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Engineering Mathematics with Tables
Description: Problem solving is an absolute essential in the learning process. The problems in this book emphasize the use of the mathematical principles so vital to a clear understanding of Engineering. They also furnish the necessary foundation for the later development of the Analytical Geometry and the Calculus.
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2 1. Introduction to GeoGebra s Spreadsheet View The dynamic mathematics software GeoGebra provides three different views of mathematical objects: a Graphics View, a, numeric Algebra View and a Spreadsheet View. They allow you to display mathematical objects in three different representations: graphically (e.g., points, function graphs), algebraically (e.g., coordinates of points, equations), and in spreadsheet cells. Thereby, all representations of the same object are linked dynamically and adapt automatically to changes made to any of the representations, no matter how they were initially created. Toolbar Algebra View Spreadsheet View Graphics View Input Bar Geometric Constructions Using the construction tools available in the Toolbar you can do geometric constructions in the Graphics View with the mouse. Select any construction tool from the Toolbar and read the Toolbar Help (next to the Toolbar) in order to find out how to use the selected tool. Any object you create in the Graphics View also has an algebraic representation in the Algebra View. Note that you are able to move objects in the Graphics View by dragging them with the mouse. At the same time, their algebraic representations are dynamically updated in the Algebra View. Every icon in the Toolbar represents a Toolbox that contains a selection of similar construction tools. In order to open a Toolbox, you need to click on the small arrow in the lower right corner of the Toolbar icon. 2
3 Hint: Construction tools are organized by the nature of resulting objects or their functionality. You will find tools that create different types of points in the Point toolbox (default icon ) and tools that allow you to apply geometric transformations in the Transformation toolbox (default icon ). Algebraic Input and Commands Using the Input Bar you can directly enter algebraic expressions in GeoGebra. After hitting the Enter-key your algebraic input appears in the Algebra View while its graphical representation is automatically displayed in the Graphics View. For example, the input f(x) = x^2 gives you the function f in the Algebra View and its function graph in the Graphics View. In the Algebra View, mathematical objects are organized as free or dependent objects. If you create a new object without using any other existing objects, it is classified as a free object. If your newly created object was created by using other existing objects, it is classified as a dependent object. Hint: If you want to hide the algebraic representation of an object in the Algebra View, you can specify the object as an Auxiliary Object: Right click (MacOS: Ctrl-click) on the corresponding object in the Algebra View and select Auxiliary Object from the appearing context menu. By default, Auxiliary Objects are not shown in the Algebra View, but you can change this setting by selecting the item Auxiliary Objects from the View menu. Note that you are able to modify objects in the Algebra View as well: Make sure that you activate the Move tool before you double click on a free object in the Algebra View. In the appearing textbox you can directly edit the algebraic representation of the object. After hitting the Enter-key, the graphical representation of the object will automatically adapt to your changes. If you double click on a dependent object in the Algebra View, a dialog window appears allowing you to redefine the object. GeoGebra also offers a wide range of commands that can be entered into the Input Bar. You can open the list of commands in the right corner of the Input Bar by clicking on the button Command. After selecting a command from this list (or typing its name directly into the Input Bar) you can press the F1-key to get information about the syntax and arguments required to apply the corresponding command. Spreadsheet Cells Input In GeoGebra s Spreadsheet View every cell has a specific name that allows you to directly address each cell. For example, the cell in column A and row 1 is named A1. 3
4 Note: These cell names can be used in expressions and commands in order to address the content of the corresponding cell. In the spreadsheet cells you can enter not only numbers, but all types of mathematical objects that are supported by GeoGebra (e.g., coordinates of points, functions, commands). If possible, GeoGebra immediately displays the graphical representation of the object you enter into a spreadsheet cell in the Graphics View as well. Thereby, the name of the object matches the name of the spreadsheet cell used to initially create it (e.g., A5, C1). Note: By default, spreadsheet objects are classified as Auxiliary Objects in the Algebra View. You can show or hide these Auxiliary Objects by selecting Auxiliary Objects from the View menu. Customize the User Interface and Toolbar The user interface of GeoGebra can be customized by using the View menu. For example, you can hide different parts of the interface (e.g., the Algebra View) by unchecking the corresponding menu item in the View menu (e.g., Algebra View ). Note: In this way, you can also show or hide the coordinate axes and a coordinate grid. The Toolbar can be customized by selecting Customize Toolbar from the Tools menu. Highlight the tool or toolbox you want to remove from the Toolbar in the list on the left hand side of the appearing dialog window and click button Remove > in order to remove the tool/toolbox from the Toolbar. You can restore the default Toolbar by clicking on the button Restore Default Toolbar in the left lower corner of the dialog window. 2. Record to Spreadsheet Feature Preparations Open a new GeoGebra file. Show the Spreadsheet View (View menu). Hide the Algebra View (View menu). 4
