Original Site Title: Appetizers and Lessons for Mathematics and
Reason, June 1995 to April 2012. New site title:
Logic and Mathematics Skill & Concept
Building Site
Map || Français: 26
pages
for college students, gifted teens, home-tutoring and K1-12 schooling, with chapters
on
Logic
and Pattern
Based Reason to inform and amuse thinkers and avid readers, studying or not. Enjoy.
|
Logic mastery strengthens comprehension and improve home,
work & study habits. |
About: Site material shows how common troubles
stem from steps too large or missing. Site material may develop
critical thinking, improve reading and writing, and build
mathematics and pattern based reasoning skills. Online Volumes
1, 1A and
2 give avid readers in school and out the best places to begin.
If one site element is not to your liking, try another. Each is
different. Many are unique |
Location: Site Entrance < Volume 3 Why Slopes - A Calculus Intro Etc << Chapter 2. Slopes and Ski Trails
[1] [2] [3] [4][5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
Chapter 2 . Slopes and Ski TrailsVolume 3, Why Slopes and More Math.
Slopes of Line SegmentsA tour of calculus begins. Recall how the slope to a straight line is computed. The slope m of a straight line segment between two points ( x1, y1) and ( x2, y2) may be calculated as follows.
The point-slope formula for a line
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)Meet the cross-country skier, Barbara:
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.
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
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.
The shorthand for the word positive is +ve. The
shorthand for negative is similarly -ve. A Skier in MotionImmediately 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:
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
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.
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
Low points of depressions are called 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.
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
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.
Notations for Slopes and DerivativesCalculus has several expressions for the slope m of the ski or hill function y = h( x). Some follow. \[m = g(x)=h'(x)=\frac{dy}{dx} = lim_{\Delta x \to 0} \frac{\Delta x}{\Delta y} \] Still more may be seen. Different notations exist because calculus was discovered and employed by different people in the past four or five centuries. |
|
Location: Site Entrance < Volume 3 Why Slopes - A Calculus Intro Etc << Chapter 2. Slopes and Ski Trails
[1] [2] [3] [4][5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
|
Logic-Reason for all Careful Thinking Chains of Reason Mathematical Induction Responsibility Bodies-of-Knowledge |
Arithmetic - Ages 10+ |
Geometry 1 Maps + Plans Use 2 Euclidean Geometry 3 Rct +Polr Coordinates 4 Lines-Slopes [I] 5. What is Similarity |
Algebra Starters - the base 1. Better Work Format 2. Solve Linear Eqns 3. Computation Rules 4. Axioms, Item 3 Viewpnt 5. Formulas Backwards |
More Algebra Logarithms-ax & m/nth roots Five Polynomial Operations Quadratics Geometrically Functions || Vectors too Arith. Skill Check+Answers |
Calculus Prep/Preview What is a Variable Why study slopes Why factor polynomials Complex Numbers Limits + Continuity |
All trademarks and copyrights in this are owned by their
respective owners.
Copyright to comments & contributions are owned by the Poster.
The Rest
© 1995-2011, by site author, Alan Selby, Ph. D., Montreal,
All Rights Reserved ---
Skype
or Email to contact.