1.5 Differentiability and Continuity. In a previous lecture, we defined differentiability of f(x) at x = a in terms of a limit. Recall that if f(x) is.

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Presentation transcript:

1.5 Differentiability and Continuity

In a previous lecture, we defined differentiability of f(x) at x = a in terms of a limit. Recall that if f(x) is differentiable at x = a, we can evaluate the following limit to determine f’(a). Conversely, if this limit does not exist, then f(x) is nondifferentiable at x = a.

There are many geometric representations of f(x) for functions that are nondifferentiable at x = a. These can result if f(x) has no tangent line at x = a, or if f(x) has a vertical tangent line at x = a.

A railroad company charges $10 per mile to haul a boxcar up to 200 miles and $8 per mile for each mile exceeding 200. In addition, the railroad charges a $1000 handling charge per boxcar. Graph the cost of sending a boxcar x miles.

If x is at most 200 miles, then the cost C(x) is given by: C(x) = x dollars If x exceeds 200 miles, then the cost will be C(x) = (x – 200) = x So the function C(x) is given by

The graph of C(x) is

Continuity

Continuity is closely related to the concept of differentiability. We say that a function is continuous at x = a if its graph has no breaks or gaps as it passes through the point (a, f(a)). If a function f(x) is continuous at x = a, it should be possible to sketch its graph without lifting the pencil from the paper at the point (a, f(a)).

The following graphs depict functions that are not continuous.

If f(x) is differentiable at x = a, then f(x) is continuous at x = a. So, a function that is differentiable at x = a will be continuous at x = a. Note however, it is still possible for a function to be continuous at x = a, but not differentiable.

Expressing continuity in terms of limits, we have the Limit Definition of Continuity A function f(x) is continuous at x = a provided the following limit relation holds:

In order for this to hold, three conditions must be fulfilled. 1. f(x) must be defined at x = a 2. must exist 3.The limit must have the value f(a)

Which of the graphs represent continuous functions?