Tuesday, April 17, 2018
Friday, April 13, 2018
Saturday, March 31, 2018
Wednesday, March 21, 2018
Notes for Homework 8, due March 28
Notes on determinants
A 2x2 matrix is an array of four numbers put in two rows and two columns, such as
| a c |
| b d |
The determinant of this matrix is ad - bc, the product of the main diagonal minus the product of the opposite diagonal. If you use Kramer's rule to solve a pair of simultaneous equations, you will need to know how to calculate determinants.
For example, let's consider the following set of equations.
3x - 2y = 7
2x + 4y = 12
As an augmented matrix , these equations become
| 3 -2 : 7 |
| 2 4 : 12|
Matrix_1 is
| 3 -2 |
| 2 4 |
The determinant is 3(4) - 2(-2) = 12 - -4 = 16
Matrix_x is
| 7 -2 |
| 12 4 |
The determinant is 7(4) - 12(-2) = 28 - -24 = 52
Matrix_y is
| 3 7 |
| 2 12 |
The determinant is 3(12) - 2(7) = 36 - 14 = 22
From here, Kramer's rule then has x = 52/16 = 13/4 or 3 1/4. The value for y = 22/16 = 11/8 = 1 3/8
Notes on solving simultaneous equations
A 2x2 matrix is an array of four numbers put in two rows and two columns, such as
| a c |
| b d |
The determinant of this matrix is ad - bc, the product of the main diagonal minus the product of the opposite diagonal. If you use Kramer's rule to solve a pair of simultaneous equations, you will need to know how to calculate determinants.
For example, let's consider the following set of equations.
3x - 2y = 7
2x + 4y = 12
As an augmented matrix , these equations become
| 3 -2 : 7 |
| 2 4 : 12|
Matrix_1 is
| 3 -2 |
| 2 4 |
The determinant is 3(4) - 2(-2) = 12 - -4 = 16
Matrix_x is
| 7 -2 |
| 12 4 |
The determinant is 7(4) - 12(-2) = 28 - -24 = 52
Matrix_y is
| 3 7 |
| 2 12 |
The determinant is 3(12) - 2(7) = 36 - 14 = 22
From here, Kramer's rule then has x = 52/16 = 13/4 or 3 1/4. The value for y = 22/16 = 11/8 = 1 3/8
Notes on solving simultaneous equations
Thursday, March 15, 2018
Thursday, March 8, 2018
Tuesday, March 6, 2018
Tuesday, February 27, 2018
Thursday, February 15, 2018
Friday, February 9, 2018
Thursday, February 1, 2018
Thursday, January 25, 2018
Wednesday, November 29, 2017
Wednesday, November 22, 2017
Notes for Homework 12, due Mon., Nov 27
Notes on the metric system
Changing Celsius to Fahrenheit
F = 9/5 C + 32
Changing Fahrenheit to Celsius
C = 5/9(F - 32)
Wednesday, November 8, 2017
Wednesday, November 1, 2017
Wednesday, October 25, 2017
Wednesday, October 18, 2017
Thursday, October 12, 2017
Monday, October 9, 2017
Thursday, September 28, 2017
Wednesday, September 20, 2017
Wednesday, September 13, 2017
Wednesday, September 6, 2017
Fractional part of a day
A year is defined by how long it takes a planet to make a single orbit around the sun. An Earth year is 365.2422 days. To make up for the decimal part we have a leap year every four years and skip the leap year if the number of the year is divisible by 100. This means we had leap years in 2004, 2008, 2012 and 2016, but we did not have one in 2000. What this lesson is about is to take the decimal part of this time and turn it onto hours, minutes and seconds.
Step 1. Multiply the decimal part by 24 to get the number of hours.
In this case, .2422 x 24 = 5.8128, which means 5.8128 hours.
Step 2. If there is still a decimal part, multiply it by 60 to get the number of minutes. We know know a year is 365 days, 5 hours and some number of minutes. That number is 60 x .8128 = 48.768
Step 3. If there is still a decimal part, multiply it by 60 to get the number of seconds.
We know know a year is 365 days, 5 hours, 48 minutes and some number of seconds. That number is 60 x .768 = 46.08. In these problems, it is okay to round to the nearest tenth of a second, so the final answer is 365 days, 5 hours, 48 minutes and 46.1 seconds.
Let's do another example. A year on Venus is 224.65 Earth days.
Step 1. .65 x 60 = 15.6, so the year on Venus is 224 days, 15.6 hours.
Step 2. 60 x .6 = 36, so that makes the year on Venus 224 days, 15 hours and 36 minutes. There is no more decimal part, so we don't have to add any seconds to our answer.
Here are two more practice problems. The answers are in the comments.
1. It takes Mercury 87.969 Earth days to travel around the sun. Write this number in days, hours, minutes and seconds.
2. It takes Mars 686.98 Earth days to travel around the sun. Write this number in days, hours, minutes and seconds.
Thursday, August 31, 2017
Wednesday, August 23, 2017
Monday, August 21, 2017
Friday, July 21, 2017
Tuesday, July 18, 2017
Notes for Homework 8, duy Wed. July 19
A quick reminder: this homework will not be accepted late because the answer sheet will be given out at the end of class to help study for Thursday's midterm.
Notes on scientific notation
Notes on Set Theory
Notes on Venn Diagrams
Notes on contingency tables and probabilities
Thursday, July 13, 2017
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