UBC Wiki - User contributions [en]

Course:MATH110/Archive/2010-2011/003/Teams/Aargau/Homework 13

2011-02-04T14:46:20Z

YihongChen: Created page with " <center><big>pH</big><center> The pH level determines how acidic a liquid solution is. It is a important tool to use in any science, especially biology. Water has a n..."

<center><big>pH</big><center>
 The pH level determines how acidic a liquid solution is. It is a important tool to use in any science, especially biology. Water has a neutral level of pH at 7.0. Anything with a pH level lower than 7 is acidic, and anything with a higher level is basic.

 Like the Richter scale used to measure earthquakes, the pH scale is logarithmic. When we say a pH scale is logarithmic it means a difference of one pH measurement unit represents a tenfold, or ten times increase or reduction of hydrogen ion activity in the solution.A low pH indicates a high concentration of hydronium ions, while a high pH indicates a low concentration. This explains how a solution’s aggressiveness increases with the distance from the neutral point. The pH scale is a logarithmic scale which extends over 15 orders of magnitude, measuring the concentration of H ions in a solution. pH 0 is 10,000,000, De-ionised water is pH 7, 1 and pH 14 is 1/10,000,000

 The pH equation is as follows:
pH=-log(H+) where H+ is ion concentration
 We can rewrite this equation to look like:
H+ = 10^(pH)
 This is where we can see that the pH is actually an exponent, explaining why the pH scale is logarithmic. The result of the scale being logarithmic is that there are huge differences between the pH levels. A liquid with a pH level of 5 is 10 times more acidic then a liquid with a pH level of 6.

 On a calculator, the antilog or 10^x button is used when you are converting from pH back to H+so if the pH=5.3you would enter -5.3 in the calculator and hit the 10^x button this would enable you to find out that [H+]=0.000005 M or 5 x 10^-6 M

 in general terms, we can define pX=-log10[X]
log (A x B)= log A + log B

 with this equation in chemistry there is the formula known as the ion product which is as follows:Kw=[H+][OH-]
 this can be rewritten as a logarithm log Kw = log[H+][OH-]
 -log Kw= -(log[H+][OH-])
 pKw=pH+pOH
 if Kw=10^-14 then pKw=14=pH+pOH

Course:MATH110/Archive/2010-2011/003/Teams/Aargau

2011-02-04T14:45:12Z

YihongChen: /* Subpages */

{{
Infobox MATH110 Teams
| team name = Aargau
| member 1 = Daisy Gobina
| member 2 = Dominic Sunga
| member 3 = Shauna Maty
| member 4 = Yihong Chen
}}
In workshop G.

===Subpages===
#[[/Homework 11|Homework 11]]
#[[/Homework 12|Homework 12]]
#[[/Homework 13|Homework 13]]

===Jan 24, Monday - Keywords===
Shauna - Hockey
Dominic - Aargau
Daisy - Math

Course:MATH110/Archive/2010-2011/003/Teams/Aargau

2011-02-04T14:43:49Z

YihongChen: /* Subpages */

{{
Infobox MATH110 Teams
| team name = Aargau
| member 1 = Daisy Gobina
| member 2 = Dominic Sunga
| member 3 = Shauna Maty
| member 4 = Yihong Chen
}}
In workshop G.

===Subpages===
#[[/Homework 11|Homework 11]]
#[[/Homework 12|Homework 12]]
#<a href="http://wiki.ubc.ca/Documentation:Aargau/Homework_13">Homework 13</a>

===Jan 24, Monday - Keywords===
Shauna - Hockey
Dominic - Aargau
Daisy - Math

Course:MATH110/Archive/2010-2011/003/Teams/Aargau

2011-02-04T14:39:16Z

YihongChen: /* Subpages */

{{
Infobox MATH110 Teams
| team name = Aargau
| member 1 = Daisy Gobina
| member 2 = Dominic Sunga
| member 3 = Shauna Maty
| member 4 = Yihong Chen
}}
In workshop G.

===Subpages===
#[[/Homework 11|Homework 11]]
#[[/Homework 12|Homework 12]]

[[http://wiki.ubc.ca/Documentation:Aargau/Homework_13|Homework 13]]

===Jan 24, Monday - Keywords===
Shauna - Hockey
Dominic - Aargau
Daisy - Math

Course:MATH110/Archive/2010-2011/003/Teams/Aargau

2011-02-04T14:38:13Z

YihongChen: /* Subpages */

{{
Infobox MATH110 Teams
| team name = Aargau
| member 1 = Daisy Gobina
| member 2 = Dominic Sunga
| member 3 = Shauna Maty
| member 4 = Yihong Chen
}}
In workshop G.

===Subpages===
#[[/Homework 11|Homework 11]]
#[[/Homework 12|Homework 12]]

#[[http://wiki.ubc.ca/Documentation:Aargau/Homework_13|Homework 13]]

===Jan 24, Monday - Keywords===
Shauna - Hockey
Dominic - Aargau
Daisy - Math

Course:MATH110/Archive/2010-2011/003/Teams/Aargau

2011-02-04T14:37:47Z

YihongChen: /* Subpages */

{{
Infobox MATH110 Teams
| team name = Aargau
| member 1 = Daisy Gobina
| member 2 = Dominic Sunga
| member 3 = Shauna Maty
| member 4 = Yihong Chen
}}
In workshop G.

===Subpages===
#[[/Homework 11|Homework 11]]
#[[/Homework 12|Homework 12]]

[[http://wiki.ubc.ca/Documentation:Aargau/Homework_13|Homework 13]]

===Jan 24, Monday - Keywords===
Shauna - Hockey
Dominic - Aargau
Daisy - Math

Course:MATH110/Archive/2010-2011/003/Teams/Aargau

2011-02-04T14:37:28Z

YihongChen: /* Subpages */

{{
Infobox MATH110 Teams
| team name = Aargau
| member 1 = Daisy Gobina
| member 2 = Dominic Sunga
| member 3 = Shauna Maty
| member 4 = Yihong Chen
}}
In workshop G.

===Subpages===
#[[/Homework 11|Homework 11]]
#[[/Homework 12|Homework 12]]

#[[http://wiki.ubc.ca/Documentation:Aargau/Homework_13|Homework 13]]

===Jan 24, Monday - Keywords===
Shauna - Hockey
Dominic - Aargau
Daisy - Math

Course:MATH110/Archive/2010-2011/003/Teams/Aargau

2011-01-28T14:51:30Z

YihongChen:

{{
Infobox MATH110 Teams
| team name = Aargau
| member 1 = Daisy Gobina
| member 2 = Dominic Sunga
| member 3 = Shauna Maty
| member 4 = Yihong Chen
}}
In workshop G.

 SUBPAGES
 1.[[Documentation:Homework_11|Homework 11]]
2.[[Documentation:Aargau/Homework_12|Homework 12]]

Jan 24, Monday - Keywords

Shauna - Hockey

Dominic - Aargau

Daisy - Math

Course:MATH110/Archive/2010-2011/003/Teams/Aargau

2011-01-28T14:51:13Z

YihongChen:

Course:MATH110/Archive/2010-2011/003/Teams/Aargau

2011-01-28T14:50:57Z

YihongChen:

Course:MATH110/Archive/2010-2011/003/Teams/Aargau

2011-01-28T14:50:27Z

YihongChen:

Course:MATH110/Archive/2010-2011/003/Teams/Aargau

2011-01-28T14:49:59Z

YihongChen:

{{
Infobox MATH110 Teams
| team name = Aargau
| member 1 = Daisy Gobina
| member 2 = Dominic Sunga
| member 3 = Shauna Maty
| member 4 = Yihong Chen
}}
In workshop G.

 SUBPAGES
 1.[[Documentation:Homework_11|Homework 11]]
 1.[[Documentation:Aargau/Homework_12|Homework 12]]

Jan 24, Monday - Keywords

Shauna - Hockey

Dominic - Aargau

Daisy - Math

Course:MATH110/Archive/2010-2011/003/Teams/Aargau/Homework 12

2011-01-28T14:45:39Z

YihongChen:

<center>'''Homework 12 Team Problem'''</center>
<center> With the function P(t)=1/(1+e^-t), we can change the height of the horizontal asymptote on the right by changing the value of the numerator 1 to any value. We will use the value of 10, which we wil denote by K. To change the y-intercept to a number between 0 and K, we can change any part of the function to make it have a different y-value when the x-value is 0. We will multiply e^-t by 3. When we plug in 0 for t into the new function P(t)=10/(1+3e^-t), we get 2.5 for the y-intercept value, which is between 0 and K. </center>
<center>'''BONUS'''</center>
<center> After testing different things, we discovered that adding a coefficient to -t will change the slope of the curved part. It must still remain negative. The larger the value of the coefficient, the closer to vertical the slope will be.
<center> Compare the graph of the original function to the graph of P(t)=1/(1+e^-5t). You can observe from the graphs below that the windows they are graphed in are different, showing that the slope has actually become more vertical while maintaining the same y-intercept and rest of the graph. The modified function's curved slope is graphed over a smaller range of x-values than the unmodified function's curved slope, showing the change in slope. Changing the coefficient to larger values such as 1/(1+e^-10t) or even larger would make the slope even more vertical.
<center> '''P(t)=1/(1+e^-t)''' </center>
<center> [[File:MSP220819e4e417b6708ich00001768587g039a5c38.gif]]</center>
<center> '''P(t)=1/(1+e^-5t)'''</center>
<center>[[File:math grahpie 2.gif]]
<center> '''One application we could use the graph to model is the population of ants in an anthill.'''
<center>'''Equation for the population of ants:
P(t)=10/(1+3e^-t)'''
 The population is in thousands, The time is in weeks 
 '''First off, we must know what the initial population is at time=0 (T0)'''
 
P(0)=10/(1+3e^0)
 
P(0)= 2.5 (in thousands)
 '''With this equation, we can predict what the population will be in 20 weeks.'''
P(20)=10/(1+3e^-20)
P(20)=9,999

 '''Assumptions
 It is important to note that this model functions because of the assumption that the
percentage change in population per week is constant.
It is also important to remember that this anthill is in an isolated location where
there is no outside interference with the population (humans etc.)
 <center> '''Explaining the Equation'''</center>
The 10 in the numerator is the horizontal asymptote. Because this equation is modeling population, the horizontal asymptote, which is 10, shows the carrying capacity for this anthill.
The y-intercept for this equation is 2.5, which is found when t is zero. The y-intercept represents the initial population.

All the data used to make this equation is imagined and not taken from anywhere.

Course:MATH110/Archive/2010-2011/003/Teams/Aargau/Homework 12

2011-01-28T14:39:46Z

YihongChen:

<center>'''Homework 12 Team Problem'''</center>
<center> With the function P(t)=1/(1+e^-t), we can change the height of the horizontal asymptote on the right by changing the value of the numerator 1 to any value. We will use the value of 10, which we wil denote by K. To change the y-intercept to a number between 0 and K, we can change any part of the function to make it have a different y-value when the x-value is 0. We will multiply e^-t by 3. When we plug in 0 for t into the new function P(t)=10/(1+3e^-t), we get 2.5 for the y-intercept value, which is between 0 and K. </center>
<center>'''BONUS'''</center>
<center> After testing different things, we discovered that adding a coefficient to -t will change the slope of the curved part. It must still remain negative. The larger the value of the coefficient, the closer to vertical the slope will be.
<center> Compare the graph of the original function to the graph of P(t)=1/(1+e^-5t). It is hard to tell from the graphs below because the windows they are graphed in are different, but the slope has actually become more vertical while maintaining the same y-intercept and rest of the graph. Changing the coefficient to larger values such as 1/(1+e^-10t) or even larger would make the slope even more vertical.
<center> '''P(t)=1/(1+e^-t)''' </center>
<center> [[File:MSP220819e4e417b6708ich00001768587g039a5c38.gif]]</center>
<center> '''P(t)=1/(1+e^-5t)'''</center>
<center>[[File:math grahpie 2.gif]]
<center> '''One application we could use the graph to model is the population of ants in an anthill.'''
<center>'''Equation for the population of ants:
P(t)=10/(1+3e^-t)'''
 The population is in thousands, The time is in weeks 
 '''First off, we must know what the initial population is at time=0 (T0)'''
 
P(0)=10/(1+3e^0)
 
P(0)= 2.5 (in thousands)
 '''With this equation, we can predict what the population will be in 20 weeks.'''
P(20)=10/(1+3e^-20)
P(20)=9,999

 '''Assumptions
 It is important to note that this model functions because of the assumption that the
percentage change in population per week is constant.
It is also important to remember that this anthill is in an isolated location where
there is no outside interference with the population (humans etc.)
 <center> '''Explaining the Equation'''</center>
The 10 in the numerator is the horizontal asymptote. Because this equation is modeling population, the horizontal asymptote, which is 10, shows the carrying capacity for this anthill.
The y-intercept for this equation is 2.5, which is found when t is zero. The y-intercept represents the initial population.

