# Archive for July, 2010

Saturday, July 24th, 2010

### Solution:

First, we can write

Then we can substitute this into the second equation you get

We can simplify this to

If multiply both sides by 25 and expand this out we get

This can be simplified to: Or

Thus the solutions for a are 6 and -4

The solutions for b can be found by the equation (b=4 when a=6, and b=-2 then a=-4)

- Another way to solve this is by looking at the equations graphically. When you plot the two equations, you can see there they intersect.

As you can see, they intersect at two places:

You can plug the answers into both equations to confirm that they are both true using these two values of a and b.

Saturday, July 24th, 2010

### Solution:

- In the above example, y=g(w), b=1 and m=-4. If we replace each of the terms we can rewrite the above as y=b+mx

Saturday, July 24th, 2010

### Solution:

1. 3x + 4y=25

2. 2x – 5y=9

We could rewrite these as follows by multiplying the first equation by 5 and the second by 4:

1. 15x + 20y = 125

2. 8x – 20y = 36

If we add the above two equations, we can eliminate y:

23x = 161

So x =3D 7, and plugging this into the first equation we can figure out

y: **21 + 4y =3D 25**,

so **y = 1**

Saturday, July 24th, 2010

### Solution:

- 2x – =y=61
- 2x + y= -7

If you subtract the second equation from the first, you can eliminate x and you are left with **-4y = 68.**

So **y =3D -17**, and plugging this into the first equation we can figure out

y: 2x = (-51)=61,

so **x = 5**

Saturday, July 24th, 2010

## Solution:

**First some definitions:**

A **neuron** is an electrically excitable cell that processes and transmits information by electrochemical signaling. On one end of the neuron are the dendrites (receiving end) and the other end is the transmitting end (axon terminal).

**Synapse:** One neuron connects to another neuron when the dendrite of one is connected to the axon terminal of the other. This connection is called a synapse.

**Dendrites** are the branched projections of a neuron that act to conduct the electrochemical stimulation received from other neural cells to the main part of the neuron from which the dendrites project. Dendrites form the main receiving part of neurons. Dendrites collect and funnel these signals to the soma and axon.

A **neurotransmitter** is a chemical that is released from a neuron.

**Synaptic Cleft**: The tiny space between two nerve cells across which the neurotransmitter diffuses

**Receptor**: Neurotransmitters cross the synapse where they may be accepted by the next neuron at a specialized site on a dendrite called a receptor.

Here are the detailed instructions for the “lost” neurotransmitter:

- You are currently on the receptor of a dendrite

- To get to the next neuron in the chain, you must travel the length of the neuron you are current on

- Travel across the dendrite

- Travel down the entire length of the axon

- Travel across the axon terminal

- At the synaptic cleft, bind with receptor sites on the neighboring neuron’s dendrite

Saturday, July 24th, 2010

### Solution:

A number a in the domain of a given function f is called a critical number ‘(a) = 0 or f ‘ is undefined at x = a.

Relative extrema are the minimums or maximum points on a part of a curve, while absolute extrema are the minimums and maximum points along the entire curve.

Now we are looking at the following curve:

f (x) =

To find the extrema, we take the first derivative and figure out at what value of x, the first derivate of the function equals to zero:

f’(x) = 2x-6

f’(3) = 0

As you can see the function is decreasing from negative infinity to 3, and increasing from 3 to infinity. The critical number is 3. There is only one relative (or local) minimum at (3, -9).

Friday, July 23rd, 2010

### Solution:

A relation is a set of ordered pairs, such as { (0,1) , (55,22), (3,-50) } (http://www.mathwarehouse.com/algebra/relation/math-function.php). A function is a special kind of relation in which each x value has only one y value.

For example:

Friday, July 23rd, 2010

### Solution:

The line of a half angle identity is as follows:

If we plug in 75 degrees we get:

Friday, July 23rd, 2010

### Solution:

1. Out of 100 women, we expect 1.2 to actually have breast cancer (12 out of 1000)

2. However, only 88% (880 out of 1000) of these will be detected, so 1.056 (i.e. 88% of 1.2) will be detected

3. In addition, of the 98.8 women who don’t have breast cancer 9% (90 out of 1000) will trigger a false positive” (9% of 98.8 = 8.892)

4. So out of 100 women, 9.948 will have positive mammography readings (1.056+9.892)

5. Of these 9.948, only 1.056 will actually have breast cancer

6. 10.6% = (1.056/1.056+9.892) This ratio is essentially an application of Bayes Theorem

Please click on http://www.thegodofreason.com/bayesintro.pdf to download a PDF that explains how Bayes Theorem works. Page 3 gives an example that is very similar to your problem:

Friday, July 23rd, 2010

### Solution:

- http://en.wikipedia.org/wiki/Laws_of_motion (Newton, Kepler, etc.)
- http://www.schooltrainer.com/study-material.html (look at the left column under the heading Physics”)