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Home < Volume 3 Why Slopes - A Calculus Intro Etc << Chapter 10 Slopes and Units
[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 10
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That is, the units of slope is the ratio given by the unit of rise
(vertical axis) over the units of run (the horizontal axis).
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Improper Units
The unit 1 is an improper unit. The units 1.0 radian = 1, the degree 1.0 degree = [(p)/180] and the one percentage 1.0% = [1/100] are further examples of improper units, that is units that are multiples of real numbers. (The discussion of improper units is unique to this author.
Units in computations have been a concern for chemists and physicists but not so far for mathematicians.
A Slope Without Units
A portion of a ski trail y = h(x) is shown below. In this portion, the height y at position x on the horizontal axis is
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Here the slope
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Note 1% = 0.01, a number. Note that the grade of most roads varies from 0% to at most 12% in most areas. Warning signs about steep downward slopes may be seen by the sides of some roads.
Cost Versus Quantity
The metric BWGZ gravel supply business charges the amount
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For w = 2000 lbs of gravel, the cost would be
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FOOTNOTE: At the earth surface, a one pound mass and one pound weight are identical measures of material. There is a distinction between mass and weight that has to be considered for calculations away from the earth surface. Students of physics should know why.In computations, we follow the convention that units can appear before or after the number in it. In contrast, in the presentation of results, the placement of the unit before or after depends on grammatical or cultural preferences of a cosmetic nature.
Units in Slope Calculations
Suppose y = h(x). The units of the
slope m = h垄(x) are then
given by the ratio of the units of y over those of x. That
is,
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Elimination of Units
The work or accounting needed to eliminate or factor out units in computations, is indicated next.Let u1 = the unit of measurement of y. Let u2 = the unit of measurement of x. Then y = Y路u1 and x = X路u2 where X and Y are real numbers. The latter provide unit-free description of the two quantities x and y provided the units are known.
The expression y = f(x) is equivalent to an
expression Y = F(X) involving no units. In
particular y = f(x) holds if and only if
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The elimination of units has an effect on slope calculation. Slopes to the graph of a unit-free Y = F(X) curve have the units
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The elimination of units and the formulation of unit-free equations Y = F(X), for further manipulation or computation, require replacements such as y = Y路u1 and x = X路u2. This demands an explicit choice of units of measurement and algebraic representation for all the quantities present in the problem. It further ties the further computations to the choice of units. This formality represents extra work and an extra burden in computations or the solving of equations. In particular, equations y = f(x) involving quantities, can be manipulated, in the first instance, without a selection of units of measurement and algebraic representations for every quantity mentioned in the statement of a problem. Moreover, when an algebraic solution is found in terms of a subset of the quantities present in a problem, substitution of quantities into the formula leads to computations involving both numbers and units of measurements. Here different units or measures of length, mass, time etc may appear in various combinations, for instance [cm/( meter2)]. Such odd ratios can be converted as needed at the end of computations, and not necessarily before.
For complicated equations, eliminating units and obtaining a dimensionless (i.e unit) free formulation has some advantages. It may indicate some similarity between frequently done computations. But for simple once-only computations the benefits may be minimal, and the elimination of units could be an unnecessary step. Here others may disagree.
Partial Elimination of Units
Suppose y = f(x) as above. For the first case, let x = X 路u2. Then y = h(X) where h(X) = f(X路u2). In the equation y = h(X) units have been eliminated from the discussion of one quantity, but not the other. (A situation close to this occurs in the common treatment of simple interest and compound interest computations. See below.) Now slopes of the graph of y versus X has units
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For a second case of partial elimination, let y = Y路u1. Then
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Interest Rates and Units
First Example (Interest Rates Without Units). The amount A in a simple interest bank account at t years after a deposit of an amount (principal) P is given by
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Now the slope of the
above graph is
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To devise a second
approach, let
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Second Example (Interest Rate With Units). The amount A in a simple interest bank account at time T since the deposit of the amount (principal) P is also given by
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Now the slope
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Third Example (Interest Rate Without Units). With interest compounded annually, an initial deposit P grows to amount A = P(1+i)n after n years where i is the annual or yearly interest rate. The interest rate i is usually given as a percentage. Compute the final amount A in the case where an initial deposit of $100.00 compounds at 4% per year, for 3 years.