All the data used to make this equation is imagined and not taken from anywhere.

Course:MATH110/Archive/2010-2011/003/Teams/Aargau/Homework 12

2011-01-28T14:39:13Z

YihongChen:

<center>'''Homework 12 Team Problem'''</center>
<center> With the function P(t)=1/(1+e^-t), we can change the height of the horizontal asymptote on the right by changing the value of the numerator 1 to any value. We will use the value of 10, which we wil denote by K. To change the y-intercept to a number between 0 and K, we can change any part of the function to make it have a different y-value when the x-value is 0. We will multiply e^-t by 3. When we plug in 0 for t into the new function P(t)=10/(1+3e^-t), we get 2.5 for the y-intercept value, which is between 0 and K. </center>
<center>'''BONUS'''</center>
<center> After testing different things, we discovered that adding a coefficient to -t will change the slope of the curved part. It must still remain negative. The larger the value of the coefficient, the closer to vertical the slope will be.
<center> Compare the graph of the original function to the graph of P(t)=1/(1+e^-5t). It is hard to tell from the graphs below because the windows they are graphed in are different, but the slope has actually become more vertical while maintaining the same y-intercept and rest of the graph. Changing the coefficient to larger values such as 1/(1+e^-10t) or even larger would make the slope even more vertical.
<center> '''P(t)=1/(1+e^-t)''' </center>
<center> [[File:MSP220819e4e417b6708ich00001768587g039a5c38.gif]]</center>
<center> '''P(t)=1/(1+e^-5t)'''</center>
<center>[[File:math grahpie 2.gif]]
<center> '''One application we could use the graph to model is the population of ants in an anthill.'''
<center>'''Equation for the population of ants:
P(t)=10/(1+3e^-t)'''
 The population is in thousands, The time is in weeks 
 '''First off, we must know what the initial population is at time=0 (T0)'''
 
P(0)=10/(1+3e^0)
 
P(0)= 2.5 (in thousands)
 '''With this equation, we can predict what the population will be in 20 weeks.'''
P(20)=10/(1+3e^-20)
P(20)=9,999

 '''Assumptions
 It is important to note that this model functions because of the assumption that the
percentage change in population per week is constant.
It is also important to remember that this anthill is in an isolated location where
there is no outside interference with the population (humans etc.)
 <center> '''Explaining the Equation'''</center>
The 10 in the numerator is the horizontal asymptote. Because this equation is modeling population, the horizontal asymptote, which is 10, shows the carrying capacity for this anthill.
The y-intercept for this equation is 2.5, which is found when t is zero. The y-intercept represents the initial population.

Course:MATH110/Archive/2010-2011/003/Teams/Aargau/Homework 12

2011-01-28T14:38:00Z

YihongChen:

<center>'''Homework 12 Team Problem'''</center>
<center> With the function P(t)=1/(1+e^-t), we can change the height of the horizontal asymptote on the right by changing the value of the numerator 1 to any value. We will use the value of 10, which we wil denote by K. To change the y-intercept to a number between 0 and K, we can change any part of the function to make it have a different y-value when the x-value is 0. We will multiply e^-t by 3. When we plug in 0 for t into the new function P(t)=10/(1+3e^-t), we get 2.5 for the y-intercept value, which is between 0 and K. </center>
<center>'''BONUS'''</center>
<center> After testing different things, we discovered that adding a coefficient to -t will change the slope of the curved part. It must still remain negative. The larger the value of the coefficient, the closer to vertical the slope will be.
<center> Compare the graph of the original function to the graph of P(t)=1/(1+e^-5t). It is hard to tell from the graphs below because the windows they are graphed in are different, but the slope has actually become more vertical while maintaining the same y-intercept and rest of the graph. Changing the coefficient to larger values such as 1/(1+e^-10t) or even larger would make the slope even more vertical.
<center> '''P(t)=1/(1+e^-t)''' </center>
<center> [[File:MSP220819e4e417b6708ich00001768587g039a5c38.gif]]</center>
<center> '''P(t)=1/(1+e^-5t)'''</center>
<center>[[File:math grahpie 2.gif]]
<center> '''One application we could use the graph to model is the population of ants in an anthill.'''
<center>'''Equation for the population of ants:
P(t)=10/(1+3e^-t)'''
 The population is in thousands, The time is in weeks 
 '''First off, we must know what the initial population is at time=0 (T0)'''
 
P(0)=10/(1+3e^0)
 
P(0)= 2.5 (in thousands)
 '''With this equation, we can predict what the population will be in 20 weeks.'''
P(20)=10/(1+3e^-20)
P(20)=9,999

 '''Assumptions
 It is important to note that this model functions because of the assumption that the
percentage change in population per week is constant.
It is also important to remember that this anthill is in an isolated location where
there is no outside interference with the population (humans etc.)
 '''Explaining the Equation'''
The 10 in the numerator is the horizontal asymptote. Because this equation is modeling population, the horizontal asymptote, which is 10, shows the carrying capacity for this anthill.

Course:MATH110/Archive/2010-2011/003/Teams/Aargau/Homework 12

2011-01-28T14:36:41Z

YihongChen:

<center>'''Homework 12 Team Problem'''</center>
<center> With the function P(t)=1/(1+e^-t), we can change the height of the horizontal asymptote on the right by changing the value of the numerator 1 to any value. We will use the value of 10, which we wil denote by K. To change the y-intercept to a number between 0 and K, we can change any part of the function to make it have a different y-value when the x-value is 0. We will multiply e^-t by 3. When we plug in 0 for t into the new function P(t)=10/(1+3e^-t), we get 2.5 for the y-intercept value, which is between 0 and K. </center>
<center>'''BONUS'''</center>
<center> After testing different things, we discovered that adding a coefficient to -t will change the slope of the curved part. It must still remain negative. The larger the value of the coefficient, the closer to vertical the slope will be.
<center> Compare the graph of the original function to the graph of P(t)=1/(1+e^-5t). It is hard to tell from the graphs below because the windows they are graphed in are different, but the slope has actually become more vertical while maintaining the same y-intercept and rest of the graph. Changing the coefficient to larger values such as 1/(1+e^-10t) or even larger would make the slope even more vertical.
<center> '''P(t)=1/(1+e^-t)''' </center>
<center> [[File:MSP220819e4e417b6708ich00001768587g039a5c38.gif]]</center>
<center> '''P(t)=1/(1+e^-5t)'''</center>
<center>[[File:math grahpie 2.gif]]
<center> '''One application we could use the graph to model is the population of ants in an anthill.'''
<center>'''Equation for the population of ants:
P(t)=10/(1+3e^-t)'''
 The population is in thousands, The time is in weeks 
 '''First off, we must know what the initial population is at time=0 (T0)'''
 
P(0)=10/(1+3e^0)
 
P(0)= 2.5 (in thousands)
 '''With this equation, we can predict what the population will be in 20 weeks.'''
P(20)=10/(1+3e^-20)
P(20)=9,999

 '''Assumptions
 It is important to note that this model functions because of the assumption that the
percentage change in population per week is constant. 
 
It is also important to remember that this anthill is in an isolated location where
there is no outside interference with the population (humans etc.)

Course:MATH110/Archive/2010-2011/003/Teams/Aargau/Homework 12

2011-01-28T14:33:19Z

YihongChen:

<center>'''Homework 12 Team Problem'''</center>
<center> With the function P(t)=1/(1+e^-t), we can change the height of the horizontal asymptote on the right by changing the value of the numerator 1 to any value. We will use the value of 10, which we wil denote by K. To change the y-intercept to a number between 0 and K, we can change any part of the function to make it have a different y-value when the x-value is 0. We will multiply e^-t by 3. When we plug in 0 for t into the new function P(t)=10/(1+3e^-t), we get 2.5 for the y-intercept value, which is between 0 and K. </center>
<center>'''BONUS'''</center>
<center> After testing different things, we discovered that adding a coefficient to -t will change the slope of the curved part. It must still remain negative. The larger the value of the coefficient, the closer to vertical the slope will be.
<center> Compare the graph of the original function to the graph of P(t)=1/(1+e^-5t). It is hard to tell from the graphs below because the windows they are graphed in are different, but the slope has actually become more vertical while maintaining the same y-intercept and rest of the graph. Changing the coefficient to larger values such as 1/(1+e^-10t) or even larger would make the slope even more vertical.
<center> '''P(t)=1/(1+e^-t)''' </center>
<center> [[File:MSP220819e4e417b6708ich00001768587g039a5c38.gif]]</center>
<center> '''P(t)=1/(1+e^-5t)'''</center>
<center>[[File:math grahpie 2.gif]]
<center> '''One application we could use the graph to model is the population of ants in an anthill.'''
<center>'''Equation for the population of ants:
P(t)=10/(1+3e^-t)'''
 The population is in thousands, The time is in weeks 
 '''First off, we must know what the initial population is at time=0 (T0)'''
 
P(0)=10/(1+3e^0)
 
P(0)= 2.5 (in thousands)
 '''With this equation, we can predict what the population will be in 20 weeks.'''
P(20)=10/(1+3e^-20)
P(20)=9,999

Course:MATH110/Archive/2010-2011/003/Teams/Aargau/Homework 12

2011-01-28T14:28:23Z

YihongChen:

Course:MATH110/Archive/2010-2011/003/Teams/Aargau/Homework 12

2011-01-28T14:25:39Z

YihongChen:

<center>'''Homework 12 Team Problem'''</center>
<center> With the function P(t)=1/(1+e^-t), we can change the height of the horizontal asymptote on the right by changing the value of the numerator 1 to any value. We will use the value of 10, which we wil denote by K. To change the y-intercept to a number between 0 and K, we can change any part of the function to make it have a different y-value when the x-value is 0. We will multiply e^-t by 3. When we plug in 0 for t into the new function P(t)=10/(1+3e^-t), we get 2.5 for the y-intercept value, which is between 0 and K. </center>
<center>'''BONUS'''</center>
<center> After testing different things, we discovered that adding a coefficient to -t will change the slope of the curved part. It must still remain negative. The larger the value of the coefficient, the closer to vertical the slope will be.
<center> Compare the graph of the original function to the graph of P(t)=1/(1+e^-5t). It is hard to tell from the graphs below because the windows they are graphed in are different, but the slope has actually become more vertical while maintaining the same y-intercept and rest of the graph. Changing the coefficient to larger values such as 1/(1+e^-10t) or even larger would make the slope even more vertical.
<center> '''P(t)=1/(1+e^-t)''' </center>
<center> [[File:MSP220819e4e417b6708ich00001768587g039a5c38.gif]]</center>
<center> '''P(t)=1/(1+e^-5t)'''</center>
<center>[[File:math grahpie 2.gif]]
<center> One application we could use the graph to model is the population of ants in an anthill.
<center>Equation for the population of ants:
P(t)=10/(1+3e^-t)

Course:MATH110/Archive/2010-2011/003/Teams/Aargau/Homework 12

2011-01-28T14:24:11Z

YihongChen:

