg}(t-t_1) \] where the symbol
$\approx$ 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 $t_1$ and $t_2$, or the distance between them. For some
functions, the approximation error may become smaller if the interval
$t_2$ to $t_1$ is made smaller. The error in this approximation is
zero, that is, it vanishes, at the two times $t=t_2$ and $t=t_1$.
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The average speed or velocity of travel between $t_1$ and $t_2$ is
given (defined by) \[v=m_{avg}=\frac{d_2-d_1}{t_2-t_1}\cdot\] The
velocity at the time or instant $t_1$ is obtained if the second time
$t_2$ in this calculation is allowed to approach $t_1$. 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 $d_2=f(t_2)$. In it, $t_2=t_1+\Delta t$ with
$t_1$ fixed, that is non-moving, while the difference $\Delta t
=t_2-t_1 \to 0$. The arrow $\to$, as before, indicates goes to
\rm or approaches. \rm For smaller and smaller values of
$\Delta t$, the slope of the line segment through $(t_1,d_1)$ and
$(t_2,d_2)$ should approach that of a tangent line touching the curve
at the point $(t_1,d_1)=(t_1,f(t_1))$. This expectation provides the
motivation for the following definition of the slope $m$ to the curve
at the fixed, non-moving point $(t_1,d_1)=(t_1,f(t_1))$.
The slope m = mtangent of the tangent
line through (t1,d1) =
(t1,f(t1)) is defined
by
The slope $m=m_{tangent}$ of the tangent line through
$(t_1,d_1)=(t_1,f(t_1))$ is defined by
\\begin{eqnarray*} m_{{tangent}}&=&f'(t_1) \\ &=&
\lim_{\Delta t \to 0}\frac{f(t_2)-f(t_1)}{t_2-t_1} \\ &=& \lim_{\Delta t \to
0}\frac{\Delta d }{\Delta t} \end{eqnarray*} The units of $m_{tangent}$ here
are those of { $\frac{\mbox{distance}}{\mbox{time}}$}, that is,
distance over time. The slope $m_{tangent}$ gives or should give the
limiting value of average velocity over the time interval $t_1$ and
$t_2=t_1+\Delta t$ as $\Delta t$ 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
\begin{eqnarray*}
v=m_{\mbox{tangent}}&=& f'(t_1) \\ &=& \lim_{\Delta t \to
0}\frac{f(t_2)-f(t_1)}{t_2-t_1} \\ &=& \lim_{\Delta t \to 0}\frac{\Delta d
}{\Delta t} \end{eqnarray*} and the speed \rm at instant $t_1$ 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 Tangent Line Approximation
For times $t$ near $t_1$, the distance \small\[ d=f(t) \approx
d_1+m_{tangent}\cdot(t-t_1)=d_1+v\cdot(t-t_1) \]\normal The error
$|f(t) -[d_1+m_{tangent}\cdot(t-t_1)]|$ in this linear
approximation vanishes when $t=t_1$. This approximation is
best \rm if the velocity between $t$ and $t_1$ is almost
constant. Otherwise, the approximation error could be large.
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For home-tutoring or -schooling, or for schools or colleges
with course content control: Secondary
Mathematics for Ages 11+, A Practical Approach.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
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areas [proportional amounts too] directly or by using maps and
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ingredients, pots and pans for a dish or a meal, along with
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McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
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Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
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Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
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Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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