Mean Value Theorem

f'(c) = [f(b) - f(a)]/(b - a)

1. Explain what this means GRAPHICALLY by showing a good example!


The average rate of change of a function is the slope of the secant line. the secant line connects the function values at the endpoint of the interval. endpoints of the function on [a,b]are (a,f(a)) and (b,f(b)).
m=y2-y1/x2-x1 = f(b)-f(a)/b-a

the instantaneous rate of change at x=c is the slope of the tangent line to the function.


The function must be continuous on [a,b] and differentiable on (a,b) then the mean value theorem must have at least one value x=c between a and b == f'(c)=f(b)-f(a)/b-a. soooo at least one x value on the interval where the tangent line is parallel to the secant line that connects the endpoints of the interval.






the function= x^3
secant line= 3x ===> f'(x)=3x [0,2]
tangent line= 3x-2

tangent line is parallel to the secant line



Explain why this only works for continuous and differentiable functions.




sqr root of (x^2)-2
this is an example of a function that is not differentiable because it has a corner at x=0 within the interval of (-1,1) it is not differentiable but this function is continuous.

this function does not satisfy the theorem because it does no have a tangent that can be parallel to secant.




this function has a discontinuity but is differentiable.
piecewise function of
f(x)= {x^3+3 for x<1
{x^2+1 for x>(or equal to) 1
close hole at one and at the same time of open hole at one.

the function is not continuous within the interval of [-1,1]