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Wednesday, December 17, 2008

The Absence of Evidence

We started off the class with a pretty funny clip which featured Calculus and the Numa, Numa song. Basically the person in the video showed two pieces of paper, one a function and one an anti-derivative.

Following that, we got into the stuffing of the class. We reviewed derivatives and what they are in order to grasp the concept of an anti-derivative. We were told to remember something when we do anti-derivatives, which is "If you can do something, you can undo it." For anti-derivatives, this means that any derivative can be anti-differentiated.

A key point is to know that an anti-derivative is a family of functions, however, if you have a given point, you can find one particular function within the family of functions.

An Indefinite Integral is not like a definite integral. It picks our a family and if a point is known, it picks a particular one. This sounds a lot like an Anti-Derivative.

Moving along, we are told that for every derivative rule, there is an anti-derivative rule, as this is automatic when a derivative rule is made.

So, knowing that, can we find the parent function with first and second derivatives?

Let's use an example.

If you know the derivative of sine, you know the anti-derivative of cosine.
Therefore if you know the derivative of sine, you know the anti-derivative of sine.

In words: The derivative of sinx is cos x, so the anti-derivative of cosx is sin x + C where C is an arbitrary constant. This basically means that C can be any value, because the function can exist in a number of places in the "family of functions."

The derivative of cos x is negative sin x, so the anti-derivative of sinx is negative cosx + C.

After this, Mr. K gave us a question so that we can apply this.

Consider the function (not the way it's supposed to be written but sitmo won't let me write it properly.) * "k" is a parameter and can be any value.*Using the first and second derivatives, describe the features of the family of functions generated for different values of "k".

Bench did the honours of doing this question.

Using the quotient rule and the chain rule...and factoring out...and reducing...and then we let it equal 0, so that we can find x.

o = 1 - kx
kx = 1

*Remember that critical numbers are at zero and undefined points, however, for this function there are no undefined points.
Therefore, this function will have no discontinuities (cusps, corners, magic tricks, hat tricks, pen tricks, kick flips etc.)

So by the first derivative test, when x = 1/k, the function is at a max.

Okay, it's 12 AM. I need to study for Physics... Ugh.

Next scribe will be Zeph!

Rence ~ Out

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