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Monday, January 12, 2009

Accumulating All We Learned About Accumulation Functions Accumulatively

Hi everyone,

I'll be your scribe for today.

Today we began with talking about Googling ourselves. Remember that every time we post something anywhere at anytime on the internet, we are leaving an online "footprint". Mr.K used an example when someone would search your name on Google or on any other online search engine. If an employer or future employer were to search your name online, would they find anything that you do not want to be known about yourself? So becareful with what you leave online, just incase someone searches you up online.

Okay back to math.

We began with a little reminder about what the Inequality Rule of Integrals was about. The Rule says that if a function, for example, f(g) ≤ g(x), than between the same intervals, the integral of f(x) is also less than or equal to the integral of g(x). Refer to the image above for a visual.

Introducing the mighty Obi-Wan Kenobi. Just so everyone knows Darth Vader is better, and if he had Darth Maul's Double Lightsaber, he would own everyone.

Look at all of his strength.

On this slide we applied the Fundamental Theorem of Calculus (FTC). If you want more info on the FTC then click here. But I'll explain it anyways. The fundamental theorem says that the integral of any function is equal to the total change in output (y-values) of the antiderivative function. For example if you take the integral of a velocity function/curve, it is also equal to the change in position of the object moving. The downside to this is that we won't know where the object is or where it began, but only how far it moved.

So to solve the problem we have to realize that we are looking for a change in values on the antiderivative of th given function. That is the Fundamental Theorem of Calculus. So what we can do is apply the constant multiple rule to make some antidifferentiating a little light for us. Then we apply the Fundamental Theorem of Calculus. So we have to find the antiderivative of the function. Once that is done we can find the total sales between year 2 and year 4. Remember that we cannot have part sales so we have to round down to the nearest integer for the final answer.

This next part is the introduction of a new topic in the unit of integrals. But it isn't really a new topic because it is putting some visuals to the antiderivatives. The function on the above slide may be a bit ugly but I'll pick out the little things to make bit nicer. Well the function involves 2 variables, x and t. t is the independent variable in the inner function, f, and x is the independant variable for the outside function, A. Since f is a function that is a nice and smooth function so we don't have to worry about any conflicts with arguements. So is the interval of the integral of the integrand, f(t). The variable, x, increases or decreases the size of the interval of the integral. So we were told to fill in the tables on the slide. For each value of x the size of the interval increases or decreases changing the integral of f(t). Since f(t) is a straight line, we can just count squares and triangles.
Oh no you have just ran into an area below the x-axis. What do you do? I'll tell you. Notice that there is a negative change in x and also a negative change in y. So the area can be multiplied by multiplying the y value and x value depending on either the area is a triangle or rectangle. So when you multiply a negative by a negative, you get a positive.
Now let`s graph the numbers found. Notice that it forms a parabola. This should make sense because the antiderivative of a line is a parabola.

That little experiment brought us to the Sign Convention. No there is no event at the MTS Centre about signs, but a sign rule. The convention says that the integral of a function is equal to the negative of the integral of the same function if the interval is reversed.

The next two questions involve the same process as the experiment from earlier. Notice that the fixed interval endpoint is changed in each question.

That was my scribe. As you can see on the last image written by Lawrence with his left hand, homework is Exercise 6.2 all odd questions and omit questions 7 and 11.

The next scribe will be Joyce.

Good night, I got to get studying. I have a lot of APs to study for.

1 comment:

Rence said...

Left hand, Yeaaaah!
*Raises Left-hand fist into air*