<center>'''Homework 12 Team Problem'''</center>
<center> With the function P(t)=1/(1+e^-t), we can change the height of the horizontal asymptote on the right by changing the value of the numerator 1 to any value. We will use the value of 10, which we wil denote by K. To change the y-intercept to a number between 0 and K, we can change any part of the function to make it have a different y-value when the x-value is 0. We will multiply e^-t by 3. When we plug in 0 for t into the new function P(t)=10/(1+3e^-t), we get 2.5 for the y-intercept value, which is between 0 and K. </center>
<center>BONUS</center>
<center> After testing different things, we discovered that adding a coefficient to -t will change the slope of the curved part. It must still remain negative. The larger the value of the coefficient, the closer to vertical the slope will be.
<center> Compare the graph of the original function to the graph of P(t)=1/(1+e^-5t). It is hard to tell from the graphs below because the windows they are graphed in are different, but the slope has actually become more vertical while maintaining the same y-intercept and rest of the graph. Changing the coefficient to larger values such as 1/(1+e^-10t) or even larger would make the slope even more vertical.
<center> '''P(t)=1/(1+e^-t)''' </center>
<center> [[File:MSP220819e4e417b6708ich00001768587g039a5c38.gif]]</center>
<center> '''P(t)=1/(1+e^-5t)'''</center>
<center>[[File:math grahpie 2.gif]]
<center> One application we could use the graph to model is the population of ants in an anthill.
<center>Equation for the population of ants:

Course:MATH110/Archive/2010-2011/003/Teams/Aargau/Homework 12

2011-01-28T14:17:09Z

YihongChen:

Course:MATH110/Archive/2010-2011/003/Teams/Aargau/Homework 12

2011-01-28T14:08:56Z

YihongChen:

Course:MATH110/Archive/2010-2011/003/Teams/Aargau/Homework 12

2011-01-28T14:07:32Z

YihongChen:

Course:MATH110/Archive/2010-2011/003/Teams/Aargau/Homework 12

2011-01-28T14:04:01Z

YihongChen:

File:Math grahpie 2.gif

2011-01-28T14:02:17Z

YihongChen:

Course:MATH110/Archive/2010-2011/003/Teams/Aargau/Homework 12

2011-01-28T14:01:23Z

YihongChen:

File:Math graphie.gif

2011-01-28T13:56:37Z

YihongChen:

Course:MATH110/Archive/2010-2011/003/Teams/Aargau/Homework 12

2011-01-28T13:56:12Z

YihongChen:

<center>'''Homework 12 Team Problem'''</center>
<center> With the function P(t)=1/1+e^-t, we can change the height of the horizontal asymptote on the right by changing the value of the numerator 1 to any value. We will use the value of 10, which we wil denote by K. To change the y-intercept to a number between 0 and K, we can change any part of the function to make it have a different y-value when the x-value is 0. We will multiply e^-t by 3. When we plug in 0 for t into the new function P(t)=10/1+3e^-t, we get 2.5 for the y-intercept value, which is between 0 and K. </center>
<center>BONUS</center>
<center> After testing different things, we discovered that adding a coefficient to -t will change the slope of the curved part. It must still remain negative. The larger the value of the coefficient, the closer to vertical the slope will be.
<center> Compare the graph of the original function to the graph of P(t)=1/1+e^-3t and P(t)=1/1+e^-20t
<center> P(t)=1/1+e^-t [[File:math graphie.jpg]]

Course:MATH110/Archive/2010-2011/003/Teams/Aargau/Homework 12

2011-01-28T13:30:22Z

YihongChen:

<center>'''Homework 12 Team Problem'''</center>
 With the function P(t)=1/1+e^-t, we can change the height of the horizontal asymptote on the right by changing the value of the numerator 1 to any value. We will use the value of 10, which we wil denote by K. To change the y-intercept to a number between 0 and K, we can change any part of the function to make it have a different y-value when the x-value is 0. We will multiply e^-t by 3. When we plug in 0 for t into the new function P(t)=10/1+3e^-t, we get 2.5 for the y-intercept value, which is between 0 and K.

Course:MATH110/Archive/2010-2011/003/Teams/Aargau/Homework 12

2011-01-28T13:20:06Z

YihongChen:

<center>'''Homework 12 Team Problem'''</center>
 With the function <math>P(t)=1/1+e^(-t)</math>, we can change the height of the horizontal asymptote on the right by changing the value of 1 to any value. We will use the value of 10, which we wil denote by K. To change the y-intercept to a number between 0 and K,

Course:MATH110/Archive/2010-2011/003/Teams/Aargau/Homework 12

2011-01-28T13:18:56Z

YihongChen:

<center>==Homework 12 Team Problem==</center>
 With the function <math>P(t)=1/1+e^-t</math>, we can change the height of the horizontal asymptote on the right by changing the value of 1 to any value. We will use the value of 10, which we wil denote by K. To change the y-intercept to a number between 0 and K,

Course:MATH110/Archive/2010-2011/003/Teams/Aargau/Homework 12

2011-01-28T13:08:05Z

YihongChen: Created page with " <center><bold>Homework 12 Team Problem</center></bold> With the function below: File:whatsthefunction.gif"

<center><bold>Homework 12 Team Problem</center></bold>
 With the function below:
 [[File:whatsthefunction.gif]]

User:YihongChen

2011-01-28T12:50:49Z

YihongChen:

Hi, I'm Yihong. I'm a first year student in the faculty of arts planning to switch into commerce.

Pythagorean Theorem

The Pythagorean theorem is a very useful theorem named after Pythagoras. Although it is named after him, it was also known by Indian, Greek, Chinese and Babylonian mathematicians before he was born. The theorem states that the square of the hypotenuse in a right angle triangle is equal to the sum of the squares of the other two sides. The theorem is typically represented as a^2+b^2=c^2, where c is the length of hypotenuse and a and b are the other two sides. Knowing this, you can find out the length of any side of a right angle triangle if you know the length of the other two sides. This theorem is used in everyday applications. An example is: a window cleaner needs to find out how long a ladder to get to clean windows that are 10 feet above the ground. He has to place the ladder 10 feet away from the building in order to avoid flowers. 10^2+10^2=200, which is the square of the length of the ladder needed. The square root of 200 is aabout 14.142, so the window cleaner needs a ladder that is 15 feet. The relevance and usefulness of the Pythagorean theorem in many applications has been recognized in Greece, Japan, San Marino, Sierra Leone, and Surinam with postage stamps depicting Pythagoras and the Pythagorean theorem as well as a Ugandan coin released in 2000 in the shape of a right triangle and an image of Pythagoras and the Pythagorean theorem. This theorem has proven to be and will continue to be very practical and useful in many applications.

--[[User:YihongChen|YihongChen]] 02:29, 20 September 2010 (UTC)

<center> Calculus in Economics</center>
 Calculus provides the means by which economists solve problems. One important topic in economics is elasticity. Elasticity measures the responsiveness of a function to changes in parameters and is the ratio of the percent change in one variable to the percent change in another variable. We can use basic calculus to calculate elasticity. When we are given a formula such as Z = f(X), elasticity = (percentage change in Z) / (percentage change in Y). (percentage change in Z) / (percentage change in Y) = (dZ / dY)*(Y/Z) where dZ/dY is the partial derivative of Z with respect to Y. Thus, elasticity of Z with respect to Y = (dZ / dY)*(Y/Z) 
 Calculus is also often used to examine functional relationships in economics, such as the relationship between the dependent variable income and independent variables, such as education and experience. If average income increases as amount of education and work experience increase, then a positive relationship exists between the variables. Derivatives in calculus are identical to the economic concepts of marginalism, which examines the change in an outcome that results from a single-unit rise in another variable, such as the average change in income relative to a single year’s rise in education and/or experience. Marginal changes relate to an important principle in economics: that people tend to think at the margin. According to Harvard economist Greg Mankiw, economists use the term “marginal changes” to describe small, incremental changes, such as incremental changes in work hours or factory output. If benefits (B) and costs (C) depend on the level of an activity (x), then the derivative of B with respect to x represents marginal benefit and the derivative of C with respect to x represents marginal cost. If revenue depends on the quantity of a good sold, then the derivative of revenue with respect to quantity represents marginal revenue. If production depends on labor input, then the derivative of production with respect to labor input represents the marginal product of labor. If resource cost depends on labor input, then the derivative of resource cost with respect to labor input represents the marginal resource cost of labor.
 Calculus can help business managers maximize their profits and measure the rate of increase in profit that results from each increase in production. The firm increases its profits as long as marginal revenue exceeds marginal cost. For a given product, the slope is defined as the rate of change in the Y variable (total cost of a product for example) for a given change in the X variable (Q, or units of the good). Taking the first derivative, or calculating the formula for the slope can determine the marginal cost for a particular good. We can not only evaluate costs at a particular level, but also see how our marginal costs are changing as we increase or decrease our level of production. The change in marginal cost or change in slope can be calculated by taking the second derivative. 
 Finally, calculus provides a means for determining the amount of interest paid over the life of a loan. It is helpful for the interest rates for a number of things, including mortgages, loans and advances. 

<center>[[File:econn.gif]]

User:YihongChen

2011-01-28T12:49:52Z

YihongChen:

Hi, I'm Yihong. I'm a first year student in the faculty of arts planning to switch into commerce.

Pythagorean Theorem

The Pythagorean theorem is a very useful theorem named after Pythagoras. Although it is named after him, it was also known by Indian, Greek, Chinese and Babylonian mathematicians before he was born. The theorem states that the square of the hypotenuse in a right angle triangle is equal to the sum of the squares of the other two sides. The theorem is typically represented as a^2+b^2=c^2, where c is the length of hypotenuse and a and b are the other two sides. Knowing this, you can find out the length of any side of a right angle triangle if you know the length of the other two sides. This theorem is used in everyday applications. An example is: a window cleaner needs to find out how long a ladder to get to clean windows that are 10 feet above the ground. He has to place the ladder 10 feet away from the building in order to avoid flowers. 10^2+10^2=200, which is the square of the length of the ladder needed. The square root of 200 is aabout 14.142, so the window cleaner needs a ladder that is 15 feet. The relevance and usefulness of the Pythagorean theorem in many applications has been recognized in Greece, Japan, San Marino, Sierra Leone, and Surinam with postage stamps depicting Pythagoras and the Pythagorean theorem as well as a Ugandan coin released in 2000 in the shape of a right triangle and an image of Pythagoras and the Pythagorean theorem. This theorem has proven to be and will continue to be very practical and useful in many applications.

--[[User:YihongChen|YihongChen]] 02:29, 20 September 2010 (UTC)

<center> Calculus in Economics</center>
 Calculus provides the means by which economists solve problems. One important topic in economics is elasticity. Elasticity measures the responsiveness of a function to changes in parameters and is the ratio of the percent change in one variable to the percent change in another variable. We can use basic calculus to calculate elasticity. When we are given a formula such as Z = f(X), elasticity = (percentage change in Z) / (percentage change in Y). (percentage change in Z) / (percentage change in Y) = (dZ / dY)*(Y/Z) where dZ/dY is the partial derivative of Z with respect to Y. Thus, elasticity of Z with respect to Y = (dZ / dY)*(Y/Z)
 Calculus is also often used to examine functional relationships in economics, such as the relationship between the dependent variable income and independent variables, such as education and experience. If average income increases as amount of education and work experience increase, then a positive relationship exists between the variables. Derivatives in calculus are identical to the economic concepts of marginalism, which examines the change in an outcome that results from a single-unit rise in another variable, such as the average change in income relative to a single year’s rise in education and/or experience. Marginal changes relate to an important principle in economics: that people tend to think at the margin. According to Harvard economist Greg Mankiw, economists use the term “marginal changes” to describe small, incremental changes, such as incremental changes in work hours or factory output. If benefits (B) and costs (C) depend on the level of an activity (x), then the derivative of B with respect to x represents marginal benefit and the derivative of C with respect to x represents marginal cost. If revenue depends on the quantity of a good sold, then the derivative of revenue with respect to quantity represents marginal revenue. If production depends on labor input, then the derivative of production with respect to labor input represents the marginal product of labor. If resource cost depends on labor input, then the derivative of resource cost with respect to labor input represents the marginal resource cost of labor.
 Calculus can help business managers maximize their profits and measure the rate of increase in profit that results from each increase in production. The firm increases its profits as long as marginal revenue exceeds marginal cost. For a given product, the slope is defined as the rate of change in the Y variable (total cost of a product for example) for a given change in the X variable (Q, or units of the good). Taking the first derivative, or calculating the formula for the slope can determine the marginal cost for a particular good. We can not only evaluate costs at a particular level, but also see how our marginal costs are changing as we increase or decrease our level of production. The change in marginal cost or change in slope can be calculated by taking the second derivative.
 Finally, calculus provides a means for determining the amount of interest paid over the life of a loan. It is helpful for the interest rates for a number of things, including mortgages, loans and advances.

