Today's Slides January 25, 2010

Hello All,

Here are today's slides on building continuous and differentiable functions.

CRMS Calculus 2010 January 25, 2010

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Cheers, Bru

Differentiability Implies Continuity

Good day to all.

Today we studied Differentiability Implies Continuity a property that will help in simplifying the process of deciding whether a function is differentiable or continuous.

WHO IS THIS???

(*answer at the end of the post along with citation)

Continuity

Opening up class today we reviewed Continuity at a point, on an interval, and even in a function. Continuity at a point has three requirements; do you remember them?

1. There must exist a value for f(c)

2. There must be a limit for f(x) at x = c

3. The limit must equal the value at x = c

Continuity at an interval is not far off from that: f(x) is continuous on an interval only if it is continuous at each x value in the interval. And for a function to be continuous it must be continuous for every x value in it’s domain.

Differentiability

We then continued reviewing by covering differentiability. We started with the definition of derivative at a point:

We also reviewed the graphical and physical meaning of derivative, which are the slope of the tangent line and the instantaneous rate of change of the function respectively.

Then, like with continuity we reviewed differentiability of f(x): at a point x = c if there is a derivative at x = c; on an interval if it is differentiable for

every x-value in the interval; and in a function if it is differentiable of each x-value in it’s domain.

Syllogisms

After we had reviewed the basics needed for the day we dove into the important information of the day. Definition (from the dictionary on my computer) : syllogism |’silə,jizəm| Noun • deductive reasoning as distinct from induction. Do you remember these from geometry?

Ex. of the positive: If pajamas are flannel then they are comfortable -- P => Q

Ex. of the contrapositive: If pajamas are not comfortable then they are not flannel -- ˜Q => ˜P

Ex. of the inverse: If pajamas are not flannel then they are not comfortable -- ˜P => ˜Q

Ex. of the converse: If pajamas are comfor

table then they are flannel -- Q => P

Think about which ones of these are true, then look at the mathematical examples and ask yourself the same question.

If function f is differentiable at x = c then f is continuous at x = c

Ex: just think of any continuous line, and

it’s derivative

If the function f is not continuous at x = c then f is not differentiable at x = c

Ex: any sort of discontinuity, leap, and infinite, removable… will fill this example

If the function f is not differentiable at x = c then f is not continuous at c

Counter Ex: a cusp is continuous, but not differentiable. The reason it is not differentiable is because there are infinite different derivatives at a cusp.

If the function f is continuous at x = c then it is differentiable at x = c

Counter Ex: again a cusp fits this counter example because it is continuous, but has infinite tangent lines, and thus is not differentiable.

Take a look at these graphs, Blue represents that a function that is not continuous is not differentiable. the reason that this is not continuous is because of the step, the limits from the two different sides are different, thus there is no limit, and it cannot be continuous. The red represents a cusp. The cusp is continuous, there is a limit, and a value, and they are equal, but it is not differentiable because at the cusp there are an infinite number of lines tangent to the tip of the cusp.

On a side note, the easiest way to graph a cusp is to use the absolute values, for example: y = |x| + 1 would be a V-shaped graph with a y intercept of 1. To make it negative you can just add a negative sign in front of the absolute value sign because that will flip the whole thing into the negative.

And what we draw from that experience is that:

Differentiability implies Continuity

If a function f is differentiable at x = c, then f is continuous at x = c

If a function f is not continuous at x = c, then f is not differentiable at x = c

Example:
If the equation is :
Is the function continuous at the point x = 2?
Not it is not because when x = 2 the numerator and denominator are zero, and thus there is a hole at x = 2 because 0/0 can be any number.
Is the function differentiable at the point x = 2?
No because the function is not continuous at the point.

*The picture was a statue of Socrates

http://heroworkshop.files.wordpress.com/2009/11/socrates.jpg

The next scribe will be (drum roll please): Mc Casper