## Tuesday, January 29, 2013

Remember that integration is the opposite of differentiation. It's also known as the "antiderivative"

If f(x) = x^3 + 4x, the derivative is 3x^2 + 4

When integrating you will get x^3 + 4x + C

This is how it's done

Take 1 over (exponent +1) times the variable raised to the (n+1) power.  Do this for each term

Thus  we get  3(1/3)x^(2+1) + 4/1(x)^(0 + 1)

= x^3 + 4x

We add a C due to a constant of differentiation

## Saturday, January 26, 2013

When thinking of the rectangular coordinate system, imagine a city that is set up with all streets running north to south and east to west. The streets running north to south are equal distance and parallel to each other as are the streets running east to west. Drawing the streets on the graph would show a grid with equal sized squares. In algebra, the vertical (north to south) line straight through the middle of the is known as the y-axis and the horizontal (east to west) line straight through the middle is known as the x-axis. The center of the graph is known as the origin.

There are four sections of the rectangular coordinate system created by the intersection of the x and y axis. Each section is known as a quadrant. It's easy to remember by thinking of "quad", which means four. The quadrants are named I, II, III and IV, starting in the upper right and rotating counter clockwise.

Think of the layout of a city. The north to south streets start at 1 and go to 10 and are avenues. The east to west streets start at 1 and go to 10 and are boulevards. Suppose John lives at the intersection of East 5th Boulevard and North 8th Aenue and Susie lives at the intersection of West 6th Boulevard and South 2nd Avenue. You can plot John's location on the graph by staring at the origin, moving right to East 5th boulevard and then up to North 8th street. Likewise, you can plot Susie's location by starting at the origin and moving left to West 6th boulevard and South to 2nd street. Notice how John's location is labeled as (5, 8) in the following graph. This point is called an ordered pair. Each number in the ordered pair is known as a coordinate.

The first number in an ordered pair is the x- coordinate and the second number in an ordered pair is the ycoordinate. Note that a coordinate is in the form (x, y). Coordinates on the x-axis are negative left of the origin and positive right of the origin. Coordinates on the yaxis are negative below the origin and positive above the origin. The coordinate of the origin is (0,0) since it is the middle of the graph. From this information, we can tell what quadrant a point falls. If both coordinates are positive, the point falls in the first quadrant. If both coordinates are negative, the point falls in the third quadrant. If x is negative and y is positive, the point falls in the second quadrant. If x is positive and y is negative, the point falls in the fourth quadrant.

Examples:
(-2, 4) falls in the second quadrant.
(4, 10) falls in the first quadrant.
(-8, -3) falls in the third quadrant.
(9, -6) falls in the fourth quadrant

This guide should provide a good introduction and ease any confusion about the rectangular coordinate system.

## Tuesday, January 22, 2013

Here's a good example of a problem involving integers as might be seen on an SAT.

If n is an integer and n2 is a positive integer, which must also be positive?

a. n2 + n
b. n2 – m3
c. 2n2 – n
d. n3 + n
e. 2n3 + n

At first glance a,b and c all seem reasonable because squaring any number will give a positive answer. But answer b wont always work because if n is positive, n3 will be greater than or equal to n2. So when subtracting it from n2 you will either get 0, if n is 1 or negative if n >1.

For answer a, if n = -1, then n2 + n = 0, which does not give a positive answer either.

Any positive number squared times 2 will be greater than the number itself so 2n2 – n > 0 for any positive number n.

If n is negative, 2n2 is positive and subtracting a negative is like adding a positive, so 2n2 – n > 0 for any negative number n as well.

## Friday, January 18, 2013

You have a two digit number.  When the digits are added, the sum is 7.  When you reverse the digits and subtract from the original number, the difference is 9. What is the 2 digit number?

Let's examine what numbers add to 7.

1 and 6
2 and 5
3 and 4

You know the original number must be larger than the second number, so the original number must be 61, 52 or 43.

When reversing the numbers you get 16, 25 and 34.

Now let's subtract

61 - 16 = 45
52 - 25 = 27
43 - 34 = 9

The original number is 43.

The easy way to know it must be 43 is to get a 9 when subtracting, the second number must be 1 greater in the one's place than the original number.  4 is 1 greater than 3.

## Tuesday, January 15, 2013

When verifying trigonometric identities, it's best to convert everything to sine and cosine if at all possible.