<center>[[File:econn.gif]]

File:Econn.gif

2011-01-28T12:48:57Z

YihongChen:

User:YihongChen

2011-01-28T12:48:09Z

YihongChen:

Hi, I'm Yihong. I'm a first year student in the faculty of arts planning to switch into commerce.

Pythagorean Theorem

The Pythagorean theorem is a very useful theorem named after Pythagoras. Although it is named after him, it was also known by Indian, Greek, Chinese and Babylonian mathematicians before he was born. The theorem states that the square of the hypotenuse in a right angle triangle is equal to the sum of the squares of the other two sides. The theorem is typically represented as a^2+b^2=c^2, where c is the length of hypotenuse and a and b are the other two sides. Knowing this, you can find out the length of any side of a right angle triangle if you know the length of the other two sides. This theorem is used in everyday applications. An example is: a window cleaner needs to find out how long a ladder to get to clean windows that are 10 feet above the ground. He has to place the ladder 10 feet away from the building in order to avoid flowers. 10^2+10^2=200, which is the square of the length of the ladder needed. The square root of 200 is aabout 14.142, so the window cleaner needs a ladder that is 15 feet. The relevance and usefulness of the Pythagorean theorem in many applications has been recognized in Greece, Japan, San Marino, Sierra Leone, and Surinam with postage stamps depicting Pythagoras and the Pythagorean theorem as well as a Ugandan coin released in 2000 in the shape of a right triangle and an image of Pythagoras and the Pythagorean theorem. This theorem has proven to be and will continue to be very practical and useful in many applications.

--[[User:YihongChen|YihongChen]] 02:29, 20 September 2010 (UTC)

<center> Calculus in Economics</center>
 Calculus provides the means by which economists solve problems. One important topic in economics is elasticity. Elasticity measures the responsiveness of a function to changes in parameters and is the ratio of the percent change in one variable to the percent change in another variable. We can use basic calculus to calculate elasticity. When we are given a formula such as Z = f(X), elasticity = (percentage change in Z) / (percentage change in Y). (percentage change in Z) / (percentage change in Y) = (dZ / dY)*(Y/Z) where dZ/dY is the partial derivative of Z with respect to Y. Thus, elasticity of Z with respect to Y = (dZ / dY)*(Y/Z)
 Calculus is also often used to examine functional relationships in economics, such as the relationship between the dependent variable income and independent variables, such as education and experience. If average income increases as amount of education and work experience increase, then a positive relationship exists between the variables. Derivatives in calculus are identical to the economic concepts of marginalism, which examines the change in an outcome that results from a single-unit rise in another variable, such as the average change in income relative to a single year’s rise in education and/or experience. Marginal changes relate to an important principle in economics: that people tend to think at the margin. According to Harvard economist Greg Mankiw, economists use the term “marginal changes” to describe small, incremental changes, such as incremental changes in work hours or factory output. If benefits (B) and costs (C) depend on the level of an activity (x), then the derivative of B with respect to x represents marginal benefit and the derivative of C with respect to x represents marginal cost. If revenue depends on the quantity of a good sold, then the derivative of revenue with respect to quantity represents marginal revenue. If production depends on labor input, then the derivative of production with respect to labor input represents the marginal product of labor. If resource cost depends on labor input, then the derivative of resource cost with respect to labor input represents the marginal resource cost of labor.
 Calculus can help business managers maximize their profits and measure the rate of increase in profit that results from each increase in production. The firm increases its profits as long as marginal revenue exceeds marginal cost. For a given product, the slope is defined as the rate of change in the Y variable (total cost of a product for example) for a given change in the X variable (Q, or units of the good). Taking the first derivative, or calculating the formula for the slope can determine the marginal cost for a particular good. We can not only evaluate costs at a particular level, but also see how our marginal costs are changing as we increase or decrease our level of production. The change in marginal cost or change in slope can be calculated by taking the second derivative.
 Finally, calculus provides a means for determining the amount of interest paid over the life of a loan. It is helpful for the interest rates for a number of things, including mortgages, loans and advances.

[[File:econn.jpg]]

User:YihongChen

2011-01-28T12:47:38Z

YihongChen:

Hi, I'm Yihong. I'm a first year student in the faculty of arts planning to switch into commerce.

Pythagorean Theorem

The Pythagorean theorem is a very useful theorem named after Pythagoras. Although it is named after him, it was also known by Indian, Greek, Chinese and Babylonian mathematicians before he was born. The theorem states that the square of the hypotenuse in a right angle triangle is equal to the sum of the squares of the other two sides. The theorem is typically represented as a^2+b^2=c^2, where c is the length of hypotenuse and a and b are the other two sides. Knowing this, you can find out the length of any side of a right angle triangle if you know the length of the other two sides. This theorem is used in everyday applications. An example is: a window cleaner needs to find out how long a ladder to get to clean windows that are 10 feet above the ground. He has to place the ladder 10 feet away from the building in order to avoid flowers. 10^2+10^2=200, which is the square of the length of the ladder needed. The square root of 200 is aabout 14.142, so the window cleaner needs a ladder that is 15 feet. The relevance and usefulness of the Pythagorean theorem in many applications has been recognized in Greece, Japan, San Marino, Sierra Leone, and Surinam with postage stamps depicting Pythagoras and the Pythagorean theorem as well as a Ugandan coin released in 2000 in the shape of a right triangle and an image of Pythagoras and the Pythagorean theorem. This theorem has proven to be and will continue to be very practical and useful in many applications.

--[[User:YihongChen|YihongChen]] 02:29, 20 September 2010 (UTC)

<center> Calculus in Economics</center>
 Calculus provides the means by which economists solve problems. One important topic in economics is elasticity. Elasticity measures the responsiveness of a function to changes in parameters and is the ratio of the percent change in one variable to the percent change in another variable. We can use basic calculus to calculate elasticity. When we are given a formula such as Z = f(X), elasticity = (percentage change in Z) / (percentage change in Y). (percentage change in Z) / (percentage change in Y) = (dZ / dY)*(Y/Z) where dZ/dY is the partial derivative of Z with respect to Y. Thus, elasticity of Z with respect to Y = (dZ / dY)*(Y/Z)
 Calculus is also often used to examine functional relationships in economics, such as the relationship between the dependent variable income and independent variables, such as education and experience. If average income increases as amount of education and work experience increase, then a positive relationship exists between the variables. Derivatives in calculus are identical to the economic concepts of marginalism, which examines the change in an outcome that results from a single-unit rise in another variable, such as the average change in income relative to a single year’s rise in education and/or experience. Marginal changes relate to an important principle in economics: that people tend to think at the margin. According to Harvard economist Greg Mankiw, economists use the term “marginal changes” to describe small, incremental changes, such as incremental changes in work hours or factory output. If benefits (B) and costs (C) depend on the level of an activity (x), then the derivative of B with respect to x represents marginal benefit and the derivative of C with respect to x represents marginal cost. If revenue depends on the quantity of a good sold, then the derivative of revenue with respect to quantity represents marginal revenue. If production depends on labor input, then the derivative of production with respect to labor input represents the marginal product of labor. If resource cost depends on labor input, then the derivative of resource cost with respect to labor input represents the marginal resource cost of labor.
 Calculus can help business managers maximize their profits and measure the rate of increase in profit that results from each increase in production. The firm increases its profits as long as marginal revenue exceeds marginal cost. For a given product, the slope is defined as the rate of change in the Y variable (total cost of a product for example) for a given change in the X variable (Q, or units of the good). Taking the first derivative, or calculating the formula for the slope can determine the marginal cost for a particular good. We can not only evaluate costs at a particular level, but also see how our marginal costs are changing as we increase or decrease our level of production. The change in marginal cost or change in slope can be calculated by taking the second derivative.
 Finally, calculus provides a means for determining the amount of interest paid over the life of a loan. It is helpful for the interest rates for a number of things, including mortgages, loans and advances.

[[File:economics.jpg]]

User:YihongChen

2011-01-28T12:45:51Z

YihongChen:

Hi, I'm Yihong. I'm a first year student in the faculty of arts planning to switch into commerce.

Pythagorean Theorem

The Pythagorean theorem is a very useful theorem named after Pythagoras. Although it is named after him, it was also known by Indian, Greek, Chinese and Babylonian mathematicians before he was born. The theorem states that the square of the hypotenuse in a right angle triangle is equal to the sum of the squares of the other two sides. The theorem is typically represented as a^2+b^2=c^2, where c is the length of hypotenuse and a and b are the other two sides. Knowing this, you can find out the length of any side of a right angle triangle if you know the length of the other two sides. This theorem is used in everyday applications. An example is: a window cleaner needs to find out how long a ladder to get to clean windows that are 10 feet above the ground. He has to place the ladder 10 feet away from the building in order to avoid flowers. 10^2+10^2=200, which is the square of the length of the ladder needed. The square root of 200 is aabout 14.142, so the window cleaner needs a ladder that is 15 feet. The relevance and usefulness of the Pythagorean theorem in many applications has been recognized in Greece, Japan, San Marino, Sierra Leone, and Surinam with postage stamps depicting Pythagoras and the Pythagorean theorem as well as a Ugandan coin released in 2000 in the shape of a right triangle and an image of Pythagoras and the Pythagorean theorem. This theorem has proven to be and will continue to be very practical and useful in many applications.

--[[User:YihongChen|YihongChen]] 02:29, 20 September 2010 (UTC)

<center> Calculus in Economics</center>
 Calculus provides the means by which economists solve problems. One important topic in economics is elasticity. Elasticity measures the responsiveness of a function to changes in parameters and is the ratio of the percent change in one variable to the percent change in another variable. We can use basic calculus to calculate elasticity. When we are given a formula such as Z = f(X), elasticity = (percentage change in Z) / (percentage change in Y). (percentage change in Z) / (percentage change in Y) = (dZ / dY)*(Y/Z) where dZ/dY is the partial derivative of Z with respect to Y. Thus, elasticity of Z with respect to Y = (dZ / dY)*(Y/Z) 
 Calculus is also often used to examine functional relationships in economics, such as the relationship between the dependent variable income and independent variables, such as education and experience. If average income increases as amount of education and work experience increase, then a positive relationship exists between the variables. Derivatives in calculus are identical to the economic concepts of marginalism, which examines the change in an outcome that results from a single-unit rise in another variable, such as the average change in income relative to a single year’s rise in education and/or experience. Marginal changes relate to an important principle in economics: that people tend to think at the margin. According to Harvard economist Greg Mankiw, economists use the term “marginal changes” to describe small, incremental changes, such as incremental changes in work hours or factory output. If benefits (B) and costs (C) depend on the level of an activity (x), then the derivative of B with respect to x represents marginal benefit and the derivative of C with respect to x represents marginal cost. If revenue depends on the quantity of a good sold, then the derivative of revenue with respect to quantity represents marginal revenue. If production depends on labor input, then the derivative of production with respect to labor input represents the marginal product of labor. If resource cost depends on labor input, then the derivative of resource cost with respect to labor input represents the marginal resource cost of labor.
 Calculus can help business managers maximize their profits and measure the rate of increase in profit that results from each increase in production. The firm increases its profits as long as marginal revenue exceeds marginal cost. For a given product, the slope is defined as the rate of change in the Y variable (total cost of a product for example) for a given change in the X variable (Q, or units of the good). Taking the first derivative, or calculating the formula for the slope can determine the marginal cost for a particular good. We can not only evaluate costs at a particular level, but also see how our marginal costs are changing as we increase or decrease our level of production. The change in marginal cost or change in slope can be calculated by taking the second derivative.
 Finally, calculus provides a means for determining the amount of interest paid over the life of a loan. It is helpful for the interest rates for a number of things, including mortgages, loans and advances.

User:YihongChen

2011-01-28T12:35:11Z

YihongChen:

Hi, I'm Yihong. I'm a first year student in the faculty of arts planning to switch into commerce.

