Why Slopes
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| Vol 2, Three Skills for Algebra covers many topics in algebra and logic that students starting calculus should have mastered or will have to master. Also includes arithmetic review problems to catch common mistakes. A fourth skill gives a unifying theme for high school maths. |
Units in Calculations:
Appendices:
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.
Motion with the Same Velocity
Two small problems follow, to further examine the situation where two graphs have the same slope everywhere. The slope in distance versus time graph is given by speed or velocity.
Information for the first problem. Paul starts at 3 kilometers north of his home. From there, for two hours, he walks northward at 5 kilometers per hour. Then he stops for a one hour rest. Next for one hour, he travels southward at 6 kilometers per hour.
First Problem: How far is Paul from his starting point?
Information for the second problem. John starts at a distance d0 north of Paul's home. He matches Paul's speed and movements. From his starting point, like Paul, he walks for two hours northward at 5 kilometers per hour. Then he stops for a one hour rest. Next for one hour, he travels southward at 6 kilometers per hour. Second Problem: How far is John from his starting point?
Solution to both problems. A graph of Paul's motion is easily drawn with the help of the following table:
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In this graph
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Paul thus travels 7 km-3 km = 4 km from his starting point.
A graph of both Paul's and John's respective motions can be obtained from the next table.
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The distance of John from his starting point is also 4 km. The scale has been omitted from the vertical axis in the following graph since d0 is unknown.
This graph shows the case d0 > 3 km. How would the graph change if d0 was £ 3 km?
Questions
- What happens when two different graphs have the same slope over the same interval?
- Suppose you see two cars, one following the
other and matching its speed exactly.
(a) What would happen to the distance between the two cars?
(b) Would the two cars travel the same distance in any given period?
(c) How far would the following car travel if both started and stopped at the same time as the followed car?
(d) How would you make the distance between the two cars change?
- How would the graphs of the motions of the two cars be related if the motion of one always matched the motion of the other, but with a thirty second delay? In the same length of time, how far would each travel? Hint: How is the graph of d = f(t) related to the graph of d = f(t-a)?
- Two cars travel along a straight road with the first car traveling twice as fast as the second, but still within the legal speed limit. Suppose the distance traveled by the first car is d1 = f(t). Find a formula for the distance traveled by the second car d2. Hint: First consider the case where both cars start moving at the same time.