Suppose you have an region bounded by the curves y = x^3 an y = x^2. Now you rotate the region about the x axis to for a three dimension figure. What is the volume of this newly formed figure?
To calculate the volume, you need to know the equation of the curve that forms the outer edge of the shape minus the equation of the curve that forms the inner edge of the shape. The bounds of integration are the points of intersection of the curves.
Set the 2 equations equal to each other and solve for x.
x^3 = x^2
x^3 - x^2 = 0
x^2(x - 1) = 0
x = 0 and x = 1. This can also be seen by a graph of the two equations.
Now that we have the points of intersection, we take pi times the integral of the outer radius squared minus the inner radius squared
integrate from 0 to 1 of Pi[(x^2)^2 - (x^3)^2]
integral from 0 to 1 of Pi[x^4 - x^6]
Pi[x^5/5 - x^7/7] evaluated from 0 to 1
Pi(1/5 - 1/7) = Pi(7/35 - 5/35) = Pi(2/35)
Saturday, April 27, 2013
Thursday, April 25, 2013
Tuesday, April 23, 2013
A way to measure the spread of data about the mean is by using the
standard deviation. If the spread is small, the standard deviation is
small. If the spread is large, the standard deviation is large. The
proportion of the data that falls within a certain number of standard
deviations from the mean is very definite and precise with a bell shaped
curve. But what can be said about the proportion of data spread about
the mean for other distributions such as skewed, symmetric, or other
shapes? Chebyshev's Theorem will solve this problem.
The basis of Chebyshev's Theorem is that no matter how large or small a data set is from a population or a sample, the proportion of data that lies within k standard deviations is at least 1 - 1/k2. For k = 2, the proportion is 1 - 1/4 = .75. For k = 3, the proportion is 1 - 1/9 = .889. For k = 4, the proportion is 1 - 1/16 = .938. These results mean that at least 75% of the data must fall within 2 standard deviations from the mean, 88.9% must fall within 3 standard deviations from the mean, and at least 93.8% must fall within 4 standard deviations from the mean.
Many distributions will have much greater percentages of the data falling within specified intervals. For example, in the well known normal distribution, which is bell shaped, 95% of the data falls within 2 standard deviations, 99.7% falls within 3 standard deviations from the mean, and virtually 100% falls within 4 standard deviations from the mean.
Here's an example using Chebyshev's Theorem.
Suppose students at a local college volunteer to work on community projects, such as cleaning parks, renovating playgrounds, and planting trees. A professor in charge of the program kept track of the time in hours that each student worked. Suppose a random sample of x students in the program were picked and the mean hours the students worked was 24.5 hours, and the standard deviation was 1.4 hours. From this information we can find an interval which at least 75%, 88.9% and 93.8% of the students worked.
Interval which at least 75% worked is 24.5 +/- 2(1.4) = 21.7 to 27.3 hours.
Interval which at least 88.9% worked is 24.5 +/- 3(1.4) = 20.3 to 28.7 hours.
Interval which at least 93.8% worked is 24.5 +/- 4(1.4) = 18.9 to 30.1 hours.
This guide should help students better understand Chebyshev's Theorem and how it can be applied.
The basis of Chebyshev's Theorem is that no matter how large or small a data set is from a population or a sample, the proportion of data that lies within k standard deviations is at least 1 - 1/k2. For k = 2, the proportion is 1 - 1/4 = .75. For k = 3, the proportion is 1 - 1/9 = .889. For k = 4, the proportion is 1 - 1/16 = .938. These results mean that at least 75% of the data must fall within 2 standard deviations from the mean, 88.9% must fall within 3 standard deviations from the mean, and at least 93.8% must fall within 4 standard deviations from the mean.
Many distributions will have much greater percentages of the data falling within specified intervals. For example, in the well known normal distribution, which is bell shaped, 95% of the data falls within 2 standard deviations, 99.7% falls within 3 standard deviations from the mean, and virtually 100% falls within 4 standard deviations from the mean.
Here's an example using Chebyshev's Theorem.
Suppose students at a local college volunteer to work on community projects, such as cleaning parks, renovating playgrounds, and planting trees. A professor in charge of the program kept track of the time in hours that each student worked. Suppose a random sample of x students in the program were picked and the mean hours the students worked was 24.5 hours, and the standard deviation was 1.4 hours. From this information we can find an interval which at least 75%, 88.9% and 93.8% of the students worked.
Interval which at least 75% worked is 24.5 +/- 2(1.4) = 21.7 to 27.3 hours.
