limits calculus 2.1b part 2 of 2. 9/13/2012 – today’s learning objective: differentiability and...

Limits

Calculus 2.1b part 2 of 2

9/13/2012 – Today’s Learning Objective: Differentiability and Continuity

Find the derivative with respect to t for the function y = 2/t.

Differentiability and ContinuityThe derivative does not exist (the function is not differentiable) at any point where the function is not continuous. In other words, wherever the function is not continuous it’s also not differentiable.

But even if the function is continuous at a point, it still might not be differentiable there. These three conditions must be fulfilled for the function to be differentiable at a given point:

1. The function must be continuous there. 2. The graph must be smooth there, with no sharp bend. 3. The tangent line must be oblique (slanted) or horizontal, not vertical.

Wherever any of those conditions is violated, the derivative does not exist at that point. Turning it around, if the derivative does exist at a given point, then you know that all three of the above are true at that point.

Graphs of functions and their derivatives. xf xf 1

x -4 -3 -2 -1 0 1 2 3 4

f

f-1

Graphs of functions and their derivatives. xf xf 1

x -4 -3 -2 -1 0 1 2 3 4

f

f-1

Graphs of functions and their derivatives. xf xf 1

x -4 -3 -2 -1 0 1 2 3 4

f

f-1

Alternative Form of Derivative

requires that one-sided limits to exist and be

=equal

derivative from the left derivative from the right

If f is differentiable on (a,b) and if the derivative from the right at a and the derivative from the left at b both exist, then:

f is differentiable on the closed interval [a,b].

Sharp Turns

Vertical Tangent Line

Differentiability implies Continuity

If f is differentiable at x = c, then f is continuous at x = c.

Continuity does NOT imply differentiability.

#111 p104 33-37 odds, 39-42, 73, 74, 83-90