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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 
  

  mchord  =

  Dy Dx

  =

  {

  slope of a chord between two points on a graph of y = f(x)

• 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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