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Mean Value Theorem

Date: Fri, 18 Nov 1994 08:24:43 -0800 (PST)
From: Winnie So
Subject: question

     I am a senior at Monta Vista High School in Cupertino, California.  As of 
right now, I am a Calculus student.  The problem that I have as a student
right now is that I know how to work the problems that my teacher gives;
however, I don't know what the purpose of doing them is.  Basically I just 
memorize the formulas, then I plug in numbers for the different variables.  

     For example, there is the Mean Value Theorem for Integrals.  This 
Theorem states that "If f(X) is continuous in [a,b], then there is a 'z' 
in (a,b) such that the integral from a to b f(x)dx = f(z)(b-a).

     I realize that there are sample problems in the textbook that show how to
work the problems.  As I mentioned earlier, I don't understand WHAT the 
purpose of doing them is.  How does the Mean Value Theorem apply to life?  
Why do we need this theorem?

     Another question that I have is what fields or professions does Calculus 
lead to?  

     Please reply.  Thanks!

Winnie So

Date: Fri, 18 Nov 1994 08:10:10 -0800 (PST)
From: Winnie Fan
Subject: question


     My name is Winnie Fan.  I am a student in Calculus.  I have 
always been wondering about how people come up with the theorems, rules, 
formulas, etc.  I don't think I will ever figure that out.  Let's get to 
the main point.  Right now in my math class, we are learning to find area 
under a curve or area between two or more intersected curves and lines.  I 
don't know how these that we are doing can be related to real life.  I 
have taken a lot of math classes because I like them, but sometimes I have 
questions for myself that what is the point of learning this and that.  I 
hope I can get a specific answer for my questions.

P.S.  Do people in the Dr. Math service get pay for helping people, or is 
it a volunteer work?

                                                        Winnie Fan

Date: Sat, 19 Nov 1994 15:52:47 -0500 (EST)
From: Dr. Ken
Subject: Re: question

Hello there!

The two Winnies there (Winnie So and Winnie Fan) wrote basically the same
question to us, so I thought I'd just do a single reply to both of you.

In answer to your question about the Mean Value Theorem, I find that it's
quite useful in life, but let's get straight what we mean by the Mean Value
Theorem, because there are two of them.

A) When most people say "Mean Value Theorem", they mean the following: if a
function f is differentiable on the interval [a,b], then there exists a value
'z' in (a,b) such that f'(z) = (f(b) - f(a))/(b-a).  Here's one application
that you may have seen in class:

If policewoman Betsy sees me at the corner of Humboldt and Emerson Avenues in
Minneapolis at precisely 12:00 noon, and then her husband policeman Bubba
sees me sixty miles away at the corner of Grand and Fairview Avenues in St.
Paul at precisely 1:00 pm, can they give me a speeding ticket?  You bet they
can, because of the Mean Value Theorem.  It says that at some point in the
trip, my speed must have been equal to my average speed for the whole trip.
Since my average speed was a mile per minute, i.e. 60 mph, I must have been
speeding at some point.  I believe that the Commonwealth of Pennsylvania
actually uses this technique to nail speeders on the Pennsylvania Turnpike.
All they need to do is put a time-stamp on your ticket when you get on, and
look at the time when you get off.

B) The other Mean Value Theorem (which I always [perhaps erroneously] call
the Intermediate Value Theorem, or the Water-Leveling Theorem) is as you
stated it:  If a function f is continuous on the interval [a,b], then there
exists a 'z' in (a,b) such that (b-a)*f(z) = Integral[f(x) dx] from a to b.

The way to think about this theorem is pretty neat.  Think of the right side
as the area under the curve f(x) between a and b, and think of it as a bunch
of water.  Then the theorem says that when you let that water settle, the
line at the top of the water is going to intersect the graph of f(x) between
a and b.  It makes sense, doesn't it?  It makes more sense if you draw a
picture than if you don't.  See, the left side of the equation is the area
of a rectangle with base (b-a) and height f(z).  Neat stuff.

In answer to your final question, Calculus leads to all sorts of fields.
For instance, if you go to college and study any science field, you'll
almost certainly be required to take Calculus.  But don't get the impression
that it's the be-all end-all of mathematics.  There's plenty beyond that,
like Linear Algebra, Topology, Knot Theory, Differential Geometry, Set
Theory, Number Theory, and the list goes on and on.  Calculus is _certainly_
a prerequisite for an in-depth look at all of these fields of study.

I hope this helps you folks out!

-Ken "Dr." Math

Sorry, I forgot to answer this question in that last message!

:P.S.  Do people in the Dr. Math service get pay for helping people, 
:or is it a volunteer work?
:                                                       Winnie Fan

Right now, it's strictly a volunteer effort.  Nobody gets paid except 
our supervisor, Steve Weimar, who has tacked this on to his already 
busy internet schedule.  We just do it for fun, believe it or not!

-Ken "Dr." Math
Associated Topics:
High School Calculus

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