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Starter Guide (Views)
Real Player Videos
4. Linear Approximation
4. Second Derivative Test
4. Sketch y = x^3 - 6x^2- 12x
4. Sketch y = x^3 - 3 x^2 - 9x
4. Sketch y = 1 - 1/(1+x^2)
Starter & Warm Up Lessons
1. Usual Review/Starter Lessons
2. Limits [13]
3. Differentiation Rules[28]
4. Applications of Derivatives [5]
5. Definite Integrals - Preview [5]
6. Integration Applications [6]
Advanced Material
| | More on the Slope Based Interpretation of Derivative
Linear Approximation of functions
Suppose for a given number k > 0, there exist an n > 0 such
that
|
ê
ê
ê |
L- |
Dy
Dx |
ê
ê
ê |
£ e
= |
½ |
·10-k |
|
whenever |Dx| £
d = [1/2]·10-n,
then the following holds whenever the inequality |x2-x1|
= |Dx| £
[1/2]·10-n is satisfied.
The difference L-[(Dy)/(Dx)]
= c for some number c with magnitude |c|
£ e = [1/2]·10-k.
(The number c will depend on x2.)
The foregoing implies
and hence that
The latter in turn implies
| f(x2)-y1
= y2-y1
= LDx-
cDx |
|
and
and hence
| f(x2) = y1+L(Dx)+an
error |
|
where the error is -cDx
and its magnitude
| |c·Dx|
£ |Dx|· |
½ |
·10-k
£ |
½ |
·10-n· |
½ |
·10-k |
|
The last inequality provides information about the error behavior in the
approximation of y = f(x2) by the linear
function y = y1+L(Dx)
= y1+mski(x2-x1).
Since x2 is arbitrary, the letter which plays it role is not
important. It can be replaced. In particular, x2 in the above
exposition can be replaced by a number x.
Theorem: [Consequences of a Non-Zero Slope] If the slope m = f¢(x1)
= L of f(x) at x = x1 is nonzero,
then there exist a d > 0 such that the sign of f(x)-f(x1)
equals the sign of L·(x-x1)
whenever |x-x1|
£ d.
Proof: In the previous discussion, choose k such [1/2]10-k
< |L| and let d
= [1/2]10-n.
This theorem implies if m = f¢(x1)
¹ 0 then no interior maximum nor minimum can occur
at x = x1. Finding all solutions x = a of
the equation f¢(x) = 0 identifies
locations x = a at which interior maximums and minimums might be
found. The latter can also occur at points where the slope or derivative f¢(x)
is not defined. The points x where
- f(x) is undefined, and
- f¢(x) is zero or undefined are
called critical points. On finite and infinite intervals, the
maximums and minimums of functions f(x) are located
- at critical points inside that interval, and/or
- at included endpoints.
So finding the critical points locates some, if not all, of the maximums and
minimums. This an extremum, that is, a maximum and minimum locating principle
for functions.
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Volumes (orders)
1, Elements of
Reason. 1996
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Curriculum Notes 1996
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3 .Why.Slopes.&.More.Math.1995
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5. Proportionality,
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19 Maps,
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