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Chapter 2 . Slopes and Ski Trails

Volume 3, Why Slopes and More Math.

• [Play Video]80 seconds: Slope Sign Interpretation for Linear Functions. (Appeared earlier in Why Slopes Appetizer)

Slopes of Line Segments

Remark. A quantity Q₂ is said to be proportional to a quantity Q₁ when and only when there is a constant k such that Q₂ = k· Q₁. If a quantity Q₂ is proportional to a quantity Q₁ then the graph of Q₂ versus Q₁ is a straight line through the origin whose slope m = k is the constant of proportionality.

A Cross-Country Skier and Her Trail

[Play Video] 2¼ minutes: Slope Interpretation for a 2D ski hill y = f(x). (Appeared earlier in Why Slopes Appetizer)

She has only one ski. Alternatively, you can imagine she always travels with both skis parallel. Travel with one ski was the way in which both alpine and cross-country skiing began. Also meet the Jack Rabbit ski trail y = h( x) (see below) which she skis, always in the direction -> from left to right.⁵ That is, she travels in the direction of increasing x.

^{5 Footnote:} The slope to a curve at point can be approximated by taking the slope of a short line segment which has one end at the point and another end also on the curve. This approximation should get better as the line segment gets shorter. The finite limiting value of this approximation, should it exist, is taken to be the slope. Before discussing this approximation any further, we will make the improper assumption that the slope of a short ski placed on the graph of y = f( x), or the graph of one quantity versus another, is the slope to the graph. This will allow some exploration of why slopes are studied.

Imagine or suppose that the hill is smooth enough, so that, at most points, a ski can lie flat against the hill surface. The slope beneath a foot or ski gives what should be the slope of a tangent line to the hill. The slope of a ski can in principle be measured any time by freezing a skier in place, or equivalently taking a photograph (snapshot) and then measuring the slope from the photograph.⁶

^{6 footnote} What happens if Barbara goes up and over a sharp peak? As she gets to the top and pivots from the uphill to downhill side, the slope of her ski goes from positive to negative.

The vertical line segment in the above graph represents a jump or cliff. The above diagram strictly speaking consists of the graph of a function y = h( x) plus a vertical portion to represent a ski jump.

• Above the point x = a on the horizontal axis, the height above the x axis of her ski midpoint is y = h( a). We will call h( x), the height function.
• The height function h( x) might be measured or computed from a formula, a map, or a graph, such as the one shown above. Or, in speaking of a height function h( x), we could leave its values unmeasured, not computed or unknown.

The shorthand for the word positive is +ve. The shorthand for negative is similarly -ve.

A Skier in Motion

Immediately below are a few snapshots of Barbara on another portion of the skill trails and hills y = h( x). In each snapshot, the slope of the ski is assumed to be equal to the slope of the hill at the ski midpoint.

In the diagram observe:

• At x = c, the slope m > 0 and she is moving uphill: her height is increasing.
• At x = b, the slope m < 0 and she is moving downhill: her height is decreasing.
• At x = a, the skier is approaching the top of the hill. What is the sign of the ski slope before, at and after the top of this smooth hill?

1. Local High-Points or Maximums

High points of bumps and hills are called maximums.

As the skier Barbara moves up a hill (or a local bump), and then down the other side, the slope of her ski changes from positive on the uphill side to negative on the downhill side. At the very top of hill, her ski is horizontal and its slope is zero.⁷

^7Footnote: What happens if Barbara goes up and over a sharp peak? As she gets to the top and pivots from the uphill to downhill side, the slope of her ski goes from positive to negative.

There can be several hills, with different and varying steepness on each side. From the slope of her ski, Barbara knows even without looking when she crosses over the top of a hill or a bump: the slope of her ski changes from positive to negative. The next diagram shows the sign of the slope (of the one ski) before, at and after a high point.

Knowledge of the sign of the slope does not provide all information about the ski trail y = h( x). In particular, unless she measures and records the height h( a) of each hilltop, she can not say which is the highest. Every hilltop gives a local maximum for her height. It has a height greater than or equal to ( ³ ) all nearby heights. A point with a height greater than or equal to ( ³ ) all other heights, is called an absolute maximum for the portion of the trail or curve y = h( x) being examined.

