Transcript Aim: How can we use the Mean Value Theorem to solve problems?

Aim: How can we use the
Mean Value Theorem to solve
problems?
By: Nihir Shah
The Mean Value Theorem
The Mean Value Theorem is a very simple phenomenon.
Here is a defination of the Mean Value Theorem from
Wikipedia:
In calculus, the mean value theorem states, roughly,
that given a section of a smooth curve, there is a point
on that section at which the derivative (slope) of the
curve is equal (parallel) to the "average" derivative of
the section. It is used to prove theorems that make
global conclusions about a function on an interval
starting from local hypotheses about derivatives at
points of the interval.
You must be like “what in the world is this?” Allow me to
explain in simple terms:
My definition (the simple one)
By this point in the so-called “calculus” era, many
of are probably familiar with the first derivative.
We have also used slopes before. So here is my
defination of the MVT.
Mean Value Theorem: Find the slope using the
slope formula, then find the first derivative of
the function. Then, connect the two . Easier
than the abstruse definition of those Wikipidians!
Example 1:
Problem 1:Find a value of c such that the
conclusion of the mean value theorem is
satisfied for
f(x) = -2x 3 + 6x - 2
on the interval [-2 , 2]
(Borrowed from
http://www.analyzemath.com)
Give it a shot, the answers on the next
slide:

Solution:
What did I tell you? Slope time is first. Slope is simply the
(f(b)-f(a))/ b-a. So let’s find the “y” values first:
f(-2)= -2(-2)3 + 6(-2)-2= 2
f(2)= -2(2)3 + 6(2)-2= -6
Plug that into the slope formula, and you’ll get the slope as
-2
Now, find the first derivative: f '(x) = -6x2 + 6
Get that… I knew you would! Now simply put them
together. -2= -6x2 + 6. Solve for that, and you’ll get
C=+ or - 2radical2/3 element of [-2,2]
Another one, Please!!!
Okay then, your wish is granted: this is simply a change of numbers
Just be careful: Borrowed from the Visual Calculus website.
f(x)= x3 -6x2+9x+2 [0,4]
Now, a bell should ring in your head, it should tell you,
SLOPEEEEEEEEE!!!
So…
f(0)=2 and f(2)= 6. You know the formula for the slope. You will simply
use the formula and get, the slope as 1. Now, you find the first
derivative: it is: f’(x)= 3x2 -12x+9. Now, comes the best part, the
connection. So, it follows: 1=3x2 -12x+9. Now just solve for x. You
will find that x = 3.154700539 and x = 0.845299461. So the
answer should be c= 3.154700539 and 0.845299461 over
the interval of [0, 4] Simple enough right, now well get a bit
more complex!!!
The trigonometry
Sometimes, intimidation arrives with the presence
of the trig king. Well, don’t worry, we use the
same steps
So…
Use the mean value theorem (MVT), to find c.
f’(x)= sin(2x)+cos(x) [0, 2pi]
Take a deep breath relax), it’s not going to bite!
Just try it as if it were a regular problem.
The answer:
Did you try the problem?
Here it goes…
find the slope: f(pi)-f(0)/ pi-0. You should wind up with a
slope of -2/pi
Now, remember, what did I tell you to do? FIND THE FIRST
DERIVATIVE.
f’(x)=cos(2x)-sin(x). Now its time for the connection. Is it
sticking to your head now??? Ok. So it’s -2/pi=cos(2x)sin(x).
After the equation is solved, you should get:x =
0.7704383989 and x = 2.371154255. So, your final
answer would be c= 0.7704383989, 2.371154255
element of the interval [0, 2pi]