5 Instructions 1 Create a slider with default interval and increment 1. Hint: Select tool Slider and click in the Graphics View to set the position for the slider. In the appearing dialog window change the increment to 1 and click the Apply button. 2 A = (a, 2a) Create point A by entering A = (a, 2a) into the Input Bar. Hint: The value of slider a determines the x-coordinate of point A while the y-coordinate is a multiple of this value. 3 Show the label of point A in the Graphics View. 4 5 Change the value of slider a to examine different positions of point A. Use tools Move Drawing Pad, as well as Zoom In and Zoom Out to adjust the visible part of the Graphics View and make point A visible in all positions. 5
6 6 Turn on the trace of point A. 7 Hint: Right click (MacOS: Ctrl-click) on point A and select Trace On from the appearing Context Menu. Change the value of slider a to examine the trace point A leaves for every slider position. 8 Set the value of slider a to Record the coordinates for different positions of point A to the spreadsheet: (1) Select tool Record to Spreadsheet. Then, click on point A in order to highlight it. Note: The coordinates for the actual position of point A are immediately entered into cells A1 (x-coordinate) and B1 (y-coordinate) of the spreadsheet. (2) Now, change the value of slider a in order to record the coordinates of all other possible positions of point A to the spreadsheet as well. Note: Do not switch to another tool before moving the slider. Additional Tasks Task 1: Examine the pattern of y-values in column B You could give this construction to your students and let them explore the pattern in column B, which is created by the y-coordinates of different positions of point A. Encourage your students to make a prediction about a function graph that runs through all different positions of point A. Have your students enter the corresponding function into the Input Bar in order to check whether their prediction was correct (e.g., students enter f(x) = 2x to create a line through all points). Task 2: Create a new problem Change the y-coordinate of point A in order to create a new problem: Right click (MacOS: Ctrl-click) on point A and select Properties from the appearing Context Menu. In tab Basic you can change the y-coordinate of point A in the text field Definition to, for example, a^2. Use the other tabs of the Properties Dialog in order to change the color (tab Color ) or size (tab Style ) of point A. Close the Properties Dialog when you have made all desired changes. 6
7 Repeat steps 7 to 9 of the instructions above in order to record the coordinates of the new positions of point A to the spreadsheet. Note: If you did not delete the old values in columns A and B, GeoGebra automatically uses the next two empty columns (e.g., columns C and D) in order to record the new values for x-coordinates and y-coordinates. 3. Relative Copy and Linear Equations Preparations Open a new GeoGebra file. Show the Spreadsheet View (View menu). Hide the Algebra View (View menu). 7
8 Instructions 1 2 (0, 0) 3 (1, 1) Activate tool Move Drawing Pad and drag the origin of the coordinate system close to the lower left corner of the Graphics View. In the Spreadsheet View, click on cell A1 and enter the point coordinates (0, 0). In the Spreadsheet View, click on cell A2 and enter the point coordinates (1, 1). Select tool Show/Hide Label and show the labels of both points in the Graphics View. Relative copy the inserted point coordinates to other cells in column A: (1) Highlight both cells A1 and A2 by using the mouse. (2) Click on the little square at the lower right corner of the highlighted cell range. (3) Hold the mouse button down and drag the pointer down to cell A11. Use tools Move Drawing Pad, as well as Zoom In and Zoom Out to adjust the visible part of the Graphics View and make all points visible. Additional Tasks Task 1: Examine the coordinates of the point sequence What sequence of numbers is created if you apply the relative copy feature of the GeoGebra spreadsheet the way it is described above? Hint: Examine the x-coordinates of all created points and come up with a conjecture about how they are related. Then, check your conjecture using the y- coordinates of the points. Task 2: Find the matching equation Make a prediction about an equation that would create a graph going through all points of this sequence. Enter this equation into the Input Bar in order to check your prediction. 8
9 Task 3: Create a new problem Change the coordinates of the initial points in order to create a sequence of points that can be examined by your students. Version 1: Change the initial points in the Spreadsheet View. Double click in cell A2 and change the coordinates of the corresponding point to (1, 2). After hitting the Enter-key, all points that depend on point A2 automatically adapt to this change, both in the Spreadsheet View as well as in the Graphics View. Version 2: Change the initial points in the Graphics View. Activate tool Move and drag point A2 to a different position in the coordinate system. Immediately, all dependent points dynamically adapt to these changes both in the Graphics View as well as in the Spreadsheet View. Note: In order to restrict the coordinates of the points to be integers, you can change the option Point Capturing in the Options menu to On (Grid). You may also display the coordinate grid by selecting Grid from the View menu. Hint: By changing the coordinates of point A1 as well, you are able to create problems that result in linear equations of the form y = m x + b which do not run through the origin of the coordinate system. 4. Investigating Number Patterns Let s investigate how the surface of a cube changes depending on the length of its edges. Preparations with Paper and Pencil Calculate the surface of a cube for the given length e of its edges. Pick at least two edge lengths from each table but do not pick the same numbers as your neighbor. 9
