differentiation

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Loki123

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If the function f is continuous on a closed interval [a, b], and differentiable on an open interval (a, b); he gives:

a function f, monotonically non-decreasing for every x from the interval (a, b), where f '(x) is greater than or equal to zero.
function f, monotonically non-increasing for each x from the interval (a, b), where f '(x) is less than or equal to zero.


I am confused about the first part. What does f being continuous have to do with it being differentiable on an open interval? What does it mean when it's differentiable? An example maybe?
 
You can't have a derivative at the endpoints of a closed interval. Why? Suppose you have a function defined on the closed interval [a,b]. f'(a) and f'(b) does not exist by the very definition of the derivative. Recall that the derivative, f'(a), is defined as a limit and for a limit to exist, it must have the same limit approaching from the left of a and to right of a. Well, there is no left of x=a and no right of x=b. This is f'(a) and f'(b) does not exist.

The fact that f(x) is continuous on [a,b] has nothing to do with f(x) being continuous anywhere on (a, b). In fact, the reverse is true. If f'(x) is differentiable at a point it follows that f(x) is continuous at that point.

A function is differentiable at a point a means that the limit of the difference function exists. It means that there is just one tangent line to the function at x=a.

Also, who is he?
 
Steven G said:
You can't have a derivative at the endpoints of a closed interval. Why? Suppose you have a function defined on the closed interval [a,b]. f'(a) and f'(b) does not exist by the very definition of the derivative. Recall that the derivative, f'(a), is defined as a limit and for a limit to exist, it must have the same limit approaching from the left of a and to right of a. Well, there is no left of x=a and no right of x=b. This is f'(a) and f'(b) does not exist.

The fact that f(x) is continuous on [a,b] has nothing to do with f(x) being continuous anywhere on (a, b). In fact, the reverse is true. If f'(x) is differentiable at a point it follows that f(x) is continuous at that point.

A function is differentiable at a point a means that the limit of the difference function exists. It means that there is just one tangent line to the function at x=a.

Also, who is he?
Click to expand...
it*
 
Steven G said:
You can't have a derivative at the endpoints of a closed interval. Why? Suppose you have a function defined on the closed interval [a,b]. f'(a) and f'(b) does not exist by the very definition of the derivative. Recall that the derivative, f'(a), is defined as a limit and for a limit to exist, it must have the same limit approaching from the left of a and to right of a. Well, there is no left of x=a and no right of x=b. This is f'(a) and f'(b) does not exist.

The fact that f(x) is continuous on [a,b] has nothing to do with f(x) being continuous anywhere on (a, b). In fact, the reverse is true. If f'(x) is differentiable at a point it follows that f(x) is continuous at that point.

A function is differentiable at a point a means that the limit of the difference function exists. It means that there is just one tangent line to the function at x=a.

Also, who is he?
Click to expand...
Why can't there be more tangent lines?
 
Limits have one answer (assuming it has one). That limit is the slope of the tangent line. So only one tangent line.
 
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