Interval which at least 88.9% worked is 24.5 +/- 3(1.4) = 20.3 to 28.7 hours.
Interval which at least 93.8% worked is 24.5 +/- 4(1.4) = 18.9 to 30.1 hours.
This guide should help students better understand Chebyshev's Theorem and how it can be applied.
Saturday, April 20, 2013
Knowing the graphs of "parent functions", such as f(x) = x^2 and f(x) = |x| makes it simple to graph variations of such functions.
For example,
f(x) = |x| + 3 has the same graph as f(x) = |x| except it's translated 3 units up. Similarly f(x) = |x| - 3 would be the same as f(x) = |x| except it is translated 3 units down.
For f(x) = (x - 3)^2, the graph is the same as for f(x) = x^2 except it is translated 3 units to the right. Similarly, f(x) = (x + 3)^2 has the same graph as f(x) = x^2 except it is translated 3 units to the left.
You can also have translations horizontally and vertically in the same function
f(x) = (x + 4)^2 - 3 is translated 4 units left and 3 units down from the graph of f(x) = x^2
For example,
f(x) = |x| + 3 has the same graph as f(x) = |x| except it's translated 3 units up. Similarly f(x) = |x| - 3 would be the same as f(x) = |x| except it is translated 3 units down.
For f(x) = (x - 3)^2, the graph is the same as for f(x) = x^2 except it is translated 3 units to the right. Similarly, f(x) = (x + 3)^2 has the same graph as f(x) = x^2 except it is translated 3 units to the left.
You can also have translations horizontally and vertically in the same function
f(x) = (x + 4)^2 - 3 is translated 4 units left and 3 units down from the graph of f(x) = x^2
Thursday, April 18, 2013
Graphs Used to Display Qualitative Data
Bar graph
A bar graph can be horizontal or vertical that are uniformly spaced and have uniform width. The length of each bar show the frequency or percentage of occurrence, depending on what is to be displayed in the data. The lengths of the bars show the variable and the values of the variable being displayed. The graph must be titled with labels for each bar and a scale or precise value for the length of each bar.
For example: Suppose you wish to display the population of Chicago for years 1982, 1992, 2002, and 2012. The population is on the vertical axis and the years are on the horizontal axis. The length of the bars represent the population for each year. The title of the graph might be "Population For Chicago".
Suppose you wish to break down the population into male and female. This type of graph is known as a clustered bar graph because there are two bars for each year the population is measured. One bar will be for male population, and one bar for female population.
A type of bar graph in which the height of each bar represents frequency is known as a Pareto chart. A distinguishing feature of this bar graph is that the bars are arranged from left to right in decreasing order of frequency. Therefore, the graph will have the highest bar at the far left and lowest bar at the far right.
Circle graph
In a circle graph, also known as a "pie chart", wedges of a circle represent a percent of a population which have a common trait.
For example, suppose you want to know how much time Americans aged 25 to 50 watch television after 7 pm on a weeknight. Suppose in a sample of 200, 25 people watch up to 1 hour of television, 50 people watch between 1 and 2 hours of television, 100 watch between 2 and 3 hours of television, and 25 people watch over 3 hours of television.
That means there are 4 wedges of the pie, 12.5% represent up to 1 hour of television, 12.5% represent 3 or more hours, 25% represent 1 to 2 hours, and 50 percent represent 2 to 3 hours.
Draw the circle and make appropriate sized wedges to show the various percentages, and notice that the total percentages is 100. Label each piece and mark each piece of the chart with the designated numbers of degrees.
Time-series graph
When you want to track a change over time, the best type of graph to use is a time-series graph. For example, suppose you start exercising and you ride a bike for 45 minutes. You want to monitor your progress over a month period of time to see how much farther you can bike at the end of the month compared to the beginning of the month.
The time-series graph shows data measurements in chronological order. Time is placed on the horizontal axis, and the variable of interest is placed on the vertical axis. The basic time-series graph is made by connecting the data points by lines.
This article should give students a solid foundation for understanding graphs that can used to display qualitative data, as well as quantitative data.
Monday, April 15, 2013
When quantitative data is used in statistics, a histogram is a great way
to provide a useful visual display. But when the data is qualitative,
other types of graphs must be used. These graphs may also be used to
display quantitative data. The graphs are the bar graph, circle graph,
and time-series graph.
