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1. Derivatives.
Before beginning to actually discuss derivatives (the key to this entire course btw), let’s go over some
background.
I can not emphasize enough how important it is for you to grasp these basic concepts that I will
present.
If you need help, please ask.
If not in class, then during my office hours, or get a tutor, study in
groups etc.
Background material
Functions
.
Consider the function
.
If x=1, 2, 3, then…
Putting this into a chart yields….
x
f(x)
1
1
2
4
3
9
Graphing this gives us…
A (1,1)
C (3,9)
B (2,4)
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View Full DocumentLimits.
A limit of a function essentially gives you the value of a function as the variable approaches a
certain number.
Using the same function
consider the following example…
What this question is actually asking us to do is to find the value of f(x) as x “approaches” 2.
Consider the
following table….
x
f(x)
3
9
2.3
5.29
2.1
4.41
2.05
4.2025
2.01
4.0401
2.001
4.004001
2.0001
4.00040001
Hopefully you can see that as the value of x approaches 2 the resulting value of f(x) approaches 4.
Frankly
in this case the simple thing to do is to just plug in the 2!
Note that f(2)=4.
However, it is NOT always that
easy!
An Introduction to Derivatives.
Now let’s consider how to find the “average rate of change” (i.e. slope) at various places on the above graph.
Find the average slope of the curve between A and B, between B and C, and finally between A and C:
Slope between A and B:
Slope between B and C:
Slope between A and C:
Now let’s suppose you wanted to figure rate of change when x was EXACTLY 2.
We could think of (2,4) as
our first point and then use other points that get progressively closer to this point.
We found the slope above
between B and C (i.e. between when x=2 and x=3).
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 Spring '08
 KUSTIN
 Derivative

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