



By now, you have seen
equations like these:






Y = X^{2}
 3X + 5



and 


F(X)
= X^{2}  3X + 5 





The difference between these
two is just picky details. 


Just think of
them as the same. 


In these pages, we will often
talk about F(X) or
G(X)
or something like it. 


This is just a
way of saying: "some equation where X is the variable." 


Like the X ^{2}  3X +
5 stuff we just had. 





If we talk about F(t) or
G(t),
that just means we are talking about some stuff 


where the variable
name is t. 


Like maybe: 





F(t)
= 16t^{3}  4t^{2} + 2 





When we solve a
function for some X (or t) value, it's usually no big deal. 


We just put the
number into the equation in place of the variable 


and work out the
arithmetic. 





If we wanted to
solve the function above for t = 2, we just put in 2 for t. 





F(t)
= 16t^{3}  4t^{2} + 2 


F(t)
= 16(2)^{3}  4(2)^{2} + 2 


F(t)
= 16 x 8  4 x 4 + 2 


F(t)
= 128  16 + 2 


F(t)
= 114 





No biggie, eh? 





What if we have
this: 








First, we can
use algebra to simplify it: 





F(x)
= 
X^{2}
x X
^{1} 

X
^{1} 






F(x) = X^{2} 





Maybe you
remember this function from algebra as a parabola. 


The graph of
the function looks like this: 











BUT, we have a
problem. 


Our original
function had a denominator, X. 


When the
denominator equals ZERO, 


the function will not work out to some nice
value: 





Putting 0 in
for X in our original function, we get: 

















And THAT puppy
is undefined. 





We can't deal
with a zero in the denominator... 


YET! 


If we
graphed the original function, we get: 








X 
 1 
 0.7 
 0.3 
 0.2 
 0.1 
0 
0.1 
0.2 
0.3 
0.7 
1 
F(X) 
1.00 
0.49 
0.09 
0.04 
0.01 
undef 
0.01 
0.04 
0.09 
0.49 
1 






Up to now, if a
denominator on one side of a function was zero, 


we just said that
the function was undefined. 





WAIT A MINUTE! 


Are we saying
that there is a way to get around that zero in the denominator 


and
find an answer? 


YUP! 





Differential
Calculus could also be called: 


"How to
get an answer when you have a zero in the denominator" 





Here's how it
works. 


If you try to
solve an equation for a value of X that makes the denominator zero, 


you get nowhere fast. 




But... 


You CAN solve
the equation for values of X 


a tiny bit bigger or smaller than the
problem value. 





We want to see
what happens to the value of the function 


as we get really, really,
close to the value of X 


that gives us the denominator zero. 


Sometimes, the
value of the function gets closer and closer to some number. 





That's what
happened in our example. 


As X got closer
and closer to the problem value, 


the function got closer and closer
to a value. 


It just so
happened in this case, 


that both the problem X value and the
function value 


we got closer and closer to were the same number.
Zero. 





It usually
doesn't happen that both numbers are the same. 





Here's the big
leap to calculus... 


We say: 


If the closer
we get to some undefined X value, 


the closer the function value gets
to some number. 


And the
function value we get closer to is the same 


if the close X value is
bigger or smaller than the undefined X value. 





WE JUST GO
AHEAD AND SAY THAT NUMBER 


IS THE VALUE OF THE FUNCTION AT THE X
VALUE 


THAT MAKES THE FUNCTION UNDEFINED. 





That's
Calculus! 





It's basically
saying: 


Yeah I KNOW
that the function is undefined at that X value, 


but if it wasn't
what would the value be. 





It's an amazing
thing. 


If you said
something like that in Algebra, 


the teacher would have probably
smacked you around. 


With words,
anyway. 


But it's the
guts of Calculus. 





Math types have
invented all kinds of fancy symbols and terms 


to try to hide behind. 


But that's the
deal. 





Differential
Calculus is division where the denominator is zero 


and we use the
little "What would the answer have been" scam 


to get the
answers. 





The same scam
makes Integral Calculus work too, but we get to that later. 





The scam has an
official math name. 


It's called
Finding the Limit. 





They say: 


We are finding
the limit as X approaches 


(the value that makes the function
undefined). 





Example: 


Find out what
the function value is when X = 2 if it wasn't undefined for ... 


Oh wait, we're
in calculus now 


OK, Find the
limit of F(X) as x approaches 2 for the following function: 











x 
1.80 
1.90 
1.95 
1.99 
2 
2.01 
2.05 
2.10 
2.20 
F(x) 
3.24 
3.61 
3.80 
3.96 
undef 
4.04 
4.20 
4.41 
4.84 






So when X
equals 2 we have big problems. 


The denominator
equals zero and the function is undefined. 


But it sure
looks like the closer X gets to 2, the closer the function gets to 4 


Two things are
really important here. 


One is that the
function value gets closer and closer to 4 


if X is bigger than 2 or
if X is smaller than 2. 


The other is
that the value the function is getting closer to 


is an actual
number, not infinity. 





If those two
things are true, 


same function value with X values bigger and
smaller than the problem value. 


And the
function value we are closing in on is a number, not infinity. 





We get to use
the scam and say that is the Limit of the function 


as X approaches
the problem value. 


And treat that
function value as if it were the answer we would actually calculate. 


If one or the
other of the two things are not true, 


math types say the limit does
not exits. 


That means
we're stopped again. 


At least for
now. 





Of course math
types invented notation for this. 





If you wanted
to find the limit of something as X approached 2, you would write: 











in front of the
thing. 


Lim is the
abbreviation for Limit 


WOW We saved
two whole letters! 





So for the
example we did, 


we would take the limit of both sides as X
approached 2. 


We would write: 


Lim 
F(X)
= 
Lim 
X^{2}(X2) 

X
®
2 
X
®
2 
(X2) 



Now just
simplify: 


Lim 
F(X)
= 
Lim 
X^{2} 
X
®
2 
X ®
2 






And if the zero
problem is gone from the denominator, 


substitute in the limit value
for X and solve. 





F(2) = 2^{2} 


F(2) = 4 





Here's the
graph of the whole thing: 








If we can't
factor away the terms that will make the denominator equal to zero, 


we're stopped. 


The limit does
not exist. 


If we can
factor those terms away, 


the next step is to put the "Lim"
notation up next to the function. 


Lim is very
powerful. 


You can use it
with any type of math function there is. 


Yes it slices,
it dices, it makes mounds and mounds of julienne fries 


and it's not
available in any stores. 





Lim is also
kind of blind. 


It doesn't
notice anything in a function except the variable 


named right below
the Lim. 


If we had the
expression: 





3X + 2Y + 5Z 





and we wanted
to find the limit as Z approached 3: 











The Lim will
ignore the X and the Y and go right for the Z. 











It substitutes
the number below it for Z 





3X + 2Y +
5 x 3 





and simplifies
the expression as much as possible. 





3X + 2Y +
15 





copyright 2005 Bruce
Kirkpatrick 
