Chapter 9
About First Courses in Calculus
Physics, chemistry, engineering disciplines and some business disciplines
all employ calculus concepts, to efficiently describe their computations
and theories.
Formulas for slope function m = g(x) =
h¢(x) (derivative) can be
obtained by applying rules for differentiation when the function
y = h(x) is given by a simple enough formula. Rules
for differentiation along a collection of functions to which they apply,
are typically explained in a 12 to 16 week first course on
calculus.
The differentiation process can be reversed sometimes. Given a formula
for the slope m = g(x), ad hoc integration (that is,
anti-differentiation methods) may identify functions h(x)
with slope h¢(x) =
g(x). This reversal provides or justifies the common
formulas for areas, volumes, weights and masses met in geometry and
physics. Anti-differentiation methods (alias integration methods) and its
applications are typically discussed briefly at the end of a first course
in calculus, and then discussed more completely in a second
course.
A third course in calculus may talk about the slopes (directional
derivatives) of 3D hills z = h(x,y) in place
of 2D hills y = f(x). Here above a point
(x,y) in the plane, the height of the hill z =
h(x,y) depends on the coordinates
(x,y). Imagine a sledge in place of a ski traveling across
this terrain. The sledge in its direction of travel, its longitudinal
direction, has a slope (rise/run). Perpendicular to the direction of
travel, that is across the sledge, there is another slope. This traverse
slope will be horizontal or zero if the sledge is directly uphill or
downhill in the direction of steepest ascent or descent. Otherwise, it
will be nonzero. At the top of a large smooth hill or the bottom of large
smooth depression), a sledge will be horizontal. The longitudinal and
traverse slopes of a horizontal sledge are both zero. When the sledge is
not horizontal, there are directions of ascent and descent. So there are
nearby points which are higher or lower than the sledge height (or its
midpoint). The foregoing analysis holds for small hills and small
depressions as well, providing the sledge is made smaller or microscopic.
Imagine now that the sledge lies on the hill z =
f(x,y) and its direction is perpendicular to the
y axis. Further suppose that the x coordinate of the
front-end of the sledge is greater than that of the other end. The
longitudinal rise/run of the sledge then gives the slope of the hill
y = f(x,y) in the x direction. Rules
for differentiation, essentially mastered in a first course in calculus,
say how this x-slope (or x-direction derivative) of the
hill shape z = f(x,y) can be computed. The
slope (or derivative) of the hill in the y direction is defined
and computed similarly. As indicated, directional slopes (or derivatives)
are often the subject of a third course.
The Limit Definition of Slope
Rules for slope computation, that is rules for obtaining g(x) =
f'(x) from formulas for f(x) are studied in the first weeks or
months of a first course on calculus ALONG with limits. Calculus do not
use pictures of skiers to define slopes to nonlinear formula or
functions f(x). Limits are used instead. Rules for the reversal of
slope computation (anti-differentiation) may be met say after three
month of calculus. With these rules, formulas for a functions can
sometimes be more easily obtained from formula for their slopes. In
particular, by the use of functions, and anti-differentiation, formulas
are obtained in calculus for areas, weights and masses of common shapes
and objects.
See Chapter 14, Limits,
Error Control and Continuity and the essay Decimal Insights besides this page
Suppose you are walking or skiing along a smooth path, a path with no
vertical steps or drops. At any point, the slope can be measured or
estimated by placing a straight rod (a ski?) on the path. The rod should
be pivoting at the point or provide a bridge between two points on the
hill, on either side of the point where the slope is to be found. Then
the slope of the hill at the point is approximately equal to the slope of
the rod = its rise over its run. Shorter rods are better for slope
approximation than longer ones. A more precise definition or explanation
of how to compute slopes requires the notions of a limit. Using a short
rod (or ski) to estimate the slope at a point on a path is enough to
understand the first geometric or physical interpretation of slopes.
