Preparing for the Chain Rule
and for derivatives of inverse functions
Theorem on Linear Approximation: If z = f(y) and
f ¢(b) = lim
h ® 0f(b+h) -f (b)
hthen
f(y) - f(b) = f '(b) (y-b) + Rb(y)(y-b)
where Rb(b) = 0 = lim y® b Rb(y)
Proof:
Let E(y) = f(y) - f(b) - f'(b) (y-b)
Then E(b) = 0
and
| lim y ® b |
E(y) y-b |
= |
lim y ® b |
f(y) -f (b) |
- f '(b) | = | f '(b)- f '(b) | = 0 |
Now let
| Rb(y) = |
{ | E(y) y-b |
for y =\= b |
| 0 | for y = 0 |
Then Rb(y) is continuous at y = b. Moreover
E(y) = Rb(y) (y-b) when y =\= b and when y = 0.
Tangent Line Revisited
Illustration of Above Theorem
The above limit definition is motivated by the graphical expectation or suggestion that the slope of the line segment joining the non-moving point (x1,y1) to the moving point (x2,y2) should approach the slope of a tangent line at the non-moving point (x1,y1). The tangent line should be the limiting position of the line extending this chord. See the previous diagram.
Tangent Line Equation
When a skier is located on a curve y = f(x) at (x1,y1) = (x1,f(x1)), the slope of his or her ski is assumed to lie on a tangent line. This tangent line has (or is now assumed to have) the equation
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The linear function
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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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