Bar graph
A bar graph can be horizontal or vertical that are uniformly spaced and have uniform width. The length of each bar show the frequency or percentage of occurrence, depending on what is to be displayed in the data. The lengths of the bars show the variable and the values of the variable being displayed. The graph must be titled with labels for each bar and a scale or precise value for the length of each bar.
For example: Suppose you wish to display the population of Chicago for years 1982, 1992, 2002, and 2012. The population is on the vertical axis and the years are on the horizontal axis. The length of the bars represent the population for each year. The title of the graph might be "Population For Chicago".
Suppose you wish to break down the population into male and female. This type of graph is known as a clustered bar graph because there are two bars for each year the population is measured. One bar will be for male population, and one bar for female population.
A type of bar graph in which the height of each bar represents frequency is known as a Pareto chart. A distinguishing feature of this bar graph is that the bars are arranged from left to right in decreasing order of frequency. Therefore, the graph will have the highest bar at the far left and lowest bar at the far right.
Circle graph
In a circle graph, also known as a "pie chart", wedges of a circle represent a percent of a population which have a common trait.
For example, suppose you want to know how much time Americans aged 25 to 50 watch television after 7 pm on a weeknight. Suppose in a sample of 200, 25 people watch up to 1 hour of television, 50 people watch between 1 and 2 hours of television, 100 watch between 2 and 3 hours of television, and 25 people watch over 3 hours of television.
That means there are 4 wedges of the pie, 12.5% represent up to 1 hour of television, 12.5% represent 3 or more hours, 25% represent 1 to 2 hours, and 50 percent represent 2 to 3 hours.
Draw the circle and make appropriate sized wedges to show the various percentages, and notice that the total percentages is 100. Label each piece and mark each piece of the chart with the designated numbers of degrees.
Time-series graph
When you want to track a change over time, the best type of graph to use is a time-series graph. For example, suppose you start exercising and you ride a bike for 45 minutes. You want to monitor your progress over a month period of time to see how much farther you can bike at the end of the month compared to the beginning of the month.
The time-series graph shows data measurements in chronological order. Time is placed on the horizontal axis, and the variable of interest is placed on the vertical axis. The basic time-series graph is made by connecting the data points by lines.
This article should give students a solid foundation for understanding graphs that can used to display qualitative data, as well as quantitative data.
Bar graph
A bar graph can be horizontal or vertical that are uniformly spaced and have uniform width. The length of each bar show the frequency or percentage of occurrence, depending on what is to be displayed in the data. The lengths of the bars show the variable and the values of the variable being displayed. The graph must be titled with labels for each bar and a scale or precise value for the length of each bar.
For example: Suppose you wish to display the population of Chicago for years 1982, 1992, 2002, and 2012. The population is on the vertical axis and the years are on the horizontal axis. The length of the bars represent the population for each year. The title of the graph might be "Population For Chicago".
Suppose you wish to break down the population into male and female. This type of graph is known as a clustered bar graph because there are two bars for each year the population is measured. One bar will be for male population, and one bar for female population.
A type of bar graph in which the height of each bar represents frequency is known as a Pareto chart. A distinguishing feature of this bar graph is that the bars are arranged from left to right in decreasing order of frequency. Therefore, the graph will have the highest bar at the far left and lowest bar at the far right.
Circle graph
In a circle graph, also known as a "pie chart", wedges of a circle represent a percent of a population which have a common trait.
For example, suppose you want to know how much time Americans aged 25 to 50 watch television after 7 pm on a weeknight. Suppose in a sample of 200, 25 people watch up to 1 hour of television, 50 people watch between 1 and 2 hours of television, 100 watch between 2 and 3 hours of television, and 25 people watch over 3 hours of television.
That means there are 4 wedges of the pie, 12.5% represent up to 1 hour of television, 12.5% represent 3 or more hours, 25% represent 1 to 2 hours, and 50 percent represent 2 to 3 hours.
Draw the circle and make appropriate sized wedges to show the various percentages, and notice that the total percentages is 100. Label each piece and mark each piece of the chart with the designated numbers of degrees.
Time-series graph
When you want to track a change over time, the best type of graph to use is a time-series graph. For example, suppose you start exercising and you ride a bike for 45 minutes. You want to monitor your progress over a month period of time to see how much farther you can bike at the end of the month compared to the beginning of the month.
The time-series graph shows data measurements in chronological order. Time is placed on the horizontal axis, and the variable of interest is placed on the vertical axis. The basic time-series graph is made by connecting the data points by lines.
This article should give students a solid foundation for understanding graphs that can used to display qualitative data, as well as quantitative data.