Units in Calculus
Unit are too often forgotten in teaching computation
The calculation of slope = rise/run results in a pure number when the
rise and run are measured in the same units. The practice in physics and
chemistry deal with quantities - numbers times a unit of measurement. If
you keep the units in formulas y=h(x) for the graph of one quantity
versus another, the rise and run will have different units. Speed and
velocity are measured in terms of distance/time. This leads to units such
as 60 miles/hour or 100 Kilometer/hour (the slash / is read per). In flow
measurement, there 10 kilograms of matter passing a given point per
second. The graph of material passed (measured in kilograms) versus time
(measured in seconds) would have a slope =rise/run = (10 kilograms)/(1
second) = 10 * (kilograms/second).
If you are averse to kilograms, replace the 10 kilograms by 21 pounds.
In the case of distance/time or velocity, a positive speed corresponds to
forward motion, a zero speed to no motion, and a negative speed to
backward motion. Forward acceleration means the speed is increasing, and
negative acceleration (de-acceleration for the grammatically proper)
means the speed is decreasing.
To put the word increasing in proper perspective, a temperature changes
from 5 to 10 degrees Celsius and a temperature change from -15 to -3
Celsius are both temperature increases. The reverse changes would be
decreases.
More About Calculus
The pictures above show how to interpret the sign of the slope, and the
slope behavior for a walk along a path. Now if you have a smooth enough
curve y = h(x) drawn in the coordinate plane, you can use a short line
segment placed against the curve to estimate the slope at each point of
this 2D hill y = h(x). You can imagine that when your x-coordinate is
specified, the function h(x) gives a rule or formula for computing the
height.
When y = h(x) = a x + b, you should know from algebra courses how to
compute the slope m = rise/run = a. The symbol we use to represent the
slope is not important. For more complicated expressions for h(x)
involving polynomials, sines, cosines, logarithms and exponentials etc,
there are rules (justified in first courses on calculus) which say how to
obtain or derive a formula for the slope of the curve y = h(x)
from a formula for h(x). Slopes in calculus are called
derivatives, presumably because they are derived. But slopes are
not called obtainables. The latter word is not in fashion.
First courses in calculus are devoted to slope or derivative computation
and slope or derivative interpretation. We have seen one interpretation
above. Velocities, acceleration and all rates of change all give examples
of slopes to graphs y = h(x) or s = f(t). Here the letters used to define
the horizontal and vertical axis variables are not important. Whenever
you have a smooth-enough graph or hill shape y = h(x) or s = f(t) given
by a simple formula, the slope may be derived from the formula, and if
not, estimated from the graph. (Slopes are called derivatives in
calculus.)
Rules for differentiation (slope calculation) give formulas for the
slopes of functions y = f(x). The word differentiation presumably
stems from the use of differences (Delta x over Delta y) in the
estimation of slopes.
In the opposite direction, formulas for functions or formula y = f(x) may
in some instances be found by reversing the methods of slope calculation,
an ad hoc process called anti-differentiation or integration. Finding a
function f(x) from a knowledge of its slope etc., leads to and justifies
common formulas for the perimeters, areas of regions in the plane, the
length of curves and the volumes and weights and masses of solids. This
reversal of slope computation, may be met at the end of a first course in
calculus. Second courses in calculus describe and explore this reversal,
that is, the integration or anti-differentiation process, in more detail,
ad nauseum. Third courses in calculus will further describe slope
computation and the slope reversal process for three dimensional hills z
= h(x,y) in place of two dimensional hills y = h(x), among other topics.
Finally note that saying how to compute a number or quantity defines it
--- and serves as a computational definition. The computation should be
feasible, otherwise the number or quantity in question is left undefined.
In calculus, there are two main examples:
- Slopes or derivatives at a point are defined by the limiting value
of
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Dy
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slope of a chord between two points on a
graph of y = f(x)
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- Areas of regions are defined by covering the regions with small
squares or thin rectangles, adding up the areas of the squares or
rectangles within the region, and finally taking the limit of such sums
as the width of the squares or rectangles tend to zero.
Reversal of the slope computation process simply gives a shortcut
(formulas) for the computation of some common areas. It avoids the
repeated approximation of area by covering a region by small squares or
thin rectangles. It gives the limiting value of this approximations --
the number or quantity it tends to.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
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Calculus Starter Lessons
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Unsolicited Advice
Learning to do and high marks if it comes to easy is often
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