Intermediate Value Theorem: Calculus 1 Project by Dongju Park
Overview
This Project has been completed as part of a standard 10 weeks Calculus 1 asynchronous online course with optional WebEx sessions during Summer 2021 Semester at MassBay Community College, Wellesley Hills, MA.
Intermediate Value Theorem
Introduction
We can assume that if we hear from the news that today's temperature rises from 40°F to 55°F, then there will be times when the temperature is 50°F or 55°F today.
Surprisingly, whether the complex shape shown below can be cut in exactly half from any angle can be solved by a similar principle as the previous example.
The Intermediate Value Theorem can be used, from intuitive areas such as temperatures to more complex areas like this.
Picture 1. Complex shape
Body
So, What is the Intermediate Value Theorem exactly?
The Intermediate Value Theorem states that if f is a continuous function in the closed interval [a, b], there is always a function value for any value between f(a) and f(b) in the open interval (a,b).
There is a very useful corollary; Bolzano’s Theorem.
- If the function f is continuous in intervals [a, b], and f(a)f(b) < 0, then there is at least one point c that satisfies f(c) = 0.
The name Bolzano from Bolzano’s Theorem represents the great mathematician Bernard Bolzano, who proved the Intermediate Value Theorem in 1817. He tried to exclude intuitive ideas like time and motion from mathematics, and he focused on proving theorems in a purely logical manner. To extend, he introduced the rigorous definition of a mathematical limit and used it to prove the Intermediate Value Theorem.
For the ε–δ definition of limits, refer to the site: Formal Definition of Limits
For the proof of Intermediate Value Theorem: Proof of Intermediate Value Theorem(Proof part in the Wikipedia)
Later on, in 1821, French mathematician Augustin-Louis Cauchy revised the function Φ into a constant function and completed the modern formulation.
Here are some examples using the Intermediate Value Theorem and its corollary, Bolzano’s Theorem.
It is pretty hard to find out the exact value of the root of these functions below.
However, using Bolzano’s theorem will help to know where the root is located approximately. To add, using differentiation first to sketch the curve of the graph will be helpful to find how many roots there are. Using Bolzano’s Theorem after this procedure will be more efficient.
Ex) \(f(x)=3x^3+2x^2-8x+2\)
Picture 2. Function 1
There are three roots of this function.
Since f(-3)<0, f(-2)>0, f(1)<0, f(2)>0, there is one root in the interval \((-3,-2) \bigcup (-2,1) \cup (1,2)\) respectively. (Bolzano's Theorem)
If you want to know more detailed roots, you can break the intervals into smaller pieces. For example, by knowing that f(0)>0 f(0.5)<0, we can narrow the interval (-2,1) into (0,0.5).
Ex) \((x^3-2x^2+8)~e^x~\log(x+2)\)
Picture 3. Function 2
There are two roots of this function.
As we know the property of the log function, we can conclude that it has a root of x=-1. However, it is hard to find the other root exactly, so we use Bolzano's Theorem.
Since f(-2)>0, f(-1.5)<0, there is one root in the interval (-2,-1.5). (Bolzano's Theorem)
For detailed roots, we can also repeat the process of example 1.
It can also be used in real life, as I mentioned in the introduction.
1. Temperature
Picture 4. Temperature Graph
Let’s say that there is a day with a temperature like a graph above.
As an example of the Intermediate Value Theorem, we know that there must be a time when the temperature is 47°F between 3:00 PM and 9:00 PM.
Let’s assume that the time is the x-axis and the temperature is the y-axis. The temperature graph could be considered a continuous function(f), and the Intermediate Value Theorem must work in this situation. Since f(3:00 PM)=55°F and f(9:00 PM)=46°F, all temperatures corresponding to 46°F to 55°F exist at least once from 3 to 9 PM. 47°F is a temperature value between 46 and 55, so there was a time when it was 47°F at least once in that time section.
2. Cutting Complex Shape in half
Picture 1. Complex shape
Let’s define a function f(x)=tanθ(x-a). (θ is the positive angle of the straight line and the x-axis.) Also, let’s define functions \(S_L\) and \(S_R\) as the size for the left and right sides of the function \(f\). Then we could finally define the continuous function \(h(a)=S_L-S_R.\)
Picture 1-1. Function f on the left
Picture 1-2. Function f on the right
In picture 1-1, \(h=S_L-S_R\) and \(S_L=0,~S_R>0.~\) So \(h<0\). In picture 1-2, \(h=S_L-S_R\) and \(S_L>0,~S_R=0.~\) . So \(h>0\).
Due to Bolzano's Theorem derived from the Intermediate Value Theorem, there must be a root of h between the lines from each picture.
If we try this procedure with another f with any different angle other than \(\pi/2\), we could also get that there is a root of \(h\). When the θ is \(\pi/2\), the function f will be the vertical line, and still, be able to use function h to determine that there is a line that cuts the shape into half. So we can conclude that we could cut this complex shape into exact half at any angle.
Conclusion
IVT is widely used in real life and can be helpful in the case of functions that are difficult to obtain answers to. However, there is a limit to the fact that it can only help assume in certain sections and that the exact location of the answer is not found.
Links to Class Wikis
Exponential Growth & Decay - Ignacio Rendon
Proof of Differentiation Rules: The Derivative of the Sum of Two Functions - Nick Woodward
References
https://en.wikipedia.org/wiki/Intermediate_value_theorem
https://www.youtube.com/watch?v=9xgO-EJ3sr0
https://en.wikipedia.org/wiki/Bernard_Bolzano