Suppose O lies on the interior of <ABC, D lies on the interior of <ABO and E lies on the interior of < OBC.  

It is also given that m<DBO = m<EBC and m<ABC = 120, m<DBC = 90.

What is the relationships between <DBE and <OBC ?

From the information in the problem we obtain the following drawing.

Since m<DBO = m<EBC and <OBE is part of both <DBO and <OBE, it follows that

m<DBE = m<OBC.

Notice this in the following diagram

Since m<ABC = 120 and m<DBC = 90, m<ABD = 30.

We know this from the angle addition postulate, m<ABD + m<DBC = m<ABC.

Remember when you have a root that is an odd power, the range includes all real numbers. If the root is an even power, the range is all real numbers from zero to infinity.

 

The domain for a root that is an odd power is all real numbers except in the event that the radical is in the denominator. In such a case, we must make sure the value does not make the denominator equal to zero.

 

The domain for a root that is an even power is all real numbers that makes the expression under the radical greater than or equal to 0. The only exception is if the radical is in the denominator, then we must make sure the domain is all values that make the expression under the radical greater than 0.

Suppose we want to find the equation of a line that passes through the point (3, -4) and has a slope of 6. How do we find the equation?

There are two methods. You can use the equation y - y1 = m(x - x1) where (x1, y1) is (3, -4) and m = 6.

Substitute the values into the equation to get

y - (-4) = 6(x - 3)

y + 4 = 6x - 18

y = 6x - 22

We can also use the equation y = mx + b, subtitute the values for x, y and m into the equation and solve for b.

-4 = 6(3) + b

-4 = 18 + b

-22 = b

Now substitute 6 for m and -22 for b into y = mx + b to get

y = 6x - 22.

The definition of the derivative is

lim f(x + Δx) - f(x)

Δx→ 0 Δx

We can therefore get the derivative of f(x) = 2x² + 5x using the above as follows:

lim [ 2(x + Δx)² + 5(x + Δx) – (2x² + 5x)]

Δx → 0 Δx

lim [2(x² + 2xΔx + Δx²) + 5x + 5Δx – 2x² - 5x]

Δx → 0 Δx

lim 2x² + 5x + 4xΔx + 2Δx² + 5Δx - 2x² – 5x

Δx→ 0 Δx

lim 4xΔx + 2Δx² + 5Δx

Δx→ 0 Δx

We can now factor out a Δx in the numerator to get

lim Δx(4x + 2 Δx + 5)

Δx→ 0 Δx

Δx in the numerator and denominator cancel out to get

lim 4x + 2 Δx + 5

Δx→ 0

Substituting 0 for Δx gives us 4x + 5, which is the derivative.

There is a much easier way to get the derivative than using the formal definition.

Multiply the coefficient by the exponent in the first term. That result becomes the new coefficient and subtract one from the exponent to get the new exponent

For 2x² that is 2(2) = 4 (new coefficient), exponent goes from 2 to (2-1) = 1. The first term of the derivative is 4x

For 5x the is 5(1) = 5 (new coefficient), exponent goes from 1 to (1-1) = 0. x⁰ = 1, so there is no x in the second term.

The derivative is 4x + 5

Square Roots, Cube Roots and Higher Roots

Recall when raising a number to a power n, where n is an integer greater than 1, we multiply the number by itself n times.

For example, 43 = 4 ∙ 4 ∙ 4. Now suppose we want to know what number multiplied by itself 2 times equals. Problems of this kind can be represented using radicals. A radical symbol √ is used to show the square root, or principal square root of a number or expression that appears under the radical symbol. Recall that the square root is defined as a number or expression multiplied by itself twice to equal the number or expression under the radical symbol, known as the radicand.

For example, if we want to know what number multiplied by itself 2 times equals 169, we can set this up with the radical symbol as follows:

√169, read as “square root of 169”. The answer to this is 13.
- √169, read as “negative square root of 169”. The answer to this is -13.

√0.09 = 0.3 and -0.3 since (0.3)2 and (-0.3)2 equals 0.09. The principal square root is 0.3. Another way to simplify this is to change √0.09 to √(9/100) and simplify to 3/10.

√(25/49) = 5/7 and -5/7 since (5/7)2 and (-5/7)2. The principal square root is 5/7.

