Math 308 2003 Term Project
Relationship Between Area and Derivatives
By Allan Liao
#20444030
Abstract For any function f the area below the height
f(x) and above the line x = 0, will form a function F and
F' = f will be the derivative of this integral function Proof 1 If we draw
a line L through point P(x,f(x)) tangent to this function, and we
can say that the slope of L is the limit
(lim h -> 0) [f(x
+ h) – f(x)]/ h where is a small non zero number
For example, lets define P as
the point (1, 1) lets take a non zero h and define a point Q(1+h,
f(1+h))
Now lets consider the fact that we
can join these two points in a line PQ, as you can see the line
connecting PQ is not yet tangent with P(1,1) but it is getting
there! Lets now decrease the non zero number h and decrease it until it is
infinitesimally small (lim h -> 0) As h becomes
smaller, the point Q will slowly approach P, and the tangent line
will become more apparent, the line L is the true tangent line at
P(1,1) and line to the right of it, is the approximated line PQ
(mathematical explanations in the conclusion)
Consider a function f(x) = x^2 where x
is 0<=x<=1 Using n
rectangles, we can approximate the area under this graph. so in this case, the
base width of the rectangles will be 1/n and the end points of these
rectangles will be defined by i/n. So the
height of a rectangle would equate to f(i/n). And with this
particular function, f(i/n) = i^2/n^2. Thus the area of one
rectangle will be (i^2)/(n^3) (in this example),
Here are various approximations with varying sizes of "n" Thus by looking at these rectangular approximations, we can determine
the area under the graph. (mathematical explanations in conclusion)
The area under the graph is determined by the sum of these n rectangles.
Rectangles which have the area of f(Xi) Dx
where Dx is the width of a
rectangle. Hence if we add up these rectangles we will have the entire area
under a function. So ultimately we can say the area A of a rectangle is
determined by the integral F =
Table of Contents
Introduction
Abstract
Proof 1
Proof 2
Conclusion
References
In this presentation I will explain the relationships between
the derivatives and areas. More specifically, how we can use derivatives
to describe a tangent line on a graph function. And also how the derivative of
the area below a function, is the function itself.
If we take a point x = p of a function f, the
derivative of this point will define the slope of the tangent line passing
through (p, f(p)).
Let f be a function
f(x) = x^2 and P(x, f(x)) be a point on this function
n=1
The slope of a tangent line on a point P is defined by the limit
(lim h ->
0) [f(x + h) – f(x)]/ h such that when h becomes smaller, the point Q
will merge into P and ultimately cause a tangent line at P(1,1) to
be formed (in this case). The more common name for this limit is called
the derivative.
The most interesting fact between the area and derivative is this. If we use a standard method of graphing F (using x as the x coordinate and F(x) as the y coordinate) we can see that in most cases, F also spans a function. And if we take the derivative of that function, the resulting slope of a tangent line is the same as F' = f.
http://archives.math.utk.edu/visual.calculus/2/tangents.8/
http://www.ima.umn.edu/~arnold/graphics-j.html
http://ugrad.math.ubc.ca/coursedoc/math101/notes/integration/area.html