f'(c)=f(b)-f(a)/b-a
What this means is that the slope of the tangent line is parallel to the slope of the secant line. Both of the lines will have the same slope and be equal to one another.
An example would be where:
f(x)=cos(x)+1
the blue line would be the tangent line (y=2) and the green line would be the secant line (y=1) the interval would be (-2,2). they are parallel to each other and their slopes equal on another as well.2. Why does the Mean Value Theorem only work for continuous and differentiable functions?
- A continuous function has the same limit from both the negative and positive sides.
- A diferentiable function has a slope at a certain point, If f is differentiable at a point c, then f is continuous at that point c.
Here are some examples:
Differentiable but not continuous:
In the function: x^(-1)+1, there is a hole at x=0, making it differentiable but not continuous because there is a hole discontinuity in the graph.
Continuous but not differentiable:
The function:x^(1/2)+1 on the interval [-1,1] is continuous, but not differentiable because there is a cusp at x=0.
shouldn't u set interval for your first section? how does a cusp makes a function continuous but not differentiable?
ReplyDeletefor your graph of f(x)= x^(-1)+1, their is a hole of discontunity becase the limit as x approaches 0 is different from the negative side than the positive side
ReplyDeleteI like alot. I think you should explain the 2nd question more.
ReplyDelete