Pythagorean Theorem

The Pythagorean theorem is a very useful theorem named after Pythagoras. Although it is named after him, it was also known by Indian, Greek, Chinese and Babylonian mathematicians before he was born. The theorem states that the square of the hypotenuse in a right angle triangle is equal to the sum of the squares of the other two sides. The theorem is typically represented as a^2+b^2=c^2, where c is the length of hypotenuse and a and b are the other two sides. Knowing this, you can find out the length of any side of a right angle triangle if you know the length of the other two sides. This theorem is used in everyday applications. An example is: a window cleaner needs to find out how long a ladder to get to clean windows that are 10 feet above the ground. He has to place the ladder 10 feet away from the building in order to avoid flowers. 10^2+10^2=200, which is the square of the length of the ladder needed. The square root of 200 is aabout 14.142, so the window cleaner needs a ladder that is 15 feet. The relevance and usefulness of the Pythagorean theorem in many applications has been recognized in Greece, Japan, San Marino, Sierra Leone, and Surinam with postage stamps depicting Pythagoras and the Pythagorean theorem as well as a Ugandan coin released in 2000 in the shape of a right triangle and an image of Pythagoras and the Pythagorean theorem. This theorem has proven to be and will continue to be very practical and useful in many applications.

--[[User:YihongChen|YihongChen]] 02:29, 20 September 2010 (UTC)

<center> Calculus in Economics</center>
Calculus provides the means by which economists solve problems. One important topic in economics is elasticity. Elasticity measures the responsiveness of a function to changes in parameters and is the ratio of the percent change in one variable to the percent change in another variable. We can use basic calculus to calculate elasticity. When we are given a formula such as Z = f(X), elasticity = (percentage change in Z) / (percentage change in Y). (percentage change in Z) / (percentage change in Y) = (dZ / dY)*(Y/Z) where dZ/dY is the partial derivative of Z with respect to Y. Thus, elasticity of Z with respect to Y = (dZ / dY)*(Y/Z) Calculus is also often used to examine functional relationships in economics, such as the relationship between the dependent variable income and independent variables, such as education and experience. If average income increases as amount of education and work experience increase, then a positive relationship exists between the variables. Derivatives in calculus are identical to the economic concepts of marginalism, which examines the change in an outcome that results from a single-unit rise in another variable, such as the average change in income relative to a single year’s rise in education and/or experience. Marginal changes relate to an important principle in economics: that people tend to think at the margin. According to Harvard economist Greg Mankiw, economists use the term “marginal changes” to describe small, incremental changes, such as incremental changes in work hours or factory output. If benefits (B) and costs (C) depend on the level of an activity (x), then the derivative of B with respect to x represents marginal benefit and the derivative of C with respect to x represents marginal cost. If revenue depends on the quantity of a good sold, then the derivative of revenue with respect to quantity represents marginal revenue. If production depends on labor input, then the derivative of production with respect to labor input represents the marginal product of labor. If resource cost depends on labor input, then the derivative of resource cost with respect to labor input represents the marginal resource cost of labor.

User:YihongChen

2011-01-28T12:33:53Z

YihongChen:

Hi, I'm Yihong. I'm a first year student in the faculty of arts planning to switch into commerce.

Pythagorean Theorem

The Pythagorean theorem is a very useful theorem named after Pythagoras. Although it is named after him, it was also known by Indian, Greek, Chinese and Babylonian mathematicians before he was born. The theorem states that the square of the hypotenuse in a right angle triangle is equal to the sum of the squares of the other two sides. The theorem is typically represented as a^2+b^2=c^2, where c is the length of hypotenuse and a and b are the other two sides. Knowing this, you can find out the length of any side of a right angle triangle if you know the length of the other two sides. This theorem is used in everyday applications. An example is: a window cleaner needs to find out how long a ladder to get to clean windows that are 10 feet above the ground. He has to place the ladder 10 feet away from the building in order to avoid flowers. 10^2+10^2=200, which is the square of the length of the ladder needed. The square root of 200 is aabout 14.142, so the window cleaner needs a ladder that is 15 feet. The relevance and usefulness of the Pythagorean theorem in many applications has been recognized in Greece, Japan, San Marino, Sierra Leone, and Surinam with postage stamps depicting Pythagoras and the Pythagorean theorem as well as a Ugandan coin released in 2000 in the shape of a right triangle and an image of Pythagoras and the Pythagorean theorem. This theorem has proven to be and will continue to be very practical and useful in many applications.

--[[User:YihongChen|YihongChen]] 02:29, 20 September 2010 (UTC)

<center> Calculus in Economics</center>
Calculus provides the means by which economists solve problems. One important topic in economics is elasticity. Elasticity measures the responsiveness of a function to changes in parameters and is the ratio of the percent change in one variable to the percent change in another variable. We can use basic calculus to calculate elasticity. When we are given a formula such as Z = f(X), elasticity = (percentage change in Z) / (percentage change in Y). (percentage change in Z) / (percentage change in Y) = (dZ / dY)*(Y/Z) where dZ/dY is the partial derivative of Z with respect to Y. Thus, elasticity of Z with respect to Y = (dZ / dY)*(Y/Z) Calculus is also often used to examine functional relationships in economics, such as the relationship between the dependent variable income and independent variables, such as education and experience. If average income increases as amount of education and work experience increase, then a positive relationship exists between the variables. Derivatives in calculus are identical to the economic concepts of marginalism, which examines the change in an outcome that results from a single-unit rise in another variable, such as the average change in income relative to a single year’s rise in education and/or experience. Marginal changes relate to an important principle in economics: that people tend to think at the margin. According to Harvard economist Greg Mankiw, economists use the term “marginal changes” to describe small, incremental changes, such as incremental changes in work hours or factory output. If benefits (B) and costs (C) depend on the level of an activity (x), then the derivative of B with respect to x represents marginal benefit and the derivative of C with respect to x represents marginal cost. If revenue depends on the quantity of a good sold, then

User:YihongChen

2011-01-28T12:22:31Z

YihongChen:

Hi, I'm Yihong. I'm a first year student in the faculty of arts planning to switch into commerce.

Pythagorean Theorem

The Pythagorean theorem is a very useful theorem named after Pythagoras. Although it is named after him, it was also known by Indian, Greek, Chinese and Babylonian mathematicians before he was born. The theorem states that the square of the hypotenuse in a right angle triangle is equal to the sum of the squares of the other two sides. The theorem is typically represented as a^2+b^2=c^2, where c is the length of hypotenuse and a and b are the other two sides. Knowing this, you can find out the length of any side of a right angle triangle if you know the length of the other two sides. This theorem is used in everyday applications. An example is: a window cleaner needs to find out how long a ladder to get to clean windows that are 10 feet above the ground. He has to place the ladder 10 feet away from the building in order to avoid flowers. 10^2+10^2=200, which is the square of the length of the ladder needed. The square root of 200 is aabout 14.142, so the window cleaner needs a ladder that is 15 feet. The relevance and usefulness of the Pythagorean theorem in many applications has been recognized in Greece, Japan, San Marino, Sierra Leone, and Surinam with postage stamps depicting Pythagoras and the Pythagorean theorem as well as a Ugandan coin released in 2000 in the shape of a right triangle and an image of Pythagoras and the Pythagorean theorem. This theorem has proven to be and will continue to be very practical and useful in many applications.

--[[User:YihongChen|YihongChen]] 02:29, 20 September 2010 (UTC)

<center> Calculus in Economics</center>
Calculus provides the means by which economists solve problems. One important topic in economics is elasticity. Elasticity measures the responsiveness of a function to changes in parameters and is the ratio of the percent change in one variable to the percent change in another variable. We can use basic calculus to calculate elasticity. When we are given a formula such as Z = f(X), elasticity = (percentage change in Z) / (percentage change in Y).
(percentage change in Z) / (percentage change in Y) = (dZ / dY)*(Y/Z)where dZ/dY is the partial derivative of Z with respect to Y.
Thus, elasticity of Z with respect to Y = (dZ / dY)*(Y/Z)

Calculus is also often used to examine functional relationships in economics, such as the relationship between the dependent variable income and independent variables, such as education and experience. If average income increases as amount of education and work experience increase, then a positive relationship exists between the variables. Differential calculus allows economists to measure the average change in income relative to a single year’s rise in education and/or experience.

Derivatives in calculus are identical to the economic concepts of marginalism, which examines the change in an outcome that results from a single-unit rise in another variable. Marginal changes relate to an important principle in economics: that people tend to think at the margin. According to Harvard economist Greg Mankiw, economists use the term “marginal changes” to describe small, incremental changes, such as incremental changes in work hours or factory output.

Calculus can help business managers maximize their profits and measure the rate of increase in profit that results from each increase in production. For a given product, the slope is defined as the rate of change in the Y variable (total cost of a product for example) for a given change in the X variable (Q, or units of the good). Taking the first derivative, or calculating the formula for the slope can determine the marginal cost for a particular good.

User:YihongChen

2011-01-28T11:53:27Z

YihongChen:

Hi, I'm Yihong. I'm a first year student in the faculty of arts planning to switch into commerce.

Pythagorean Theorem

The Pythagorean theorem is a very useful theorem named after Pythagoras. Although it is named after him, it was also known by Indian, Greek, Chinese and Babylonian mathematicians before he was born. The theorem states that the square of the hypotenuse in a right angle triangle is equal to the sum of the squares of the other two sides. The theorem is typically represented as a^2+b^2=c^2, where c is the length of hypotenuse and a and b are the other two sides. Knowing this, you can find out the length of any side of a right angle triangle if you know the length of the other two sides. This theorem is used in everyday applications. An example is: a window cleaner needs to find out how long a ladder to get to clean windows that are 10 feet above the ground. He has to place the ladder 10 feet away from the building in order to avoid flowers. 10^2+10^2=200, which is the square of the length of the ladder needed. The square root of 200 is aabout 14.142, so the window cleaner needs a ladder that is 15 feet. The relevance and usefulness of the Pythagorean theorem in many applications has been recognized in Greece, Japan, San Marino, Sierra Leone, and Surinam with postage stamps depicting Pythagoras and the Pythagorean theorem as well as a Ugandan coin released in 2000 in the shape of a right triangle and an image of Pythagoras and the Pythagorean theorem. This theorem has proven to be and will continue to be very practical and useful in many applications.

--[[User:YihongChen|YihongChen]] 02:29, 20 September 2010 (UTC)

<center> Calculus in Economics</center>
Calculus provides the means by which economists solve problems. One important topic in economics is elasticity. Elasticity measures the responsiveness of a function to changes in parameters and is the ratio of the percent change in one variable to the percent change in another variable. We can use basic calculus to calculate elasticity. When we are given a formula such as Z = f(X), elasticity = (percentage change in Z) / (percentage change in Y).
(percentage change in Z) / (percentage change in Y) = (dZ / dY)*(Y/Z)where dZ/dY is the partial derivative of Z with respect to Y.
Thus, elasticity of Z with respect to Y = (dZ / dY)*(Y/Z)

Calculus is also often used to examine functional relationships in economics, such as the relationship between the dependent variable income and independent variables, such as education and experience. If average income increases as amount of education and work experience increase, then a positive relationship exists between the variables. Differential calculus allows economists to measure the average change in income relative to a single year’s rise in education and/or experience.

Derivatives in calculus are identical to the economic concepts of marginalism, which examines the change in an outcome that results from a single-unit rise in another variable. Marginal changes relate to an important principle in economics: that people tend to think at the margin. According to Harvard economist Greg Mankiw, economists use the term “marginal changes” to describe small, incremental changes, such as incremental changes in work hours or factory output.

Calculus can help business managers maximize their profits and measure the rate of increase in profit that results from each increase in production. The firm increases its profits as long as marginal revenue exceeds marginal cost.