In the requested computation, i = 4% = 0.04, P = $100.00 and n = 3. Therefore
A = P(1+i)n = $100.00(1+4%)3 = 录.
REMARK.
Daily, weekly, monthly and annual
interest rates are given by percentages or pure numbers (the unit free
approach). For example, a yearly or annual interest rate of 5% is given
by the number 5% = 5 ×[1/100] = [5/100]. Second, interest rates per day,
week, month or year refer to a percentage over a period of time. With the
latter, for example [5%/year] represents a 5% per year
interest.
FOOTNOTE: Some conventions like these are needed for the consistent use of units in computations. Without any such conventions, the use of units in financial computations will depart from the practice in technology and science.
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Road Safety Messages and Questions: When and why should you face traffic when walking along a road or cycle path? Is it a good idea to hang limbs outside of cars etc? What gives more protection in a crash: a car, motorbike or bicycle? See too, the BBC-Belgium story Texting and Driving - texting & the impossible test - the article links to a gruesome utube video on the subject The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
May 2012, Composition Starting: Pre-School and Primary Mathematics - Quantitative Skills, An Intellectual View, Feedback Welcome: The 8 Most Popular Site Inlinks
20 Times Table - the most popular site page - popular pages -
unexpected. Division of Labour: This site offers advice and directions with pointers to resources elsewhere, if known, when they help or lessen the need to write more. Parent Center: Help your child or teen learn: Parent-friendly Work Booklets for ages 3+ to 13 Use these or others to check or build skills. Other booklets are available but these booklets allow parents unsure of themselves in mathematics to help their children. The selection acquired in Canada is published in the USA. So it has a US orientation. In retrospect, the selection shows parents what to check with the booklets or by other ways, the choice is theirs. But in retrospect, the selection does not cover integral and fractions liquid weights and measures - ask the publishers to correct that! For ages 9 to 12 say, parents may compensate by showing boys and girls how to use weights or mass, and further measures in food preparation. Beyond that children may be shown how to measure and calculate angles, lengths and areas [proportional amounts too] directly or by using maps and plans drawns to scale. Learning how to gather and measure all the ingredients, pots and pans for a dish or a meal, along with cleaning up sets the stage for like activities or experiments in science courses, and in developing organizational skills, gives boys and girls a head start. Good luck. At the other extreme, more comprehensive than light, if your motto is McCainian: drill, drill, drill then Toronto mathematician and actor John Mighton's jump math organization has jump math workbooks for at least grades 3 to 8 for at-home and in-school use - training sessions for teachers available. Jump math has been expanding to cover older students. Jump Math Samples: plus Fractions for Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8 [unread - likely to be good]. and Mathematics Skills For Ages 3 to 14 - technical! Skills with take home value - A few ideas Basic skills include time-date-calendar Matters; money matters; map, plan and scale diagram matters;counting, measuring and figuring; decision making with logic and likelyhood; being careful and being aware of the domino effect of mistakes; reading and writing with precision. Is your child able to add, subtract and multiply amounts of money, work with fractions, work with clocks and calendars, work with maps and plans, and measure length, weight-mass and volume? Schools may promote your son or daughter without providing basic skills in reading, writing and arithmetic. Arithmetic and Number Theory Skills
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Natural-Logarithms Exponentials Powers Roots
13 Lessons on Limits and Continuity
38 Lessons on Calculating Derivatives 4 Lessons on Using Derivatives 5 Lessons on Integration 12 Webvideo Lessons on Area and Volume Calculation Calculus Lessons Elsewhere: They cover basic topics in ways likely to complement your notes, your textbooks and site material. When Goldilocks trespassed in the house of the three bears, she found three bowls of porridge, two not to her liking, and one just right. Different bears have different tastes. As invited guest here and elsewhere, if one or more explanations is not to liking, try another. It may be better or just right. Unsolicited Advice Learning to do and high marks if it comes to easy is often deceptive - light rather than deep. For that reason, students with learning difficulties determined not to let it get in their way may go deeper and farther than those with none. High marks, if the come easy, may be deceptive - provide a too light and not a deep mastery. That could have been your problem in secondary school, one that leads to comprehension shock or difficulties in calculus and more generally in the first year of college. Bon Appetite. |
Home < Volume 3 Why Slopes - A Calculus Intro Etc << Chapter 10 Slopes and Units
[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]
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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 |
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