Saturday, April 13, 2013
When dealing with the unit circle, sometimes it's easier to remember angles and convert them to radians.
The angles along the unit circle in degrees are
0, 30, 45, 60, 90, 120, 135, 150, 180, 210, 225, 240, 270, 300, 315, 330
To convert angles from degrees to radians, multiply by Pi/180. Therefore the above angles in radians are
0, Pi/6, Pi/4, Pi/3, Pi/2, 2Pi/3 3Pi/4, 5Pi/6, Pi, 7Pi/6, 5Pi/4, 4Pi/3, 3Pi/2, 5Pi/3, 7Pi/4, 11Pi/6
The values for sin and cos along the unit circle will be +/- 1/2, +/- sqrt(2)/2, +/- sqrt(3)/2. The + or - depends on what quadrant the angle falls.
The angles along the unit circle in degrees are
0, 30, 45, 60, 90, 120, 135, 150, 180, 210, 225, 240, 270, 300, 315, 330
To convert angles from degrees to radians, multiply by Pi/180. Therefore the above angles in radians are
0, Pi/6, Pi/4, Pi/3, Pi/2, 2Pi/3 3Pi/4, 5Pi/6, Pi, 7Pi/6, 5Pi/4, 4Pi/3, 3Pi/2, 5Pi/3, 7Pi/4, 11Pi/6
The values for sin and cos along the unit circle will be +/- 1/2, +/- sqrt(2)/2, +/- sqrt(3)/2. The + or - depends on what quadrant the angle falls.
Wednesday, April 10, 2013
Understanding Sampling Techniques
There are five different types of sampling techniques that are commonly used. Those techniques are as follows:
Simple Random
In a simple random sample, "n" objects are selected from a population in such a way that each sample of size "n" is equally liked to be chosen from the population. Also, every member of the population is equally likely to be chosen.
Stratified
in a stratified sample, the population is divided into subgroups called strata. All members of each strata fit a specified trait, such as height, gender, income level, education, and so on. Then random samples are drawn from each strata.
Systematic
In a systematic sample, each member of the population is numbered in order. Then from a specified point, select every so many members to be included in the sample. For example, you can choose to have every 4th, 10th, 15th member, etc, to be part of the sample.
Cluster
In cluster sampling, the population is divided into specific pre-existing groups. Many times these groups are geographic regions. Then, randomly select a group of clusters, and every member from the cluster is part of the sample.
Convenience
In convenience sampling, the sample is chosen by using data from members of the population that are most convenient to use or easiest to obtain.
Here's some examples of the different types of sampling techniques.
Simple random: Assign each teacher in the Wilson School District a number, then select the teachers to be included in the sample using a random number generator.
Stratified: Group sports franchises according to sport: baseball, football, basketball, hockey. Then select a random sample of 12 teams from each sport.
Systematic: Use the Wilson School District faculty directory. Number all of the teachers. Select a starting point in the list of teachers and select every 25th to be included in the sample. Continue selecting members of the sample in this manner until 50 teachers are selected.
Cluster: Divide a state into regions by using the counties. Pick a random sample of 10 counties and include all the businesses in each selected county.
Convenience: Choose 10 newspaper reporters from a local newspaper. Have each reporter select a neighborhood in the area and interview a business owner from any business found. The reporter is finished after interviewing 15 business owners.
This guide should help assist any student having difficulty understanding and distinguishing between the different types of sampling techniques.
Saturday, April 6, 2013
Suppose you wish to multiply a number, variable or expression a
repeated number of times. Exponents allow you to write these products in
a simpler form. For example, 5 ∙ 5 ∙ 5 ∙ 5 = 54, y ∙ y ∙ y ∙ y = y4 , (x - 3) ∙ (x - 3) ∙ (x - 3) ∙ (x - 3) ∙ (x - 3) ∙ (x - 3) = (x - 3)6
. Each of the simplified expressions are called exponential
expressions. The number being raised to the exponent is known as the
base and the exponent is known as the power. But how do we deal with the
concept of negative exponents?
The general rules for positive exponents hold true for negative exponents. If x is an integer and n ≠ 0, then x-n = 1/xn. To see why this holds true, multiply the equation by xn to get x-n ∙ xn = 1. Remember when multiplying exponential expressions, you add the exponents, so -n + n = 0 and x0 = 1.
You can think of this as the reciprocal of x-n and changing the exponent from negative to positive. Generally speaking, you want to make sure you have all positive exponents in the expression in simplest form.
Examples: Write each expression with only positive exponents.