User:YihongChen

2011-01-28T11:38:08Z

YihongChen:

Course:MATH110/Archive/2010-2011/003/Teams/Aargau

2011-01-19T11:56:14Z

YihongChen:

Course:MATH110/Archive/2010-2011/003/Teams/Aargau

2011-01-19T11:51:55Z

YihongChen:

Course:MATH110/Archive/2010-2011/003/Teams/Aargau/Homework 11

2011-01-19T11:49:15Z

YihongChen:

The marginal cost is the cost of producing one more unit.
The data that is given to us is:
It will cost 100 dollars to create 20 units, and it will cost 7 dollars per unit if you want to create more.
In this situation, we are given two pieces of information: one point on our graph (20,100) and the slope of the linear function, which is 7. We can find the equation.
 y-y1=m(x-x1)
 y-100=7(x-20)
 y-100=7x-140
 y=7x-40 
We can find how much it would cost to produce 150 units using this equation.
 Y=7(150)-40
 Y=1050-40
 Y=1010 
It will cost 1010 dollars to produce 150 units.
Because this is a linear equation, the average cost of each unit as production increases remains constant.
We can find other models that show other things about the average cost as production increases. We can do this by imagining that on the graphs, the y-axis is the average cost and the x-axis is the quantity produced.
In a situation where the average cost remains constant as production increases, the function would have to be linear and the slope 0. An example is y=5 or y=9
If the average cost diminishes as production increases, it means that the slope is negative, or that the marginal cost is negative. An example would be y=-7x-40
If the average cost is increasing, we need a positive slope. An example would be y=10x+5. It’s important to note that these functions are plotted where the y-axis is average cost and the x-axis is quantity produced.
In a situation when a company reaches economies of scale, the equation will have to have a negative slope, similar to a problem that was explored earlier.

Course:MATH110/Archive/2010-2011/003/Teams/Aargau/Homework 11

2011-01-19T11:47:33Z

YihongChen: Created page with " The marginal cost is the cost of producing one more unit. The data that is given to us is: It will cost 100 dollars to create 20 units, and it will cost 7 dollars per unit if..."

The marginal cost is the cost of producing one more unit.
The data that is given to us is:
It will cost 100 dollars to create 20 units, and it will cost 7 dollars per unit if you want to create more.
In this situation, we are given two pieces of information: one point on our graph (20,100) and the slope of the linear function, which is 7. We can find the equation.
y-y1=m(x-x1)
y-100=7(x-20)
y-100=7x-140
y=7x-40
We can find how much it would cost to produce 150 units using this equation.
Y=7(150)-40
Y=1050-40
Y=1010
It will cost 1010 dollars to produce 150 units.
Because this is a linear equation, the average cost of each unit as production increases remains constant.
We can find other models that show other things about the average cost as production increases. We can do this by imagining that on the graphs, the y-axis is the average cost and the x-axis is the quantity produced.
In a situation where the average cost remains constant as production increases, the function would have to be linear and the slope 0. An example is y=5 or y=9
If the average cost diminishes as production increases, it means that the slope is negative, or that the marginal cost is negative. An example would be y=-7x-40
If the average cost is increasing, we need a positive slope. An example would be y=10x+5. It’s important to note that these functions are plotted where the y-axis is average cost and the x-axis is quantity produced.
In a situation when a company reaches economies of scale, the equation will have to have a negative slope, similar to a problem that was explored earlier.

Course:MATH110/Archive/2010-2011/003/Groups/Group 02/Basic Skills Project

2010-12-03T16:00:54Z

YihongChen: /* Practice Problems */

{| class="wikitable"
!style="background: #F75D59; text-align: left; padding:3px;"| WELCOME TO GROUP TWO'S PAGE: DISTANCE AND LINES
|-
|style="background: #FBBBB9; padding:12px;"|
Welcome to [http://wiki.ubc.ca/Course:MATH110/003/Groups/Group_02 Group 2]`s page. Our contribution to the Basic Skills Project is the topic of '''Distance and Lines.''' On this page you will find
* detailed step-by-step examples
* tutorial videos
* tips and tricks
* practice problems
* helpful links
on the all of the sub-topics.

[[File:Index.jpg]]Please Visit [http://www.youtube.com/user/math110group2 Group 2's Youtube Page] for
[http://www.youtube.com/user/math110group2#p/u videos created by us] and [http://www.youtube.com/user/math110group2#p/f videos we find informative and helpful.]

|}

===What does it mean for two lines to be parallel and/or perpendicular?===
----

{| class="wikitable"
!style="background: #3BB9FF; text-align: left; padding:3px;"| How do you know if two lines are parallel?
|style="background: #FFE6EA; padding:12px;"|
<center>'''Their slopes are the same!'''</center>
|}

[[File:parallelpic.gif]]

{| class="wikitable"
!style="background: #F75D59; text-align: left; padding:3px;"| How do you know if two lines are perpendicular?
|style="background: #FBBBB9; padding:12px;"|
<center>'''When you multiply their slopes, you get -1!'''</center>

|}

[[File:Per.gif]]
====Examples====

{| class="wikitable"
!style="background: #F75D59; text-align: left; padding:3px;"| '''Parallel Lines Example Question 1'''
|-
|style="background: #FBBBB9;padding:12px;"|

'''
'''Problem: Determine whether the graphs of y = -3x + 5 and 4y = -12x + 20 are parallel lines.'''''
*Solve for y for both graphs
y = -3x + 5 >already solved.

4y = -12x + 20 >Solve for y

4y=-12x+20 >Divide both sides by 4
to get:

y = -3x + 5

''''''The slope-intercept equations are the same. The two equations have the same graph and the same slope; thus, they are parallel.''''''

|}

{| class="wikitable"
!style="background: #F75D59; text-align: left; padding:3px;"| '''Parallel Lines Example Question 2'''
|-
|style="background: #FFE6EA;padding:12px;"|

'''''Find the equation of the line that is: parallel to y = 2x + 1 and passes though the point (5,4)'''''

''Recall that parallel lines have the same slope!''

1.) The first step is to find the slope of <math>y = 2x + 1.</math>
Recall <math>y = mx + b\,</math>
with m being slope. Therefore, the slope of this line is 2.

The slope of <math> y=2x+1 is : 2 </math>

2.) The next step is to substitute the slope 2 into the point slope equation
of a line which is <math>y - y_1 = m( x - x_1 )\,</math>

We obtain: <math> y - y_1 = 2(x - x_1)</math>

And now we must put in the point (5,4):

<math>y - 4 = 2(x - 5)</math> '''>answer'''

This is the answer, however we can also put this answer into slope-intecept form or
<math>y = mx + b\,</math> form:

<math>y - 4 = 2x - 10</math> >solve for y

<math>y = 2x - 6 </math>'''>answer'''

'''The lines have both the same slope: [2] making them parallel.'''
|}

{| class="wikitable"
!style="background: #3BB9FF; text-align: left; padding:3px;"| '''Perpendicular Lines Example Question 2'''
|-
|style="background: #FFE6EA;padding:12px;"|

'''Determine whether the lines 5y = 4x + 10 and 4y = -5x + 4 are perpendicular.
'''

Find the slope-intercept equations for both lines
by solving for y.

y = (4/5)x + 2

y = -(5/4)x + 1

(4/5) MULTIPLIED BY -(5/4) = '''-1'''

'''The product of the slopes is -1, so the lines are perpendicular.'''
|}

{| class="wikitable"
!style="background: #3BB9FF; text-align: left; padding:3px;"| '''Perpendicular Lines Example Question 2'''
|-
|style="background: #FFE6EA;padding:12px;"|
Example:
'''
'''Find the equation of the line that is perpendicular to y = -4x + 10 and passes though the point (7,2)'''''

1.)The first step is to find the slope of <math>y= -4x + 10.</math>
Recall <math>y = mx + b\,</math>
with m being slope. Therefore, the slope of this line is -4

'''>'''The slope of<math> y=-4x+10 is: -4</math>

The negative reciprocal of that slope is:
<math> m=\frac{1}{-4}=\frac{1}{4}</math>

So the perpendicular line will have a slope of 1/4.

NOTE: For more information on negative recipricals, see tips and tricks.

2.) The next step is to substitute the slope (1/4) into the point slope equation
of a line which is <math>y - y_1 = m( x - x_1 )\,</math>

<math> y - y1 = (1/4)(x - x1)</math>

And now put in the point (7,2):

<math> y - 2 = (1/4)(x - 7)</math> '''>answer'''

This is the answer, however we can also put this answer into slope-intecept form or
<math>y = mx + b\,</math> form:

<math> y - 2 = x/4 - 7/4</math> >solve for y

<math>y = x/4 + 1/4 </math> '''>answer'''

'''When you multiply the slopes of the lines [-4] and [1/4] you get -1, making the lines perpendicular'''

|}

----

====Tips and Tricks====

{| class="wikitable"
!style="background: #F75D59; text-align: left; padding:3px;"| '''TIP: Know Your Negative Reciprocals'''
|-
|style="background: #FBBBB9;padding:12px;"|

*Knowing how to find a negative reciprocal is a useful skill for it helps you find a perpendicular line to an equation. (See Perpindicular Lines Example Question Above)

For example:
For an equation with a slope of 5x, the recriprocal would be: <math>\frac{-1}{5}</math> x

How do we get <math>\frac{-1}{5}</math> X

It`s quite simple:
*Take 5x (which can also be written as <math>\frac{5}{1}</math> X and flip the bottom and the top, so it becomes: <math>\frac{1}{5}</math> X
This is the RECIPRICOAL, but we want the NEGATIVE RECIPROCAL so we must do the next step:
*Change the positive sign to a negative sign so it becomes <math>\frac{-1}{5}</math> X

'''TIP:''' It is important to remember the difference between simply a recipricoal and a negative reciprocoal as they can be easily confused.
For practice see practice problems below.

|}

----

====Tutorial Videos====

{{#ev:youtube | Rew54K6mYUo| 400}}
{{#ev:youtube | oZg7O-3GLNI| 400}}

----

====Practice Problems====

{| class="wikitable"
!style="background: #3BB9FF; text-align: left; padding:3px;"| '''Negative Reciprocal Practice Problems'''
|-
|style="background: #FFE6EA;padding:12px;"|
'''What is the negative reciprocal of the following:'''

'''1.) 5'''

'''2.) 4/9'''''

'''3.) -7/3'''

Answers:

1.) -1/5
2.) -9/4
3.) 3/7


|}


----

====Helpful Links====

Here are some websites that have a great amount of information:

[http://www.purplemath.com/modules/slope2.html Purple Math] - A great website with lots of examples.

[http://www.beaconlearningcenter.com/documents/1750_01.pdf] - This fantastic PDF has lots of practice problems and tons of examples.

===How to compute the distance between two points===

[[File:basicskills-1.jpg]]
[[File:basicskills-2.jpg]]

----
====Examples====

[[File:basicskills-3.jpg]]
----
====Tutorial Videos====

{| class="wikitable"
!style="background: #FBBBB9; text-align: left; padding:3px;"| '''Original Tutorial Video'''
|-
|style="background: padding:12px;"|
<center>'''YOUTUBE CHANNEL: math110group2'''''
http://www.youtube.com/user/math110group2'''</center>

{{#ev:youtube | KP5xEzoABic| 400}}

|}

----

====Practice Problems====
{| class="wikitable"
!style="background: #3BB9FF; text-align: left; padding:3px;"| '''Practice Problems'''
|-
|style="background: #FFE6EA;padding:12px;"|

'''1) What is the distance between the points (-2,7) and (4,6)?''''

'''2) What is the distance between the points(5,6) and (-12,40)?''''

<center>Answers:
1)6.08

2)38.01 
|}

----
====Helpful Links====

http://www.tpub.com/math2/2.htm

http://www.purplemath.com/modules/distform.htm

===How to compute the equation of a line given two points===

----

If you have two points on a line you can construct the general equation for the that line. It is achieved easily by breaking it down into two simple tasks:

1*'''Determine the slope of the line'''. Recall that the slope is: the rise over run of a line. The slope of the line is just the change in Y (values of the y-coordinates of the points on the line) divided by the change in X (values of the x-coordinates of the points on the line). This means that you find the numerical value of the slope of any line by <math>[rise:(y_2-y_1)/run:(x_2-x_1)]</math> Let us now call this number ''m.''

2*Next, you simply '''plug in the values''' from one of the line's points and the slope into the most basic formula for a line: <math>y - y1 = m( x - x1 </math> For y1 and x1, either point can be used as long as the x-coordinate and y-coordinate are from the same point. If you want to put this into slope intercept form <math>y = mx + b,\,</math> you can plug in the y-coordinate and x-coordinate from either point as well as m (slope) into the equation to find b (the y-intercept).
====Example====
----
Let us choose two random points: (2, 1) and (4, -4). Now we will perform the steps outlined above.

