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.

Changing Units (Digression)

Multiplying by the number 1 does not affect the value of an expression, but the number 1 can be written in many ways. Some, not all, are helpful. These different ways can sometimes help in changing the units used for the expression of a quantity. A few slope or velocity based examples follow. Note that [60min/1hr] = 1 implies the following

m1 = - 4 3 km min = - 4 3 km min ×1 = - 4 3 km min × 60min hr = - 4 3 · 60 1 km min min hr = 4 1 · 20 1 km hr = -80 km hr m2 = 8 3 km min × 60min hr = 160 km hr m3 = 0 m4 = - 1 2 km min × 60min hr = -30 km hr m5 = - 3 2 km min × 60min hr = -90 km hr

In summary, multiplying by 1 = [60min/hr] does not change the speed or velocity m, but it does help to change the units from minutes to hours. Note to do the reverse change, multiply by

1 = hr 60min

instead. Note that care must be taken in selecting the ratio of units whose value is 1. A faulty choice will introduce more units instead of permitting some to cancel.

Remark.   An alternate method is to substitute 1 min = [1/60] hr into an expression. For example

m2 = 8 3 km min = 8 3   km 1 60 hr = 8 3 · 60 1 · km hr = 160 km hr

as before. Substitution replaces one unit by an expression for it in terms of another. With this method, there is little or no hazard of introducing units that don't cancel, but the algebra or arithmetic requires a little more thought. The choice of unit conversion method depends on where you would like to do the work or reasoning.

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