Intermediate Value Theorem

Calculus
Section 2.5

The authors of the book are not too clear explaining the Intermediate Value Theorem.
So, using basic jargon, what exactly is the I.V.T?

 

The idea behind the Intermediate Value Theorem is this:



When we have two points connected by a continuous curve:

• one point below the line
• the other point above the line

then there will be at least one place where the curve crosses the line!
of course we must cross the line to get from A to B!

Here is the Intermediate Value Theorem stated more formally:



When:

• The curve is the function y = f(x),
• which is continuous on the interval [a, b],
• and w is a number between f(a) and f(b),

Then there must be at least one value c within [a, b] such that f(c) = w
In other words the function y = f(x) at some point must be w = f(c)

Notice that:

• w is between f(a) and f(b), which leads to

• c must be between a and b
  It also says "at least one value c", which means we could have more.
  
  Here, for example, are 3 points where f(x)=w:

• 

• How Is This Useful?
  Whenever we can show that:
  • there is a point above some line
  • and a point below that line, and
  • that the curve is continuous,
  we can then safely say "yes, there is a value somewhere in between that is on the line".

 

I will study this reply during the week on my lunch break. Like I said in another thread, I will not solve Intermediate Value Theorem problems tonight due to extremely hot weather in NYC. My room feels like an oven even with the fan on.