Preparing the for the chain rule
Linear Function Chain Rule
The graph of y = a x + b versus x gives a line with slope a. The graph of z = cy + d versus y gives a line with slope c. Then the substitution y = ax + b into z = cy+d yields a linear function of x whose graph is a line with slope ca. Reasons why follow.
z = cy + d =
= c(ax+b)+d
= cax +cb+d
= m x + k
where k = cb+d and m = ca is the slope of the graph of z versus x.
The chain rule for linear function could (pure speculation on my part) come from the industrial revolution and the distance traveled by chains or pulleys or gears when coupled together.
In the above examples, three disks are connected by bands which which assume do not slip. As the first band moves without slipping, the distances traveled by A and B around their respective circles are equal, denote that common distance by S. Likewise, As the second band moves without slipping, the distances traveled by C and D around their respective circles are equal. Denote that second distance by T. A gear ratio for this pulley system is given by S/T or its reciprocal. Let X, Y and Z denote the angle displacement of
A, B and D, respectively, in radians. Then the angle displacement of C is
y as well. Now,
Therefore
Therefore Y = (a/b) X and Z = (d/c)Y. So substitution implies Z = (d/c)(a/b) X |
Linear Functions
The graph of y = a x + b versus x gives a line with slope a. The graph of z = cy + d versus y gives a line with slope c. Then the substitution y = ax + b into z = cy+d yields a linear function of x whose graph is a line with slope ca. Reasons why follow.
z = cy + d =
= c(ax+b)+d
= cax +cb+d
= m x + k
where k = cb+d and m = ca is the slope of the graph of z versus x.
Calculus Appetizers
& Lessons
YOU are better than YOU think. Show yourself how:
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For better work & study skills, read chapters 2 in Three Skills for Algebra. Sooner is better. Good luck. |
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for site exits, follow the dove |
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Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
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Caution: Site advice is approximately correct, for some circumstances, not all. . That leaves room for thought and refinement.. |
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