Remark (Advanced Material.) To be more precise, a point ( x0, h( x0)) is a local maximum for the portion of a curve y = f( x) where a < x < b if a < x0 < b. That is, x0 is in the interval being examined. There exists at least one interval ( x0- d, x0+ d) centered at x0 such that h( x) < h( x0) if a < x < b and x0- d < x < x0+ d. These two conditions describe precisely what is meant in talking about all nearby heights. Note that in talking about numbers and quantities, a legalistic precision is required. Otherwise, tacit assumptions will be made differently by different writers and different readers.

Definition. (Highest of the High Points). A point with height greater than all other heights in a given portion of the trail, more precisely not less than all other heights in the portion, is called an absolute maximum for the portion or interval in question.

In the event of a tie, each of those points having or sharing the greatest height is called an absolute maximum.

2. Local Low-Points or Minimums

As with high points (hill tops or maximums), several depressions or valley bottoms may be met and crossed in following a ski trail. From the slope of her ski, Barbara again knows even with her eyes closed when she crosses the valley bottom: the slope of her ski changes from negative to positive.

Note again that unless she measures and records the height h( a) of the low points in each depression or hollow, she can not say which is the lowest. The bottom of each depression or hollow gives a local minimum for her height. It has a height less than or equal to all nearby heights.

According to this definition, all the points on a horizontal straight lines are local minimums. Excluding the word equal here would yield a strict local minimum: a height less than nearby heights. See the previous discussion of nearby heights.

Definition. A point with a height less than or equal to ( < ) all other heights in a portion of the trail is called an absolute minimum for the portion of the trail or curve y = h( x) in question.

In the event of a tie, each of those points having or sharing the least height is called an absolute minimum.

x-Dependence of Slope

[Play Video] 1¾ minutes: Along a 2D ski trail, see how height y = f(x) and slope m = f'(x) both depend on the horizontal coordinate x.

As Barbara, the one-ski skier, moves along a trail y = h( x), both the height of the ski midpoint y and the slope m of the ski depend on its x coordinate. The slope of her ski at x = a, x = b, x = c and x = 2 in the following diagram are all different. The slope of her ski midpoint depends on her location.

• At each point, the slope m of the ski is determined by x and the shape of the trail y = h( x). That is, the slope depends on x and the hill y = h( x). To signal this dependence, we write m = g( x) = h'( x). As said before, the slope m = g( x) could be measured from a snapshot - freeze Barbara and her ski in place. The mathematical definition of \( m = h'(x) $ appears later.
• The slope m = g( x) depends on the shape of the hills y = h( x). The slope when x = a is m = g( a) = h¢( a). The slope when x = b is m = g( b) = h¢( b), and so on.
• In most problems that you will meet or be shown, formulas for the slope m = g( x) can usually be obtained or derived from formulas for the height y = h( x). For each new type of function added to your knowledge, there will be a differentiation rule to be learnt.
• For a given formula or function y = h( x), rules of differentiation say how to obtain or derive a formula or function g( x) = h¢( x) from a formula for height h( x). Presumably, because of these rules for deriving or obtaining the slope m = g( x) = h¢( x) from formulas for h( x), the function g( x) = h¢( x) is also called the derivative, the first derivative. Note that the use of the word obtainable for slopes is not an accepted alternative to the term derivative.
• There are several simple rules for calculation slope functions or derivatives. These differentiation rules are given in the first instance for the cases where an expression for the height function h( x) involves polynomials, logarithms, exponentials, sines or cosines. Here for each operation (addition, subtraction, multiplication, division and composition) involving functions and yielding a new function or formula, there are additional rules, all of which appear to be very simple after, but not before, they have been mastered. Meeting and mastering these rules requires or instills an understanding of the algebraic way of writing and thinking. The algebraic way of writing and thinking is seen or required here in full strength.

Notations for Slopes and Derivatives

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science