Question

Hey...how do you use the mean value theorem in calculus?

Answer 1

example? it is used in different ways

Answer 2

it is used for example to find the bound of a function at a point
it is also used to prove the fundamental theorem of calculus

Answer 3

a basic example? that would explain the idea

Answer 4

so you have a specific problem? i can make one up if you like

Answer 5

no specific problem

Answer 6

usually they come in one or two ways:
the silly ones that say for example:
find the number \(c\) in the interval \([1,3]\) guarenteed to exist by the mvt for the funciton
\[f(x)=x^2-6x+6\]
your job is to write
\[f'(x)=2x-6\]
\[f(3)=-3\]
\[f(1)=1\]
\[\frac{f(b)-f(a)}{b-a}=\frac{-3-1}{3-1}=-2\]and then solve
\[2c-6=-2\] for c to get
\[c=2\]

Answer 7

how is f(1)=1

Answer 8

or they come like this:
suppose \(f(1)=5\) and \(f'(x)<1\) for all x what is the biggest \(f(4)\) can be ?

i just made that example up. for the function i made up \(f(x)=x^2-6x+6\) we have \(f(1)=1-6+6=1\)

Answer 9

you can make up a different one, and if you want a better understanding, convince yourself that the "c" that must exist by the mean value theorem is always in the center of the interval if your function is a quadratic (like the one i made up)

Answer 10

You can use it to show
\[ |\sin(x)-\sin(y)| \le |x-y|
\]
for all x and y.
In general if you have a function such that
\[|f'(x)|\le K
\]
for all x
then
\[| f(x) -f(y) | \le K |x-y|
\]
for all x and y

Answer 11

eliassaab wrote an example like the second one i wrote, since we know that the derivative of sine is cosine and therfore can never be bigger than 1
this is probably a homework problem in your text

Answer 12

ohhh i get it! thank u so much!

Answer 13

yw