1* <math>(-4-1))/(4-2)= -5/2,\,</math>. SO, <math>m = -5/2,\,</math> or <math>-2.5,\,</math>

2* Using the first point: <math>y-1= -2.5(x-2),\,</math> or <math>y = -2.5x + 6,\,</math>
Using the second point: <math>y-(-4) = -5/2(x-4)\,</math> or <math>y = -5/2x + 6\,</math>

====Tutorial Videos====
----

{| class="wikitable"
!style="background: #FBBBB9; text-align: left; padding:3px;"| '''Original Tutorial Video'''
|-
|style="background: padding:12px;"|
<center>'''YOUTUBE CHANNEL: math110group2'''''
http://www.youtube.com/user/math110group2'''</center>

{{#ev:youtube | hsc9POhVPh8| 400}}

|}
----

====Practice Problems====

----
{| class="wikitable"
!style="background: #3BB9FF; text-align: left; padding:3px;"| '''Practice Problems'''
|-
|style="background: #FFE6EA;padding:12px;"|

<center>Here are a few sets of points that you can find equations for if you want practice. If you want the answers they can be sent by email upon request:</center>

''''1. (4, 3) and (6, 2)''''

''''2. (-8, -2) and (13, 9)''''

''''3. (2.6, 1) and (-pi, 11)''''
|}

====Helpful Links====
----
http://www.mathsisfun.com/algebra/line-equation-2points.html

http://www.tutorvista.com/content/math/geometry/straightlines/two-point-form.php

===What is the equation of a line?===

<center>'''General equation of a straight line'''</center>
<center>'''<math>Ax+By=C\,</math>'''</center>
<center>''where a,b & c are constants and a & b cannot both be zero.''</center>
<center>'''Slope-intercept form of a line'''</center>
<center>'''<math>y = mx + b\,</math> '''</center>
<center>''where m is the slope of the line and b is the y-intercept of the graph of the line.'' </center>
<center>'''Point-slope form'''</center>
<center>'''<math>y - y_1 = m( x - x_1 )\,</math> )''' </center>
<center>''for a line through a point with coordinates (x1, y1) and slope m.''</center>

<center>'''Horizontal lines'''</center>
<center>[[File:horizontal line 3.gif]]</center>
<center>''As shown in the graph above, every point on a horizontal line has the same y-coordinate so
the equation of a horizontal line is '''y=k''' (where k represents any real number that is the value
of the y-coordinate of the graph).''
<center>'''Vertical lines'''</center>
<center>[[File:vertical line.gif]]</center>
<center>'' As shown in the graph above, every point on a vertical line has the same x-coordinate so
the equation of a vertical line is '''x=k''' (where k represents any real number that is the value
of the x-coordinate of the graph).''

====Examples====
----
'''1)What is the equation of this line?'''
<center>[[File:Example.gif]]</center>
<center>The graph crosses the y-axis at zero, so the easiest equation to use for this graph is the slope intercept form. We know that b=0, so we need to find the slope m. Using the point (1,2) and the point (2,4)from the graph, we can find the slope with the equation y2-y1/x2-x1. 4-2/2-1 = 2/1 so the slope is 2. Plugging that into the slope intercept form for m,the answer is '''y=2x.'''</center>
'''2) What is the equation of this line? '''
<center>[[File:example 2.gif]]
<center> The graph shows us the y-intercept, so the slope intercept form is the easiest equation to use. We know that b=1, so we just need to find the slope m. Using the point (2,5) and the point (0,1) from the graph, we can find the slope with the equation y2-y1/x2-x1. 1-5/0-2 =-4/-2 =2 so the slope is 2. Plugging that into the slope intercept form for m, the answer is '''y=2x+1.'''</center>

====Tips and Tricks====
----
When given a graph of a line and asked to find the equation, the slope intercept form (y=mx+b)is generally the easiest to use because the y-intercept b can be easily identified from the graph.
The slope m can usually also be easily calculated by using two points from the graph and the slope formula y2-y-1/x2-x1.

When given a point and its slope, the slope intercept form is also generally the easiest because the slope and y and x can be plugged into the equation to find b.

When given the coordinates of two points and asked to find the equation, the point-slope form (y-y1=m(x-x1) is usually the easiest to use because the slope can easily be calculated by using the slope formula. After finding the formula, plug in the slope for m in the point-slope formula and x1 and y1 from either given point, as long as both coordinates are from the same point.

====Tutorial Videos====
----

{{#ev:youtube | hh6RAEPlza4| 400}}

{{#ev:youtube | GFM8NOe_XM4| 400}}
{{#ev:youtube | rNQ36DK6aBk| 400}}

====Practice Problems====
----
{| class="wikitable"
!style="background: #3BB9FF; text-align: left; padding:3px;"| '''Practice Problems'''
|-
|style="background: #FFE6EA;padding:12px;"|
'''
'''1) Determine an equation for a line through the points (5,5) and (4,-1)'''
'''

'''2) Determine an equation for a line for the point (2,6) with slope -3.
'''

'''3) Determine an equation for a line that passes through the point (1,1) with slope 2.'''

Answers:

'''1)'''y-5=6(x-5)

'''2)'''y=-3x+12

'''3)'''y-1=2(x-1)

<center> '''Practice Questions from Just-In-Time textbook'''</center>
'''Page 60 #9, #11, #13'''

|}

====Helpful Links====
----

1)[http://www.mathsisfun.com/equation_of_line.html Equation of straight line]
2)[http://www.purplemath.com/modules/strtlneq.htm Slope Intercept Form]
3)[http://www.purplemath.com/modules/strtlneq2.htm Point Slope Form]

===How to compute the equation of a line given its slope and a point.===

-----

[[File:Cartoon.math.gif]]

The equation of a line is defined by the equation
<math>y = mx + b,\,</math>

All these variables represent some specific and important part of a curve that define that shape of it:

''m = slope

b = y intercept''

If the slope is given, then the only thing that has to be done is that it must be plugged it into the equation: <math>y = mx + b,\,</math>

Now that you have a value of m, the next thing you do is plug in the values for x and y.

This means that the point given as the x value is plugged into the equation as x. The same is done with the y point into the equation.

When this is done the only thing needed is solve for b.

Now that you have m and b, you plug those values back into the original equation (<math>y = mx + b\,</math> ).

And VOILA! You have the equation for the curve!

====Examples====
----
'''Find the equation of a line given the point (2,5) and slope -1.'''
<center>We know that m = -1 and x=2 and y=5 so we can plug those into the slope intercept formula (y=mx+b), resulting in 5=-1(2)+b. Simplifying this, we get 5=-2+b-->7=b. Therefore, the answer is y=-1x+7</center>

====Tutorial Videos====

{| class="wikitable"
!style="background: #FBBBB9; text-align: left; padding:3px;"| '''Original Tutorial Video'''
|-
|style="background: padding:12px;"|
<center>'''YOUTUBE CHANNEL: math110group2'''''
http://www.youtube.com/user/math110group2'''</center>

{{#ev:youtube | WKAUmRUaai8| 400}}
|}
----

====Practice Problems====
----
{| class="wikitable"
!style="background: #3BB9FF; text-align: left; padding:3px;"| '''Practice Problems'''
|-
|style="background: #FFE6EA;padding:12px;"|

'''1) Find the equation of a line through the point (-1,3) with slope 2.'''

'''2) Find the equation of a line through (-2,-10) with slope 4.'''
<center>Answers: 1)y=2x+5 2)y=4x-2
|}

====Helpful Links====
----
<center>http://www.nipissingu.ca/calculus/tutorials/linear.html
</center>
<center> http://www.tpub.com/math2/6.htm </center>

Course:MATH110/Archive/2010-2011/003/Groups/Group 02/Basic Skills Project

2010-12-03T15:54:34Z

YihongChen: /* Practice Problems */

{| class="wikitable"
!style="background: #F75D59; text-align: left; padding:3px;"| WELCOME TO GROUP TWO'S PAGE: DISTANCE AND LINES
|-
|style="background: #FBBBB9; padding:12px;"|
Welcome to [http://wiki.ubc.ca/Course:MATH110/003/Groups/Group_02 Group 2]`s page. Our contribution to the Basic Skills Project is the topic of '''Distance and Lines.''' On this page you will find
* detailed step-by-step examples
* tutorial videos
* tips and tricks
* practice problems
* helpful links
on the all of the sub-topics.

[[File:Index.jpg]]Please Visit [http://www.youtube.com/user/math110group2 Group 2's Youtube Page] for
[http://www.youtube.com/user/math110group2#p/u videos created by us] and [http://www.youtube.com/user/math110group2#p/f videos we find informative and helpful.]

|}

===What does it mean for two lines to be parallel and/or perpendicular?===
----

{| class="wikitable"
!style="background: #3BB9FF; text-align: left; padding:3px;"| How do you know if two lines are parallel?
|style="background: #FFE6EA; padding:12px;"|
<center>'''Their slopes are the same!'''</center>
|}

[[File:parallelpic.gif]]

{| class="wikitable"
!style="background: #F75D59; text-align: left; padding:3px;"| How do you know if two lines are perpendicular?
|style="background: #FBBBB9; padding:12px;"|
<center>'''When you multiply their slopes, you get -1!'''</center>

|}

[[File:Per.gif]]
====Examples====

{| class="wikitable"
!style="background: #F75D59; text-align: left; padding:3px;"| '''Parallel Lines Example Question 1'''
|-
|style="background: #FBBBB9;padding:12px;"|

'''
'''Problem: Determine whether the graphs of y = -3x + 5 and 4y = -12x + 20 are parallel lines.'''''
*Solve for y for both graphs
y = -3x + 5 >already solved.

4y = -12x + 20 >Solve for y

4y=-12x+20 >Divide both sides by 4
to get:

y = -3x + 5

''''''The slope-intercept equations are the same. The two equations have the same graph and the same slope; thus, they are parallel.''''''

|}

{| class="wikitable"
!style="background: #F75D59; text-align: left; padding:3px;"| '''Parallel Lines Example Question 2'''
|-
|style="background: #FFE6EA;padding:12px;"|

'''''Find the equation of the line that is: parallel to y = 2x + 1 and passes though the point (5,4)'''''

''Recall that parallel lines have the same slope!''

1.) The first step is to find the slope of <math>y = 2x + 1.</math>
Recall <math>y = mx + b\,</math>
with m being slope. Therefore, the slope of this line is 2.

The slope of <math> y=2x+1 is : 2 </math>

2.) The next step is to substitute the slope 2 into the point slope equation
of a line which is <math>y - y_1 = m( x - x_1 )\,</math>

We obtain: <math> y - y_1 = 2(x - x_1)</math>

And now we must put in the point (5,4):

<math>y - 4 = 2(x - 5)</math> '''>answer'''

This is the answer, however we can also put this answer into slope-intecept form or
<math>y = mx + b\,</math> form:

<math>y - 4 = 2x - 10</math> >solve for y

<math>y = 2x - 6 </math>'''>answer'''

'''The lines have both the same slope: [2] making them parallel.'''
|}

{| class="wikitable"
!style="background: #3BB9FF; text-align: left; padding:3px;"| '''Perpendicular Lines Example Question 2'''
|-
|style="background: #FFE6EA;padding:12px;"|

'''Determine whether the lines 5y = 4x + 10 and 4y = -5x + 4 are perpendicular.
'''

Find the slope-intercept equations for both lines
by solving for y.

y = (4/5)x + 2

y = -(5/4)x + 1

(4/5) MULTIPLIED BY -(5/4) = '''-1'''

'''The product of the slopes is -1, so the lines are perpendicular.'''
|}

{| class="wikitable"
!style="background: #3BB9FF; text-align: left; padding:3px;"| '''Perpendicular Lines Example Question 2'''
|-
|style="background: #FFE6EA;padding:12px;"|
Example:
'''
'''Find the equation of the line that is perpendicular to y = -4x + 10 and passes though the point (7,2)'''''

1.)The first step is to find the slope of <math>y= -4x + 10.</math>
Recall <math>y = mx + b\,</math>
with m being slope. Therefore, the slope of this line is -4

'''>'''The slope of<math> y=-4x+10 is: -4</math>

The negative reciprocal of that slope is:
<math> m=\frac{1}{-4}=\frac{1}{4}</math>

So the perpendicular line will have a slope of 1/4.

NOTE: For more information on negative recipricals, see tips and tricks.

