Mathematics and Logic - Skill and Concept Development

A Calculus Preview

Volume 3, Why Slopes and More Math.

If you have ever been or gone skiing, if you have ever walked over hills, then you know about slopes and you have also met or felt basic ideas in calculus before the use of symbols. Calculus in the first instance is the subject of slope computation and interpretation, and the reversal of slope computation with its applications. Slopes (rises/runs) appear whenever one quantity is mapped against another. Height versus horizontal movement is just one example.

Recall the slope of a straight line or line segment is given by the rise over run of a right triangle with hypotenuse on the segment, and sides horizontal or vertical.

Slope Interpretation

 For travel along a line segment, the slope m is positive for uphill motion. It is negative for downhill motion. Finally, it is zero for horizontal motion.
 

 [play realplayer video]  80 seconds: Slope Sign Interpretation for Linear Functions. 

Meet the Skier

While skiing or walking you can observe and feel when you are walking uphill from the slope of your ski or heel. Likewise, you can feel when you are walking downhill. Alex the skier shown in the diagrams has a similar skill. It is his picture - the stick diagrams -- that you see above and below.

Formulas for Slope

 Ski hills y =f (x) usually do not consist of a single straight line segment with a single slope. In consequence, the slope m of his ski varies with his position. 

 Height y at x is given by a formula or function f(x) involving x. So we write y = f(x). Likewise, when the skier Alex is above x at height y on the hill, the slope of his ski may be given by a formula or function m = g(x). It depends on x.

Note we also write g(x) = f'(x) -- read f prime of x -- to say or suggest that the formula for slope m can obtained or derived from the formula for f(x). Rules for slope computation (differentiation) say when. Calculus courses may call formulas for slopes obtainables or derivatives -- one of these names is correct. The other is not.

For the following diagram, answer the following questions. Assume forward motion in the direction of increasing x.

1. Where (above what intervals) is the slope m m = g(x) =f'(x),
    (a) positive?
    (b) negative?
    (c) zero?
2. Where is the slope increasing? In other words, where is the slope becoming more positive or less negative?
3. Where is the slope decreasing? That is, where is the slope becoming more negative or less positive.
   Where does the hill become steeper? (That is, identify the intervals or hill portions where with x increasing, motion forward, the slope become more positive or more negative?)

 In this ski trip, when x = b, is the skier at a hilltop or at the bottom of valley (or depression)? Is the value y = f(b), the least or greatest value of f(x) for x between a and c?

[play realplayer video]  2¼ minutes:  Slope Interpretation for a 2D ski hill y = f(x).

Slope (or Derivative) Tests  for High and Low Points

In first calculus courses, you may be given a formula for y =f(x). From this formula, you may then obtain a formula for the slopem = g(x) = f'(x) at each point x on the curve. By factoring the expression for m, if that be possible, you may see where the slope m is positive, negative or zero. This allows you to say where are the maximums (greatest value) and minimums (least values) of the original function. Slope sign analysis can be done whenever one quantity y is graphed against another x.

In graphs of height versus distance, the slope has no units, but in graphs of distance versus time the slope has a unit of the form distance over time. Slopes with units appear when the abscissa and ordinate are multiples of different units of measurement.

In Summation (to say that again)

In skiing or walking you can tell where the path is going up, down or is on the level. The slope is positive on uphill portions, negative on downhill portions and zero on flat portions. Knowing the sign of the slope gives information about the hill. The slope changes from positive to negative in crossing a hilltop. It changes from negative to positive in crossing through a low point (a valley). Just knowing the sign of the slope is enough to identify the uphill, downhill and flat portions of the path, and then location of high points and low points.

 Here the sign of the slope indicates where the path is going up (ascending) or going down (descending). Positive slope corresponds to going up while negative corresponds to going down. Moreover, and this is whether matters become complicated, the slope in changing may increase or decrease. Here a positive slope may increase by becoming more positive and a negative slope may increase by becoming less negative. Likewise, a positive slope may decrease by becoming less positive and a negative slope may decrease by becoming more negative. And in all these cases, the steepness or slope of the curve changes.

 The steepness is given by the absolute value or magnitude of the slope. Problem: What can you say about the slope behavior when the steepness of the path is increasing? (The answer will depend on whether the path you are following is ascending or descending).

Slope or Derivative Tests for High Points and Low Points

Advanced Topic -- take a break before proceeding.

(???)In travelling over an interval a < x < b interval the over downhill travelling is (s)he that and negative slope b, a when uphill going positive observes ski her or his of feel from skier c,> In this ski trip, when x = b, is the skier at a hilltop or at the bottom of valley (or depression)? Is the value y = f(b), the least or greatest value of f(x) for x between a and c? How would your conclusions change if the words positive and negative were interchanged? In first calculus courses, you may be given a formula for y =f(x). From this formula, you will obtain a formula for the slope m = g(x) = f'(x). Then by factoring the expression for m, if that be possible, you may see where the slope m is positive, negative or zero. This allows you to say where are the maximums (greatest value) and minimums (least values) of the original function. This analysis can be done whenever one quantity is graphed against another.

Algebra and Logic in Calculus - A warning

Calculus employs at full strength the algebraic way of writing and reasoning. Students who have done well in previous math courses without fully understanding the algebraic way of writing and reasoning will find calculus stressful. Memorization of formulas or rules for differentiation by itself is not enough. Understanding is required.

The computations in calculus employ very finely and carefully, constants, variables and algebraic shorthand notation (formulas) to discuss and describe calculation that might be done. A few are even performed. In Volume 2, chapter 8 Three skills for Algebra and the  logic chapters before it (or chapters 4, 6, 7, 8 and 12 in Volume 1A) should be read and mastered, preferably before you take calculus. Mastering the logic appetizers should help read the definition in calculus precisely, and follow the chains of reason provided by your teacher or textbook. (Reading the text is advised -- it gives a second opinion.)

A Review

In skiing or walking you can tell where the path is going up, down or is on the level. The slope is positive on uphill portions, negative on downhill portions and zero on flat portions. Knowing the sign of the slope gives information about the hill. The slope changes from positive to negative in crossing a hilltop. It changes from negative to positive in crossing through a low point (a valley). Just knowing the sign of the slope is enough to identify the uphill, downhill and flat portions of the path, and then location of high points and low points.

Now in walking along a path, you can also tell when or where the steepness or slope of the path changes. For instance,

• Along one portion of a path that the slope could be positive and becoming more positive. On this portion, you are walking up hill and the slope is increasing.
• On another uphill portion of the path, the slope could be decreasing --- becoming less positive and less steep.
• On yet another portion of the path, the slope could be negative. So as your walking along, the height of the path is decreasing. Now as you walk along this downhill portion of the path, the slope may become more negative or less negative. Where the slope is becoming more negative, your downhill path of descent is become steeper; and where the slope of the path is becoming less negative, your downhill path is becoming less steep.

Here the sign of the slope indicates where the path is going up (ascending) or going down (descending). Positive slope corresponds to going up while negative corresponds to going down. Moreover, and this is whether matters become complicated, the slope in changing may increase or decrease. Here a positive slope may increase by becoming more positive and a negative slope may increase by becoming less negative. Likewise, a positive slope may decrease by becoming less positive and a negative slope may decrease by becoming more negative. And in all these cases, the steepness or slope of the curve changes.

The steepness is given by the absolute value or magnitude of the slope.

 Problem: What can you say about the slope behavior when the steepness of the path is increasing? (The answer will depend on whether the path you are following is ascending or descending).

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