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pdf


http://www.maths.uq.edu.au/courses/MATH2200/lectures/ode_runge.pdf


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--Last edited by saucer on 2007-03-07 10:46:04 --

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http://archives.math.utk.edu/visual.calculus/2/index.html




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Calculus: Derivatives

The Derivative of a function is another function which measures the rate at which the original function is changing. If the original function is linear then the derivative of that function is simply the slope(rate at which a linear function increases) of the original function. For non-linear functions, finding the derivative becomes a little more mathematical. The knowledge of derivatives provides a very useful tool for solving many kinds of problems.
Slopes:
Recall that the slope of a straight line is a numerical descrïption of the "direction" of the line. Another way of describing the slope is that the slope is y will change when x increases by one. The formula for finding slope is:
m = (y2 - y1) / (x2 - x1)
Where m is slope and y2 and y1 and x2 and x1 are the second and first y and x coordinates respectively.
If we know the slope of a straight line, we can tell if it goes uphill or downhill from left to right. If the slope is positive, the function goes uphill and if the slope is negative, the slope goes downill.

A horizontal line has zero (0) slope
A vertical line has undefined(infinite) slope

If f(x) is not a linear function, then unfortunately the change f(x) increases by when x
increases by 1  will not be constant.
Example:
Let f(x) = -4x + 9
If x is 5 then f(x) = -11
now let x be one higher than 5 which is six and f(x) = -15
A rate of change of -4 which is the slope.
But now lets try a NON - Linear function:
f(x) = x^2
then...
f(1) = 1
f(2) = 4
f(3) = 9
There is change of 3 between f(1) and f(2) but a change of 5 between f(2) and f(3)....
Therefore, there is no constant rate of change for a non linear function.
Tangents:
However at specific points on the graph of a non-linear function, there are "instantaneous" slopes.
But to find a slope, we need two points. There is nothing wrong with choosing two points very close together.
Example:
Find the slope of the following function at point ( 3,9)
f(x) = -x^2 + 8x - 6
We will fill in the slope formula with (3,9) as our first set and choose 3.1 as the "x" of the second set. To find the "y" of the second set, plug in 3.1 to the function and get 9.19.
With these numbers, we come up with 1.9 as the slope. We try it agin the second time but this time using 2.9 as the second "x" and 8.79 as the second "y" and we the slope to be 2.1
The slope must be between 1.9 and 2.1
Try to think of the two pairs of numbers we use as being two points on a curved line coming closer and closer to each other so we can find the "slope" of the spot right between them. If you draw this (a curved line with a straight line crossing it a two very close numbers [essentially one spot]) then you see a tangent line to the point you are finding the slope for.




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--Last edited by saucer on 2007-01-04 23:21:43 --

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Techniques of Differentiation


http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/Calcsumm4.html


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http://www.math.tamu.edu/~tom.vogel/gallery/node14.html#SECTION00041000000000000000


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good Link


http://mathdaily.com/lessons/Derivative


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http://216.239.59.104/search?q=cache:adPl_nIAbFIJ:www.tech.plym.ac.uk/maths/resources/PDFLaTeX/gradient.pdf+maths+derivatives&hl=en&ct=clnk&cd=5&client=firefox-a