### Calculus…

Saturday, March 26th, 2011What is the square function and what does it look like on a graph? f(x) = x^2

What is the reciprocal function and what does it looks like on a graph? f(x) = 1/x

What is the square function and what does it look like on a graph? f(x) = x^2

What is the reciprocal function and what does it looks like on a graph? f(x) = 1/x

A real function of one variable is a set of ordered pairs of real numbers where for every real number one of the following two things happen.

1) There is exactly one real number b for which the ordered pair (a,b) is a member of f.

2) There is no real number b for which the ordered pair (a,b) is a member of f.

If the center of the circle is on the origin then the radius squared is equal to x squared plus y squared.

x*x + y*y = r*r

The set of all points in the plane at distance r from a point P is called the circle of radius r and center P.

In Calculus you need to understand the concept of Delta X and Delta Y.

X is the position of a point along the horizontal axis.

Y is the position of a point along the vertical axis.

delta is change.

Delta X is the change along the X axis when a point’s location changes.

Delta Y is the change along the Y axis when a point’s location changes.

Delta is represented by a small triangle.

In calculus there is an xy plane. The x line is horizontal. The y line is vertical. Where the x line crosses the y line is the origin. Moving left along the the x line is lower and moving right along the x line is higher.

The upper right section is quadrant 1.

The upper left section is quadrant 2.

The lower left section is quadrant 3.

The lower right section is quadrant 4.

So it’s important to assume the all real (even negative) numbers has one cube root. For example 2*2*2 is 8 and -2*-2*-2 = -8.

According to the books it’s important to understand:

1) That all positive real numbers have a square root or 2 numbers that can be multiplied together and their product is the number. In addition there are two negative numbers that can be multiplied together and their product is the number as well.

For example the number 9 has two two square roots. The first square root is 3 and the second square root is -3. So 3*3 = 9 and -3*-3=9.

2) All negative numbers do not a a square root, so -9 has no square root.

Apparently you really need to know this if you are going to learn calculus:

1/0, 0/0, x/0 is always considered undefined.

Due to not feeling well I had a chance to look at H. Jerome Keisler’s “Elementary Calculus” book and I read the Introduction. It’s awesome.

He wrote about how a car’s speedometer is a good model for calculus concepts:

The odometer measures the integral of the distance. We learned that an integral is the area under a curve. This caused a pretty good discussion because what’s the curve.

The graph has the distance along the y axis and the time along the x axis

at time = 0 distance = 0

at time = 2 hours distance = 40

It’s a straight line that makes a triangle.

The area for a triangle is 1/2 base* height

1/2*2*40 = 40

And 40 is what would be on the odometer. It sounds complicated, but because we have been studying the definitions it all made sense to us, after some thought.

It’s cool..

The speedometer is the derivative of the distance.

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