Mean Value Theorem

so let's talk about the mean value theorem what's the basic idea behind it so let's say if we have a function f of x let's say it looks like this let's call this point a and point b now if the function f of x if it's continuous on the closed interval a b

and if it's differentiable on the open interval a b then there's some number c where f prime of c is equal to f of b minus f of a divided by b minus a so what does this mean on the left side this represents the instantaneous rate of change which is the same as the slope of the

tangent line on the right side this is the average rate of change which represents the slope of the secant line now the secant line is the line that touches two points on a curve so we're going to focus on the points a and b to draw the secant line draw a line that passes through both

points now the secant line has to be parallel to the tangent line because they're equal to each other they have the same slope and notice that here the tangent line will have the same slope as the secant line they're both horizontal in this case now keep this in mind the secant line touches the curve at two points the

tangent line only touches the curve at one point so therefore c is right in the middle between a and b in this example so anytime you have a closed interval if the function is continuous on a closed interval and if it's differentiable on the open interval there's always going to be some number c

where the slope of the tangent line is equal to the slope of the secant line where those two lines will be parallel to each other