Here's some basic identities

sin^2(x) + cos^2(x) = 1
1 + tan^2(x) = sec^2(x)
1 + cot^2(x) = csc^2(x)

You can use properties of equality on the identities as follows:

1 - cos^2(x) = sin^2(x)
1 - sin^2(x) = cos^2(x)

tan^2(x) = sec^2(x) -1
cot^2(x) = csc^2(x) - 1

Remember that tanx = sinx/cosx

cotx = cosx/sinx/

= 1/tanx

cscx = 1/sinx

secx = 1/cosx

sin(x + y) = sinxsiny + cosxcosy

## Friday, January 11, 2013

In a beginning statistics class, topics students are taught are basic counting principles, ways to order a certain number of objects and how many ways to select a certain number of objects from a group of objects.

A permutation is an ordered arrangement that each item in the arrangement is used once and the order of the items in the arrangement makes a difference. When we think about combinations , we consider that items are selected from the same group and no item may be used more than once. This is the same as in permutations. What makes combinations and permutations different is that the order does not matter in combinations.

How does one know whether to use combinations or permutations? The following examples will erase any confusion on this topic.

1. A local restaurant offers pizza with up to 20 toppings. One special they offer is any large pizza with 3 toppings for \$10. How many different pizzas are possible?

Solution:
The order in which the toppings are put on the pizza does not matter. A pizza with pepperoni, sausage and green peppers is the same as one with green peppers, pepperoni and sausage. Therefore, this is a problem involving combinations.

2. Ten lodge members are running for president and vice president. When the votes are counted, the member with the second highest number of votes is named vice president and the member with the highest number of votes is elected president. How many possible outcomes are possible?

Solution:
Lodge members are choosing a president and a vice president from ten members. The order in which they are chosen matters since the positions are different. Since order matters, this is a problem involving permutations.

3. A football team of 52 players wishes to elect a group of 4 players to be team captains for the season. How many different groups of 4 players can be chosen to be team captains?

Solution:
The order in which the 4 players are chosen to be team captains does not matter since they are not occupying different positions. Each player in the group is considered a captain and is equal to the other captains. Therefore, since order does not matter, this is a problem involving combinations.

4. How many different 7 digit passwords are possible from the numbers 1, 2, 3, 4, 5, 6 and 7 if no digits may be used more than once.

Solution:
The order the digits appear in the password is important. A password of 1, 3, 5, 2, 4, 6, 7 is different from 1, 2, 3, 4, 5, 6, 7. Since order matters, this is a problem involving permutations.
We use the notation n Pr to denote the number of permutations of n objects taken r at a time. For formula for permutations is as follows

n Pr = n! /( n - r)!
The number of combinations of n objects taken r at a time is given by

nCr = n!/ [r!(n - r)!]

Note that n! = n(n - 1)(n - 2).....1

This guide should clear any issues pertaining to what combinations and permutations are and when to use them.

## Wednesday, January 9, 2013

When graphing the sine and cosine curves, remember to write the key components first, such as

Amplitutde
Period
Max
Min
Spaces
Phase Shift
Transformation

For example, for y = sinx and y = cosx

Amplitude = 1
Period = 2Pi
Max = 1
Min = -1
Spaces = 2Pi/4 = Pi/2
Phase shift : none
Transformation: none

The absolute value of the number in front of sin or cos is the amplitude

ex,   y = 4sinx, amplitude is 4.    y = -5cosx,  amplitude is 5

Period for both sine and cosine is 2Pi/amplitude

Max and min with no transformation is amplitude times max and amplitude times min

Spaces is the Period divided by 4

There is a phase shift if there is a + or - with the angle..

For example  y = sin(x - 1), has a phase shift of 1 to the right
y = sin(x + 1), has a phase shift of 1 to the left

There is a transformation if there is a + or - outside of the angle

For example, y = sinx + 2, transformation is 2 up
y = sinx - 2 , transformation is 2 down

## Tuesday, January 8, 2013

In the quadratic formula, the expression under the radical b2 - 4ac is known as the discriminant, with the condition that a ≠ 0. This quantity determines whether the roots of a quadratic equation are rational, irrational or imaginary. If the discriminant is positive, there are 2 different real number roots. If the discriminant is negative, there are 2 different imaginary roots. If the discriminant is 0, there is one repeated rational root, known as a double root. If the discriminant is a perfect square, there are 2 different rational roots and if the discriminant is positive and not a perfect square, there are two different real, irrational roots.
Examples: Determine the number and nature of roots for each of the following quadratic equations.
1. x2 + 12x + 36 = 0
a = 1, b = 12, c = 36
b2 - 4ac = 122 - 4(1)(36
= 144 - 144
= 0

The discriminant equals 0, therefore there is one repeated rational root, a double root.If we solve the equation, we'll find that the double root is x = -6.