• Note that a square root also has a negative value since a negative times a negative equals a positive, but we will deal with only the principal square root unless otherwise noted.
• Note that you can also simplify the square root of a fraction by taking the square root of the numerator and then the square root of the denominator instead of the square root of the fraction as a whole. In the previous example, you can take the square root of 25 first, then the square root of 49.
• Note that the square root of many positive integers are not whole numbers or rational numbers. For example, √19 can be found on a calculator or by leaving the answer as √19.

Sometimes we have to find the square root of a number that is not a perfect square. In these cases, we break down the radicand into factors, one of which is a perfect square.

Examples: Find each square root.
1. √68

First, find factors of 68.
Since 68 is even, we can divide it by 2. Therefore, 68 = 2 ∙ 34. Notice 34 is also divisible by 2, therefore 34 = 2 ∙ 17.

So 68 is factored into 2 ∙ 2 ∙ 17. Notice that 17 is prime and cannot be factored further and 2 ∙ 2 = 4, which is a perfect square. Therefore √68 = √4 ∙ √17 = 2√17.

2. √108
First, find factors of 108.

Since 108 is even, we can divide it by 2. Therefore 108 = 2 ∙ 54. Notice 54 is also divisible by 2, therefore 54 = 2 ∙ 27.

Next, we know that 27 = 3 ∙ 3 ∙ 3.
The factors of 108 are 2 ∙ 2 ∙ 3 ∙ 3 ∙ 3. Notice 2 ∙ 2 = 4, which is a perfect square and 3 ∙ 3 = 9, which is also a perfect square. Therefore, 108 = 4 ∙ 9 ∙ 3 and √108 = √4 ∙ √9 ∙ √3 = 2 ∙ 3 ∙ √3 = 6√3.

• Note that 108 = 36 ∙ 3 and 36 is a perfect square. But if you can't see right away that 3 is a factor of 108, you can break down by dividing 108 by 2 first and then simplify further at that point. It's easy to determine that 108 is divisible by 3. If the sum of the digits of a number are divisible by 3, the number is divisible by 3.

The 3 measures of central tendency in a set of data is mean, median and mode.  The mean is the average of the set of numbers.  The median is the number in the middle of the set of data and the mode is the number that occurs most frequently. But how do we know which measure of central tendency is the best to use?

Take this example.

The data set is  2, 5, 6, 10, 12, 12, 13, 16, 20, 21, 23, 110

Mean is (2 + 5 + 6 + 10 + 12 + .... + 110)/12 = 20.8

Median is 12.5 (middle values are 12 and 13)

Mode is 12

Notice that the mean is much larger than the median and mode.  If you compare the average of 20.8 to the numbers in the set, only 3 numbers are higher.  The mean is not a good measure of central tendency in this case because of 110. Any outliers will greatly affect the mean.


 Generally, we use the median when the data set has an outlier, as in example above.

The mode is actually not really a measure of central tendency. It should only be used with nonnumeric data to state what item in the data set is most popular.

The distribution of the data also has some effect but I gave a simple example that will explain which to use, in general.

 Here's a few problems involving volume and comparing volumes:


The diameter of a baseball is approximately 7.37 cm. A hockey puck is cylindrical with a thickness
of 2.54 cm and a diameter of 7.6 cm. Which has the greatest volume?

Solution:
A baseball is spherical and the volume of a sphere is (4/3)πr^3. The radius of the sphere is half the diameter, so r = 3.685 cm. Using 3.14 for π, the volume of the baseball is (4/3)(3.14)(3.685)^3 = 209.5 cm^3. (rounded to one decimal place)

The volume of a cylinder is πr2h. The radius of the hockey puck is 3.8 cm, the height is 2.54 cm. Therefore the volume of the hockey puck is (3.14)(3.8)^2(2.54) = 115.2 cm^3. (rounded to one decimal place)
The volume of a baseball is nearly double the volume of a hockey puck.

Suppose you have a metal cone shaped container and a plastic container in the shape of a shoe box. The cone shaped container is 14 inches high with a diameter of 10 inches. The plastic container is 12 inches long, 6 inches wide and 4 inches high. You want to fill up one container with water. Which container will hold the most water?

Solution:
The volume of a cone is (1/3) πr^2h.
Therefore, the volume of the cone shaped container is (1/3)(3.14)(5)^2(14) = 366.3 cubic inches (rounded to one
decimal place)

The volume of a rectangular solid is length times width times height, therefore the volume of the plastic container  is (12)(6)(4) = 288 cubic inches.

The metal cone shaped container will hold the most water.