## Thursday, June 30, 2011

### Parametric

A pair or a set of parametric equations may look like
$x =2,y=t, \; t\; \in\; (-\infty,\infty)$,
or
$x =2\cos(t),y=3\sin(t), \; t\; \in\; [0,2\pi]$,
or
$x=\theta,\; \; y=2\sin \theta,\; \; \theta\; \in\; [0,2\pi]$,
or
$x=\frac{1}{2}t,\; \; y=2t,\; \; z=t,\; t\in\; (-\infty,\infty)$.

See the common feature?

Now, let's try to combine the variables:
a) $x =2,y=t, \; t\; \in\; (-\infty,\infty)\; \; \Leftrightarrow \; \; x=2$

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This is a vertical line x=2.

b) $x =2\cos(t),y=3\sin(t), \; t\; \in\; [0,2\pi] \Leftrightarrow \frac{x^2}{4}+\frac{y^2}{9}=1$

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This is a vertical ellipse centered at the origin with vertices at (0,3) and (0,-3), and endpoints of minor axis at (2,0) and (-2,0).

c) $x=\theta,\; \; y=2\sin \theta,\; \; \theta\; \in\; [0,2\pi]\Leftrightarrow y=2\sin x$

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This is a sine function with magnitude 2.

d) $x=\frac{1}{2}t,\; \; y=2t,\; \; z=t,\; t\in\; (-\infty,\infty) \Leftrightarrow z^2=xy$
This is an object in 3-dimension.

You see, parametric equations can form a lot of curve with different type:
a) function, non-function;
b) line, curve;
c) 2-D, 3-D;
etc.

Can you try to build one set of parametric equations and figure out the graph of the set of equations?

## Saturday, June 25, 2011

### Parabola

A parabola has a standard equation of

$(x-h)^2=4p(y-k)$ ------(1)

or

$(y-k)^2=4p(x-h)$ ------(2)

where (h,k) is the coordinate of the vertex (labeled as V in the graph).

For the parabola of type (1), the parabola is either opening upwards (p > 0) or downwards(p < 0). This type of parabola has a focus at F(h, k + p) and a directrix at y = k - p.

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On the other hand, for the parabola of type (2), the parabola is either opening rightwards (p > 0) or leftwards(p < 0). This type of parabola has a focus at F(h + p, k) and a directrix at x = h - p.

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Betty, Created with GeoGebra

### Hyperbola

A hyperbola which is centered at (h,k) has a standard equation

$\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$ ------(1)

if the hyperbola is opening sidewards, else, if it's opeing upwards or downwards, the standard equation will be

$\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1$ ------(2)

Let's take a look at equation type (1):

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This hyperbola in blue has
a) vertices at B1 and B2 at $(h\pm a,k)$,
b) foci at F1 and F2 at $(h\pm c,k)$ where $c=\sqrt{a^2+b^2}$,
c) asymptotes in red dotted lines $y-k=\pm \frac{b}{a}(x-h)$.

Now, let's look at equation type (2):

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This hyperbola in blue has
a) vertices at B1 and B2 at $(h,k\pm a)$,
b) foci at F1 and F2 at $(h,k\pm c)$ where $c=\sqrt{a^2+b^2}$,
c) asymptotes in red dotted lines $y-k=\pm \frac{a}{b}(x-h)$.

Betty, Created with GeoGebra

### Ellipse

An ellipse has a standard equation of

$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$

where
(h,k) is the center of the ellipse,
a is the distance between the vertex and the center, and
b is the distance between the endpoint of minor axis and the center.

Look at the graph below, point A is the center.
When a > b, points B1 and B2 are the vertices, points C1 and C2 are the endpoints of minor axis, and F1 and F2 are the foci.
When a < b, points C1 and C2 are the vertices, points B1 and B2 are the endpoints of minor axis, and F3 and F4 are the foci.

The line segment which connects the vertices is called as major axis. The line segment which is shorter, perpendicular to the major axis and passes through the center is called the minor axis.

You may slide the values of a, b, h or k to observe the changing of the graph.

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Betty, Created with GeoGebra

## Saturday, June 4, 2011

### Indefinite Integrals

Let say we have a function f(x). If we integrate f(x) with respect to x, we get the antiderivative F(x) + c as follow:

$\int f(x)\; dx =F(x)+c$

where c is a constant.

Now, consider you do not know the function f(x) but you need to sketch the graph of F(x) from the graph of f(x) only, taking c = 0. How are you going to start?

Let's take a look at the graph below first:

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Curve f (blue) is the curve for f(x) and curve F (maroon) is the curve of F(x).
Line L is the tangent at point B, with equation y = mx + c, where m is the slope of tangent and c is the y-intercept of the tangent.
Point A is a point at curve f(x) and point B is the corresponding point at curve F(x) having the same x-coordinate of point A.

Now, try to move the point A and observe the following:
(a) As point A falls above x-axis, curve F(x) is increasing.
(b) As point A falls on the x-axis, curve F(x) reaches to its extremum (maximum/minimum).
(c) As point A falls below the x-axis, curve F(x) is decreasing.
(d) As point A falls on the extremum point, point B is a point of inflection, which curve F(x) changes its concavity.

Take note that the y-coordinate of point A is exactly the slope of tangnet (m) of point B!

Perhaps you can try to draw any new curve f(x) and find its integral F(x), taking c = 0?

Betty, Created with GeoGebra