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.

First Varying Velocity Example

Problem:   Graph the distance d to the origin of a path versus time t for the following journey of Harry Snail.

1. At two o'clock in the afternoon, he is 50 km west of the origin, he travels further west at 100 km/hr. He drives at this speed for 1[1/2] hours.
2. At half past three in the afternoon, he stops for one-half hour.
3. He then drives eastward at 75 km/hr for the next two hours and then stops for another 2 hours.

Solution.

The trip has five segments. Comments on each segment or portion follow.

1. Before his trip begins, he could be stationary, that is, not moving. This possibility, a suspicion which cannot be confirmed, is represented by the horizontal dashed line. The slope of this speculative dashed line is

m = 0 km 2hr = 0

So his slope or speed m is 0 or 0 km/hr, as you like. The dashed line in the above diagram could have and probably should have been left out. 

Footnote: When in doubt leave out, is a rule to follow in solutions of problems. Or, when in doubt say so, to show what is certain and what is not. Your credibility is at stake. Indicating precisely where you are guessing in a solution, identifies a question to be answered later by yourself or your instructor. And in marking assignments or tests, I would be less severe with mistakes explicitly identified as guesses than I would be with guesses deceptively presented as sure knowledge. Caution: Not all instructors will have this opinion.

2. The first described portion of the trip starts at the point A = (2 hrs,50km). He reaches the point B = ([3½] hr, 200km) after traveling at 100 kilometer per hour for one and a half hours. The slope of this portion of the trip m = 100[ km/hr] = 100 km per hour.

3. The second described portion of the trip lasts for one half hour. By remaining stopped (stationary) for [1/2] hour, his (t,d) coordinate changes from B = (3[1/2]hr,200km) to C = (4hr,200km). The slope or speed m here is again zero.

4. By traveling at 75 kilometers per hour back towards the origin for two hours, his position coordinates (t,d) change from C = (4hr,200km) to D = (6hr,50km). The slope

m = rise run = -75 km hr = -75 km hr

5. Finally, he does not move for 2 hours. This gives the last portion of the graph with d = 50km and slope m = 0.

 [ Back ] [ Up ] [ Next ] [Top of this Page]  
www.whyslopes.com?

Road Safety Message  Do not walk on a road with your back to the traffic - rule of thumb
Please report by email,  errors in mathematics or grammar or terminology to site author
If a mathematics topic you need is not covered in site pages,  report that as well. Topics in most demand
will be covered first in site growth.  

All trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster. 
The Rest © 1995 onward by site author,   Alan Selby,  All Rights Reserved.