Why Slopes
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| Vol 2, Three Skills for Algebra covers many topics in algebra and logic that students starting calculus should have mastered or will have to master. Also includes arithmetic review problems to catch common mistakes. A fourth skill gives a unifying theme for high school maths. |
Units in Calculations:
Appendices:
Integration
& Lipschitz
Continuity
These appendices continue the
decimal viewpoint of limits, error
control and continuity begun
in Chapter 14. The One Sided
Range Theorem is a postscript,
not in printed version.
Chapter 6
Slopes and Vertical Shifts
The vertical shift or motion of a curve y = f(x) adds a constant d (the displacement) to each point. It yields a new curve y = f(x)+d. The next diagram indicates an example.
A simple vertical shift or motion may change the height of a skier, but not the slope of his or her skis.
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On the Constant Difference Theorem 4:32 minutes 404 by 408 |
Constant Difference Theorem
What can be said for sure about two functions when they have
the same slope (or derivative) everywhere? One response is
given by the following assertion.
[Constant Difference Theorem]
If the functions f1(x) and f2(x)
have the same non-infinite slope (that is, derivative) m = f¢1(x) = f¢2(x)
at every x in an interval (a,b) then the difference
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Note the assertion says that there is a constant d. It does not say how to find it. Here is an analogy: Saying there is a needle in a haystack, does not say how to find it. Note also the word every. If there is a point x1 in the interval (a,b) where the slope is not defined then no conclusions can be drawn from the Constant Difference theorem.
Remark. If f1(x) and f2(x) satisfy the conditions (hypotheses) in the Constant Difference theorem with f2(c) = A and f1(c) = B at some point c in the interval (a,b) then at x = c,
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Remark. Mathematical assertions and theorems which say that a number (or limit) is not defined or does not exist actually mean that a finite number (or finite limit) does not exist. The vertical motion theorem given above applies only when the common slope is finite in the interval (a,b) of interest.
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When the Constant Difference Theorem fails 404 by 408 3:36 minutes |
Different vertical displacements over different portions of the trail are possible. The next diagram gives examples of this. Upshifts have been made in the topmost curve of the previous diagram.
Between a and b, between b and c and between c and d, the two trails y = f1(x) and y = f2(x) shown have the same slope but not the same heights. At the ski jump and cliffs in the upper trail y = f1(x), the slope is not defined.
Where Slopes are Not Defined
On a ski trail y = h(x), there may be a few places where the slope is undefined or a single slope to the graph or trail does not exist. In the following diagram, the slope is undefined at the ski jump above the point x = a. At sharp peaks and kinks, a short ski may pivot or rotate while keeping in contacting with one point, the kink or peak on the trail. So a single slope to the trail cannot be defined there. Where the trail has some vertical jumps, the graph ceases to be the graph of a function. The slope is said to be undefined or not to exist here, even though we might say it is infinite, +¥ or -.|
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2 minute explanation of the following 1:50 minutes |
Consider the next diagram. At the points a,b,c,d and e, the slope function or derivative m = h¢(x) cannot be given a single value. In general, a single value cannot be assigned to slopes at sharp changes in the direction of a curve y = h(x).
The study of generalized slopes or gradients replaces the discussion of a single slope to a point on a curve by the discussion the set of slopes to a point on a curve. This set based discussion is too complicated to be examined further here.
But a single slope may sometimes be defined before and after such kinks or sharp turns in the graph of a function.
The ski at the sharp peak is shown pivoting, that is rotating, as the ski passes over.1 In pivoting at the peak, a ski can have many orientations or slopes without intersecting the curve y = h(x) on either side.
Problem for Advanced Students. A graph with vertical segments is not the graph of a function, but it may be the image of a parameterized curve (x(t),y(t)) where t belongs to some interval. Show that if (xj(t),yj(t)) are two continuous curves parameterized by t Î [a,b], then (x1¢(t),y1¢(t)) = (x2¢(t),y2¢(t)) for all t Î (a,b) implies there are constants d1 and d2 such that (xj(t),yj(t)) = (xj(t),yj(t))+(d1,d2).1The slope values during this rotation form a set, the slope set. The value 0 belongs to this set.
This problem is both a consequence and an extension, a generalization, of the constant difference theorem just stated. It applies to some graphs with vertical segments.