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A Preview of Calculus and Calculus Courses

Several physical interpretations of slopes are summarized next.

Slopes describe how fast the curve given by graphing one quantity versus another, rises and falls.

Slopes in daily life occur in the discussion of streets. A road or railway may rise three feet for every 100 foot traveled horizontally. The grade or slope of the road is then said to be 3%. An interesting or steep ski hill may fall one meter for every three meters traveled - a slope of -[1/3] or -33.3 percent. In buying and selling, there may be an order charge plus a cost per unit for the total amount ordered. The cost per unit is a slope. When one quantity is proportional to second, the proportionality constant is also a slope. The cost of purchase is often proportional to the amount bought. When apples cost 25 cents each, ten more apples added to a purchase will cost another 250 cents or 2.50 dollars.

In general, a slope gives the change or increase in a first quantity per or for each unit change in a second. Mathematically the slope is expressed as the ratio of the change in the first quantity to the change in a second. Speed and velocities give the change in distance per unit change in time. The acceleration which you feel as your speed or velocity changes is mathematically represented by the change in speed (or velocity) per unit change in time. These quantities are all given by the slope of the graph of one quantity or number versus another.

The slope at point on ski trail y = h(x) is first imagined to be given by the slope of a small ski located at the point, more precisely whose midpoint is located at the point. This provides an initial image or definition of the slope to a curve or ski trail, y = h(x). (The mathematical definition to be preferred is much more precise but less easily described. The slope at a point is actually taken to be the limiting value of numerical approximations to it. More will be said about this.)

In traversing the 2D (two dimensional) hill where y = f(x), the slope m of the traveler's one ski changes with (or depends on) its horizontal coordinate x. This gives a slope function m = g(x). The notation m = g(x) signals that the quantity m depends on the quantity x. Formulas for the slope derivative function m = g(x) = h¢(x) can be obtained or derived from simple formulas for the height function h(x), whenever the latter are available or given. The prime in the notation h¢(x) indicates that the formula for h¢(x) is obtained or derived from the formula for h(x).

A positive slope (when the horizontal coordinate x is increasing) corresponds to the skier going uphill. Similarly a negative slope means going downhill. With this perspective, the slope of a ski will go from positive to negative as it goes over a hill point. At the top for one instant, its slope may be zero. When a ski goes through a depression or a valley bottom, the slope of this ski is first negative on the downhill side and then positive on the uphill side. A skier may tell from the slope of a ski when or where he or she has crossed a hilltop (maximum) or low point (minimum).

A skier can recognize the intervals where the slope is increasing, and the intervals where the slope is decreasing. A slope which is becoming less negative or more positive as the skier moves forward in the positive x directions is said to be increasing. A slope which is becoming less positive or more negative as the skier moves forward in the positive x directions is said to be decreasing. On intervals where the slope its slope is increasing, a function, y = h(x), is said to be convex, and on intervals where its slope is decreasing, a function, y = f(x), is said to be concave.

Ski jumps and cliffs correspond to jumps or discontinuities in the skiers trail y = h(x). Cross-sections of rift valleys and plateaus further give examples of 2D ski hills y = h(x) with ski jumps. Snow is assumed. At these jumps the slope or derivative is not defined.

At vertical drops for instance, the slope is undefined. The slope of the ski is further undefined or not determined by the trail at kinks or sharp peaks where a short ski could pivot on its midpoint without touching the trail on either side. For instance, the top of an upside down V gives a sharp peak.

Earthquakes, or vertical motions up and down of 2D hills and curves, suggest or imply that slope functions are not affected by the upward and downward shifts of part of a curve. In consequence, different hill shapes y = h(x) and y = f(x) could have the same slope function m = g(x) = h¢(x) = f¢(x), but different heights. Yet (theorem) if the slope of a function y = h(x) is defined everywhere on an interval, any other function with the same slope will differ from y = h(x) by a constant vertical shift up or down, in the interval (for why see the advance material in the appendices). The observation that two functions with the same slope everywhere on an interval will differ by a constant provides a key to the calculation of functions or even their definition, from a knowledge of their slope. Calculating functions from formulas for their slopes is used to calculate area, volumes, and other quantities.

Slopes, areas and volumes etc may be calculated or approximated numerically by various methods. If the error in the approximations tends to zero, the approximations approach or converge to a limiting value. The limit should yield the value of the number or quantity in question. The question of what is a number or quantity may be answered precisely by saying how it is calculated or how it can be approximated with unlimited accuracy. Such an answer often, if not always, gives the accepted mathematical definition of the number or quantity in all computational disciplines. See the chapters on slope, area and velocity approximation.

In three dimensions, the direction of a perpendicular ^ to a sledge which is flat against the trail surface, can be used to locate slopes, hilltops, valley bottoms and mountain passes between valleys. The perpendicular direction to the sledge is vertical at hilltops, valley or depression bottoms and at the high point of a pass between two valleys. Ski jumps and sharp points on the terrain can be used to represent the idea of discontinuity and the occasional absence of tangent lines or planes. The rift valley in Africa with its vertical sides, the Grand Canyon in North America and holes or trenches (with vertical sides) dug or found by road repair crews, give examples of three dimensional discontinuities. Their cross-sections provide examples of two dimensional ski jumps or discontinuities.

Chapter 8 Almost A Review

A Preview of Calculus and Calculus Courses

Chapter 8
Almost A Review