Find the critical numbers of f (x) = x^2-6x. Find also the open intervals on which the function is increasing or decreasing and locate all relative extrema.

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).

What is the difference b/w relation and function. It would be great if you could give some real life example.

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:

Use a half angle identity to find the exact value- sin(75 degrees)

Solution:

The line of a half angle identity is as follows:

If we plug in 75 degrees we get:

12 out of 1000 women at age forty who participate in routine screening have breast cancer. 880 out of 1000 women with breast cancer will get positive mammographies. 90 out of 1000 women without breast cancer will also let positive mammographies. If 100 women in this age group undergo a routine screening, about what fraction of women with positive mammographies will actually have breast cancer?

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:

What is law of motion?

Solution:

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

Need to get the answer to this problem by using elimination: =x-3y=61 and 2x+y=-7

Solution:

1. 2x –  =y=61
2. 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

What is 9/10 of 80?

Solution:

The expression “9/10 of 80” can be written as

This is equal to 72.

What is the name of a 6-sided polygon?

Solution:

A six-sided polygon is known as a Hexagon:

What is the dimension of amount of a substance?

Solution:

In the physical sciences, the amount of a sample is often given in terms of moles (i.e. Avogadro’s number, or 6.02214179 x 10²³):

A particle is trapped in an infinite one-dimensional well of width 0.132nm. The electron is in the n=10 state. A) What is energy of the electron? B) What is the uncertainty in its momentum?

Solution: