Content Guide
Foreword
Chapter Descriptions
1. Introduction
Calculus Appetizer (1983)
2. Second Preview Begins
2 Skier in Motion (V)
2 The Skier (V)
2. Position Dependent (V)
3 Slope & Extrema (V)
4 Single Factor Analysis (V)
4 Two Factor (V)
4 More Factors (V)
4 With Divisors (V)
5 Maxima & Minima Tests
6 Jumps & Discontinuities
8 Review  (optional)
9 On Calculus Studies
11 Slope of Slope
13  Acceleration
14 Limits & Error Control (V)
14 Limit of a Fn.
14. Limited Error Control
14 Signif. Digits
14 Cauchy Limits
14 Sequence Limits
14 Decimal Arith.
15 What is Slope (V)
15 Slope Calculation (V)
15 Slope, a Limit
15 Tangent Lines
15 Linear Approx.,
15 Limits via Algebra (V)
15 Recap.
PS.Chain Rule for Polys
PS Chain Rule- General  (V) -
PS More Chain Rule (V)
PS - Sign Analysis (V)
16 What is Velocity
17  What is Area
18 Integration
18 Area Calculation
18  Fn DefN, 6 Ways
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc

Units in Calculations:
7 Velocity
7 Varying Velocity Example
7. Velocity Calculation
7 Changing Units
7 Same Velocity  Motions
10 Slopes without Units.
10 Units & Slopes
10  Units in Cost vs. Quantity
10  How Units  Appear
10 Unit  Elimination
10 Partial Elimination
10 Interest & Units
12 More on Units

Appendices:

Pigeon Hole Principle
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side Theorem
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 10
Slopes and Units

For a straight line graph of a first quantity Q₁ versus a second quantity Q₂, the slope

m = change in Q1 change in Q2 = DQ1 DQ2 = rise run ·

The units of the slope m are therefore given by

units of Q1 units of Q2 = units of the rise units of the run

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).
Examples of ratios of units that often occur are:

$ kg ,    km min ,   km hr ,    miles hr , 1,   etc.

Observe that when Q₁ and Q₂ are measured with the same units, the slope m has no units. It is given by a (real) number. An example follows. The purpose of the unit 1 in the previous list is to be a reminder of this. We may say that a real number has the unit 1 to include numbers in any discussion of quantities. Alternatively, we may say, as convenient, that real numbers are without units. See next example.

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.

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