3-4 , (2a)-2 , 5x-3
3-4 = 1/(34) = 1/81
(2a)-2 = 1/(2a)2 = 1/4a2
5x-3 = 5/(x3)
Note when raising a fraction to a negative exponent, take the reciprocal of the fraction and change the exponents to positive.
Example: Simplify the expression using only positive exponents.
(x/y)-3
Take the reciprocal first then change the negative -3 to 3.
(y/x)
(y/x)3
Then apply the rule for raising an exponent to an exponent (remember y is y1 and x is x1). Therefore, (x/y)-3 = y3/x3.
Example: Simplify the expression using only positive exponents.
4x-3/(x10)
There are two ways to solve this. You can apply the rule for dividing exponential expressions and subtract the exponents.
4x-3/(x10) = 4x-(3-10) = 4x-13
Then change the negative exponent to positive by changing x-13 to 1/x13. The simplified form is 4/x13. You can change the negative exponent to positive first to get 4/(x3x10), then apply the rule for multiplying exponential expressions to get 4/x13.
Example: Simplify the expression using only positive exponents.
(x5x6)/(x-4)
Keep the common base x in the numerator and add the exponents to get x11/x-4
Subtract the exponents to get x(11 + 4) = x15. You could also use the rules for negative exponents to get x5x6x4 = x15 .
The examples above should help students who are having trouble understanding the concept of negative exponents.
The general rules for positive exponents hold true for negative exponents. If x is an integer and n ≠ 0, then x-n = 1/xn. To see why this holds true, multiply the equation by xn to get x-n ∙ xn = 1. Remember when multiplying exponential expressions, you add the exponents, so -n + n = 0 and x0 = 1.
You can think of this as the reciprocal of x-n and changing the exponent from negative to positive. Generally speaking, you want to make sure you have all positive exponents in the expression in simplest form.
Examples: Write each expression with only positive exponents.
3-4 , (2a)-2 , 5x-3
3-4 = 1/(34) = 1/81
(2a)-2 = 1/(2a)2 = 1/4a2
5x-3 = 5/(x3)
Note when raising a fraction to a negative exponent, take the reciprocal of the fraction and change the exponents to positive.
Example: Simplify the expression using only positive exponents.
(x/y)-3
Take the reciprocal first then change the negative -3 to 3.
(y/x)
(y/x)3
Then apply the rule for raising an exponent to an exponent (remember y is y1 and x is x1). Therefore, (x/y)-3 = y3/x3.
Example: Simplify the expression using only positive exponents.
4x-3/(x10)
There are two ways to solve this. You can apply the rule for dividing exponential expressions and subtract the exponents.
4x-3/(x10) = 4x-(3-10) = 4x-13
Then change the negative exponent to positive by changing x-13 to 1/x13. The simplified form is 4/x13. You can change the negative exponent to positive first to get 4/(x3x10), then apply the rule for multiplying exponential expressions to get 4/x13.
Example: Simplify the expression using only positive exponents.
(x5x6)/(x-4)
Keep the common base x in the numerator and add the exponents to get x11/x-4
Subtract the exponents to get x(11 + 4) = x15. You could also use the rules for negative exponents to get x5x6x4 = x15 .
The examples above should help students who are having trouble understanding the concept of negative exponents.
Thursday, April 4, 2013
When dealing with polar coordinates, it's easy to convert to rectangular coordinates
x = rcos(angle)
y=rsin(angle)
If the polar coordinate (r,angle) is (4, pi/2)
The rectangular coordinates are
x = 4cos(pi/2) = 0
y = 4sin(pi/2) = 4
If you are given rectangular coordinates, you can convert to polar coordinates
x^2 + y^2 = r^2
tan(y/x) = angle
So if you have the rectangular coordinates (2, 3)
The polar coordinates are
2^2 + 3^2 = r^2
13 = r^2
sqrt(13) = r
tan(3/2) = angle
angle = 56.3 degrees
x = rcos(angle)
y=rsin(angle)
If the polar coordinate (r,angle) is (4, pi/2)
The rectangular coordinates are
x = 4cos(pi/2) = 0
y = 4sin(pi/2) = 4
If you are given rectangular coordinates, you can convert to polar coordinates
x^2 + y^2 = r^2
tan(y/x) = angle
So if you have the rectangular coordinates (2, 3)
The polar coordinates are
2^2 + 3^2 = r^2
13 = r^2
sqrt(13) = r
tan(3/2) = angle
angle = 56.3 degrees
Subscribe to:
Posts (Atom)