2.) The next step is to substitute the slope (1/4) into the point slope equation
of a line which is <math>y - y_1 = m( x - x_1 )\,</math>

<math> y - y1 = (1/4)(x - x1)</math>

And now put in the point (7,2):

<math> y - 2 = (1/4)(x - 7)</math> '''>answer'''

This is the answer, however we can also put this answer into slope-intecept form or
<math>y = mx + b\,</math> form:

<math> y - 2 = x/4 - 7/4</math> >solve for y

<math>y = x/4 + 1/4 </math> '''>answer'''

'''When you multiply the slopes of the lines [-4] and [1/4] you get -1, making the lines perpendicular'''

|}

----

====Tips and Tricks====

{| class="wikitable"
!style="background: #F75D59; text-align: left; padding:3px;"| '''TIP: Know Your Negative Reciprocals'''
|-
|style="background: #FBBBB9;padding:12px;"|

*Knowing how to find a negative reciprocal is a useful skill for it helps you find a perpendicular line to an equation. (See Perpindicular Lines Example Question Above)

For example:
For an equation with a slope of 5x, the recriprocal would be: <math>\frac{-1}{5}</math> x

How do we get <math>\frac{-1}{5}</math> X

It`s quite simple:
*Take 5x (which can also be written as <math>\frac{5}{1}</math> X and flip the bottom and the top, so it becomes: <math>\frac{1}{5}</math> X
This is the RECIPRICOAL, but we want the NEGATIVE RECIPROCAL so we must do the next step:
*Change the positive sign to a negative sign so it becomes <math>\frac{-1}{5}</math> X

'''TIP:''' It is important to remember the difference between simply a recipricoal and a negative reciprocoal as they can be easily confused.
For practice see practice problems below.

|}

----

====Tutorial Videos====

{{#ev:youtube | Rew54K6mYUo| 400}}
{{#ev:youtube | oZg7O-3GLNI| 400}}

----

====Practice Problems====

{| class="wikitable"
!style="background: #3BB9FF; text-align: left; padding:3px;"| '''Negative Reciprocal Practice Problems'''
|-
|style="background: #FFE6EA;padding:12px;"|
'''What is the negative reciprocal of the following:'''

'''1.) 5'''

'''2.) 4/9'''''

'''3.) -7/3'''

Answers:

1.) -1/5
2.) -9/4
3.) 3/7


|}


----

====Helpful Links

Here are some websites that have a great amount of information:

[http://www.purplemath.com/modules/slope2.html Purple Math] - A great website with lots of examples.

[http://www.beaconlearningcenter.com/documents/1750_01.pdf] - This fantastic PDF has lots of practice problems and tons of examples.

===How to compute the distance between two points===

[[File:basicskills-1.jpg]]
[[File:basicskills-2.jpg]]

----
====Examples====

[[File:basicskills-3.jpg]]
----
====Tutorial Videos====

{| class="wikitable"
!style="background: #FBBBB9; text-align: left; padding:3px;"| '''Original Tutorial Video'''
|-
|style="background: padding:12px;"|
<center>'''YOUTUBE CHANNEL: math110group2'''''
http://www.youtube.com/user/math110group2'''</center>

{{#ev:youtube | KP5xEzoABic| 400}}

|}

----

====Practice Problems====
{| class="wikitable"
!style="background: #3BB9FF; text-align: left; padding:3px;"| '''Practice Problems'''
|-
|style="background: #FFE6EA;padding:12px;"|

'''1) What is the distance between the points (-2,7) and (4,6)?''''

'''2) What is the distance between the points(5,6) and (-12,40)?''''

<center>Answers:
1)6.08

2)38.01 
|}

----
====Helpful Links====

http://www.tpub.com/math2/2.htm

http://www.purplemath.com/modules/distform.htm

===How to compute the equation of a line given two points===

----

If you have two points on a line you can construct the general equation for the that line. It is achieved easily by breaking it down into two simple tasks:

1*'''Determine the slope of the line'''. Recall that the slope is: the rise over run of a line. The slope of the line is just the change in Y (values of the y-coordinates of the points on the line) divided by the change in X (values of the x-coordinates of the points on the line). This means that you find the numerical value of the slope of any line by <math>[rise:(y_2-y_1)/run:(x_2-x_1)]</math> Let us now call this number ''m.''

2*Next, you simply '''plug in the values''' from one of the line's points and the slope into the most basic formula for a line: <math>y - y1 = m( x - x1 </math> For y1 and x1, either point can be used as long as the x-coordinate and y-coordinate are from the same point. If you want to put this into slope intercept form <math>y = mx + b,\,</math> you can plug in the y-coordinate and x-coordinate from either point as well as m (slope) into the equation to find b (the y-intercept).
====Example====
----
Let us choose two random points: (2, 1) and (4, -4). Now we will perform the steps outlined above.

1* <math>(-4-1))/(4-2)= -5/2,\,</math>. SO, <math>m = -5/2,\,</math> or <math>-2.5,\,</math>

2* Using the first point: <math>y-1= -2.5(x-2),\,</math> or <math>y = -2.5x + 6,\,</math>
Using the second point: <math>y-(-4) = -5/2(x-4)\,</math> or <math>y = -5/2x + 6\,</math>

====Tutorial Videos====
----

{| class="wikitable"
!style="background: #FBBBB9; text-align: left; padding:3px;"| '''Original Tutorial Video'''
|-
|style="background: padding:12px;"|
<center>'''YOUTUBE CHANNEL: math110group2'''''
http://www.youtube.com/user/math110group2'''</center>

{{#ev:youtube | hsc9POhVPh8| 400}}

|}
----

====Practice Problems====

----
{| class="wikitable"
!style="background: #3BB9FF; text-align: left; padding:3px;"| '''Practice Problems'''
|-
|style="background: #FFE6EA;padding:12px;"|

<center>Here are a few sets of points that you can find equations for if you want practice. If you want the answers they can be sent by email upon request:</center>

''''1. (4, 3) and (6, 2)''''

''''2. (-8, -2) and (13, 9)''''

''''3. (2.6, 1) and (-pi, 11)''''
|}

====Helpful Links====
----
http://www.mathsisfun.com/algebra/line-equation-2points.html

http://www.tutorvista.com/content/math/geometry/straightlines/two-point-form.php

===What is the equation of a line?===

<center>'''General equation of a straight line'''</center>
<center>'''<math>Ax+By=C\,</math>'''</center>
<center>''where a,b & c are constants and a & b cannot both be zero.''</center>
<center>'''Slope-intercept form of a line'''</center>
<center>'''<math>y = mx + b\,</math> '''</center>
<center>''where m is the slope of the line and b is the y-intercept of the graph of the line.'' </center>
<center>'''Point-slope form'''</center>
<center>'''<math>y - y_1 = m( x - x_1 )\,</math> )''' </center>
<center>''for a line through a point with coordinates (x1, y1) and slope m.''</center>

<center>'''Horizontal lines'''</center>
<center>[[File:horizontal line 3.gif]]</center>
<center>''As shown in the graph above, every point on a horizontal line has the same y-coordinate so
the equation of a horizontal line is '''y=k''' (where k represents any real number that is the value
of the y-coordinate of the graph).''
<center>'''Vertical lines'''</center>
<center>[[File:vertical line.gif]]</center>
<center>'' As shown in the graph above, every point on a vertical line has the same x-coordinate so
the equation of a vertical line is '''x=k''' (where k represents any real number that is the value
of the x-coordinate of the graph).''

====Examples====
----
'''1)What is the equation of this line?'''
<center>[[File:Example.gif]]</center>
<center>The graph crosses the y-axis at zero, so the easiest equation to use for this graph is the slope intercept form. We know that b=0, so we need to find the slope m. Using the point (1,2) and the point (2,4)from the graph, we can find the slope with the equation y2-y1/x2-x1. 4-2/2-1 = 2/1 so the slope is 2. Plugging that into the slope intercept form for m,the answer is '''y=2x.'''</center>
'''2) What is the equation of this line? '''
<center>[[File:example 2.gif]]
<center> The graph shows us the y-intercept, so the slope intercept form is the easiest equation to use. We know that b=1, so we just need to find the slope m. Using the point (2,5) and the point (0,1) from the graph, we can find the slope with the equation y2-y1/x2-x1. 1-5/0-2 =-4/-2 =2 so the slope is 2. Plugging that into the slope intercept form for m, the answer is '''y=2x+1.'''</center>

====Tips and Tricks====
----
When given a graph of a line and asked to find the equation, the slope intercept form (y=mx+b)is generally the easiest to use because the y-intercept b can be easily identified from the graph.
The slope m can usually also be easily calculated by using two points from the graph and the slope formula y2-y-1/x2-x1.

When given a point and its slope, the slope intercept form is also generally the easiest because the slope and y and x can be plugged into the equation to find b.

When given the coordinates of two points and asked to find the equation, the point-slope form (y-y1=m(x-x1) is usually the easiest to use because the slope can easily be calculated by using the slope formula. After finding the formula, plug in the slope for m in the point-slope formula and x1 and y1 from either given point, as long as both coordinates are from the same point.

====Tutorial Videos====
----

{{#ev:youtube | hh6RAEPlza4| 400}}

{{#ev:youtube | GFM8NOe_XM4| 400}}
{{#ev:youtube | rNQ36DK6aBk| 400}}

====Practice Problems====
----
{| class="wikitable"
!style="background: #3BB9FF; text-align: left; padding:3px;"| '''Practice Problems'''
|-
|style="background: #FFE6EA;padding:12px;"|
'''
'''1) Determine an equation for a line through the points (5,5) and (4,-1)'''
'''

'''2) Determine an equation for a line for the point (2,6) with slope -3.
'''

'''3) Determine an equation for a line that passes through the point (1,1) with slope 2.'''

Answers:

'''1)'''y-5=6(x-5)

'''2)'''y=-3x+12

'''3)'''y-1=2(x-1)

<center> '''Practice Questions from Just-In-Time textbook'''</center>
'''Page 60 #9, #11, #13'''

|}

====Helpful Links====
----

1)[http://www.mathsisfun.com/equation_of_line.html Equation of straight line]
2)[http://www.purplemath.com/modules/strtlneq.htm Slope Intercept Form]
3)[http://www.purplemath.com/modules/strtlneq2.htm Point Slope Form]

===How to compute the equation of a line given its slope and a point.===

-----

[[File:Cartoon.math.gif]]

The equation of a line is defined by the equation
<math>y = mx + b,\,</math>

All these variables represent some specific and important part of a curve that define that shape of it:

''m = slope

b = y intercept''

If the slope is given, then the only thing that has to be done is that it must be plugged it into the equation: <math>y = mx + b,\,</math>

Now that you have a value of m, the next thing you do is plug in the values for x and y.

This means that the point given as the x value is plugged into the equation as x. The same is done with the y point into the equation.

When this is done the only thing needed is solve for b.

Now that you have m and b, you plug those values back into the original equation (<math>y = mx + b\,</math> ).

And VOILA! You have the equation for the curve!

====Examples====
----
'''Find the equation of a line given the point (2,5) and slope -1.'''
<center>We know that m = -1 and x=2 and y=5 so we can plug those into the slope intercept formula (y=mx+b), resulting in 5=-1(2)+b. Simplifying this, we get 5=-2+b-->7=b. Therefore, the answer is y=-1x+7</center>

====Tutorial Videos====

{| class="wikitable"
!style="background: #FBBBB9; text-align: left; padding:3px;"| '''Original Tutorial Video'''
|-
|style="background: padding:12px;"|
<center>'''YOUTUBE CHANNEL: math110group2'''''
http://www.youtube.com/user/math110group2'''</center>

{{#ev:youtube | WKAUmRUaai8| 400}}
|}
----

====Practice Problems====
----
{| class="wikitable"
!style="background: #3BB9FF; text-align: left; padding:3px;"| '''Practice Problems'''
|-
|style="background: #FFE6EA;padding:12px;"|

'''1) Find the equation of a line through the point (-1,3) with slope 2.'''

'''2) Find the equation of a line through (-2,-10) with slope 4.'''
<center>Answers: 1)y=2x+5 2)y=4x-2
|}

====Helpful Links====
----
<center>http://www.nipissingu.ca/calculus/tutorials/linear.html
</center>
<center> http://www.tpub.com/math2/6.htm </center>