Showing posts with label mean value. Show all posts
Showing posts with label mean value. Show all posts

Saturday, May 4, 2013

Interactive: Mean Value Theorem for Integrals

Mean Value Theorem for Integrals: If \(f\) is continuous on \([a,b]\), then there exists a number \(c\) in \([a,b]\) such that \[\int_a^b {f\left( x \right)dx = f\left( c \right)\left( {b - a} \right)} \]


Instructions: Drag a and b to see c calculated dynamically. The area under the curve from a to b is the same as the area of the pink rectangle formed by b - a and f(c).



Interactive: Mean Value Theorem


Mean Value Theorem: Let \(f\) be a function that satisfies the following:
  • \(f\) is continuous on a closed interval \([a, b]\).
  • \(f\) is differentiable on the open interval \((a,b)\).
Then there is a number \(p\) in \((a,b)\) such that
\[f'\left( p \right) = \frac{{f\left( b \right) - f\left( a \right)}}{{b - a}}\]

Instructions: With the mouse, move points A and B along the function to see the Mean Value Theorem in action. Refresh browser to start over.




YouTube Video Lectures by Rob Shone