Math 308 2003 Term Project

Relationship Between Area and Derivatives

By Allan Liao

#20444030

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Table of Contents

Introduction

Abstract

Proof 1

Proof 2

Conclusion

References

**Introduction
**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.

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**Abstract
**If we take a point

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**

Let **f** be a function
**f(x) = x^2** and **P(x, f(x))** be a point on this function

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

**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(X i) Dx
**where

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