When the discriminant is 0, the radical portion of the quadratic formula is 0, leaving only a single value for x. Since the equation is a 2nd degree equation, there must be 2 solutions, therefore the solution is a double root.

2. x2 - 3x + 2 = 0
a = 1, b = -3, c = 8
b2 - 4ac = (-3)2 - 4(1)(8)
= 9 - 32
= -23
The discriminant is negative, therefore there are 2 different imaginary roots. The roots are imaginary because the square root of a negative number is an imaginary number.

3. 2x2 - 8x + 7/2 = 0
a = 2, b = -8, c = 7/2
b2 - 4ac = (-8)2 - 4(2)(7/2)
= 64 - 28
= 36

The discriminant is positive and since 36 is a perfect square, we have two different rational roots. The square root of 36 is 6 and -6, which are the two different rational roots.

4. 3x2 - 7x + 1 = 0
a = 3, b = -7, c = 1
b2 - 4ac = (-7)2 - 4(3)(1)
= 49 - 12
= 37
The discriminant is positive and not a perfect square, therefore there are two different real, irrational roots.

As you can see, the definition and use of the discriminant is fairly straight forward and easy to use. This article should have erased any confusion on this topic.

## Saturday, January 5, 2013

Suppose you wish to enclose a field for corn corn by a fence the is 2x feet long and y feet wide.  The perimeter of the fence is 800 feet.  What is x and y and what dimensions will maximize the area of the fence?

Perimeter: 2x + 2x + y + y = 800

4x + 2y = 800

Area = 2x(y)

Solve for y in the equation for perimeter and substitute in the equation for area

2y = 800 - 4x

y = 400 - 2x

Area = 2x(400 - 2x)

= 800x - 4x^2

To maximize the area, take the derivative of the area and set equal to 0 and solve for x.

Derivative of Area = 800 - 8x

Therefore, x = 100

y = 400 - 2(100)

= 200

The dimensions of the field which gives maximum area is 200 x 200 (which makes sense, a square will always maximize area)

## Friday, January 4, 2013

Difference of squares

Some binomials can be written as the difference of two squares. In order to factor a difference of two
squares, it's important to recognize some perfect squares. The first 25 perfect squares are as follows:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576 and 625.

The formula for factoring the difference of two squares is
(x^2 - y^2) = (x - y)(x + y)

It's easy to think of this as just taking √x^2 and √y^2, which is x and y. Both binomials in the factored form will have x and y, one with a “-” between them and one with a “+” between them.
Example: Factor x^2 – 16.

Notice that x^2 and 16 are both perfect squares
.
√x^2 = x and √16 = 4, therefore the factored form of x^2 – 16 is (x - 4)(x + 4).

Example: Factor 4x^2 – 25.
√4x^2 = 2x and √25 = 5, therefore the factored form of 4x^2 - 25 is (2x - 5)(2x + 5).

Example: Factor 25y^2 - 49x^2.
√25y^2 = 5y and √49x^2 = 7x, therefore the factored form of 25y^2 - 49x^2 is (5y - 7x)(5y + 7x).

## Wednesday, January 2, 2013

When proving triangles congruent, there are a few methods generally used.

SSS - If three sides of a triangle and congruent to three sides of a second triangle, the 2 triangles are congruent.

SAS - If 2 sides and the included angle of one triangle are congruent to 2 sides and the included angle of another triangle, then the triangles are congruent.

ASA - If 2 angles and the included side of one triangle are congruent to 2 angles and the included side of another triangle, then the triangles are congruent.

AAS - If 2 angles and the non included side of one triangle are congruent to 2 angles and the non included side of another triangle, then the triangles are congruent.

HL - In a right triangle, if the hypotenuse and a leg of one triangle are congruent to the hypotenuse and a leg of another triangle, then the triangles are congruent.

## Tuesday, January 1, 2013

Happy new year and thank you to everyone who has viewed my blog! Pass the word to others you know that enjoy math.

So for 2013, enjoy the

365 days

or 8,760 hours

or 525,600 minutes

or 31,536,000 seconds !!