10 Edge Surface Edge Surface Preparations in GeoGebra Open a new GeoGebra file. Show the Spreadsheet View (View menu). Hide the Algebra View (View menu). In the Options menu set the Labeling to New Points Only. 10
11 Instructions Create a Scatter Plot from your Data 1 2 Enter the following numbers into the spreadsheet cells of column A: A1: 1 A2: 2 Highlight cells A1 and A2. Relative copy the values to cell A10 in order to create a sequence of different edge lengths. Hint: This creates the integers from 1 to In column B, enter the surface values you calculated earlier next to the corresponding edge length of the cube. Hint: You may collaborate with your neighbors to complete the table. 4 Create a Scatter Plot from this data: 5 Hint: The points created from the data are displayed in the Algebra View as a list of points. By default, GeoGebra calls this list L 1. Use tool Move Drawing Pad in order to change the scale of the y-axis so that all points are visible in the Graphics View. Hint: Select tool Move Drawing Pad. Click on the y-axis and drag it down until you can see the 600 tick mark. Investigate the Number Pattern in Column B 6 In cell C2, enter the formula = B2 - B1 to compute the difference of the two successive surface values. Hint: After entering the equal sign, you may click on cell B2 in order to enter its name into the active cell C2. 7 Select cell C2 and relative copy the formula down to cell C10. 8 In cell D3, enter the formula = C3 - C2 to compute the difference of the two successive differences. 11
12 9 Select cell D3 and relative copy the formula down to cell D10. Task 1 Examine the number sequences in columns C and D. Make a conjecture about the polynomial function that runs through all points plotted in the Graphics View and allows you to compute the surface of a cube for any given edge length e. Is it possible to determine the degree of this polynomial by investigating the sequences of differences you generated in columns C and D? Explain to your neighbor why we were repeatedly calculating differences of successive values and what they actually mean. Is it possible to determine the coefficient of the polynomial by investigating the sequences of differences you generated in columns C and D? Would this also work if the values in column A are not successive integers (e.g., 1, 3, 5, )? Explain your answer. Check your Conjecture about the Polynomial Create a slider n with interval from 0 to 5 and increment 1. Change the orientation of the slider from Horizontal to Vertical. Create a slider a with interval from -10 to 10 and increment 1. Change the orientation of the slider from Horizontal to Vertical. Enter the polynomial f(x) = a * x^n in order to create a polynomial of degree n with coefficient a. Note: Both, the degree n as well as the coefficient a can be changed by using the corresponding sliders. Change the values of sliders a and n to match your conjecture. Does the polynomial run through all points plotted in the Graphics View? Enhance your Construction 14 Insert the polynomial s equation as a dynamic text in the Graphics View. Hint: Select tool Insert Text and click on the Graphics View to open the text edit dialog window. (1) Enter f(x)= into the text edit dialog window. (2) Click on the graph of the polynomial to insert it s name into the text edit dialog window. Note: GeoGebra will enter the syntax necessary for dynamic text automatically. (3) Click on the OK button. 12
13 Insert a checkbox that allows you to show/hide the polynomial s equation. Hint: Select tool Checkbox to Show/Hide Objects and click on the Drawing Pad to open the checkbox dialog window. (1) Enter the caption Show equation. (2) Click on the little arrow to open list of available objects. (3) Select text1 from this list and click the Apply button. Activate the Move tool and try out if your checkbox controls the visibility of the text. Open the Properties Dialog and enhance the layout of the objects in the Graphics View (e.g., change the color of the polynomial and the points, match the color of the text with the color of the polynomial, fix the position of the slider, checkbox, and text in the Graphics View). Task 2 Try if this concept of investigating sequences of differences of two successive function values works for all polynomials f(x) = a x n. Hint: You may enter a formula into cell B1 and relative copy it down to cell B10 in order to create a list of function values. Don t forget to start the formula with an equal sign (e.g., = x^2) What modifications in the Spreadsheet View and Graphics View are necessary to be able to easily determine the constant of polynomials f(x) = a x n + b? 5. Scatter Plot and Best Fit Line Preparations Open a new GeoGebra file. Show the Spreadsheet View (View menu). Hide the Algebra View (View menu). In the Options menu set the Labeling to New Points Only. 13
14 Instructions 1 2 Enter the following numbers into the spreadsheet cells of column A: A1: 1 A2: 5 A3: 2 A4: 8 A5: -2 Enter the following numbers into the spreadsheet cells of column B: B1: -1 B2: 2 B3: 3 B4: 4 B5: 1 3 Create a Scatter Plot from this data: 4 Use tool Best Fit Line in order to create the line that best fits your data points. Hint: Activate tool Best Fit Line and select all data points using a selection rectangle: Click in the upper left corner of the Graphics View. Hold the mouse key down while moving the pointer to the lower right corner of the Graphics View in order to specify the 14
15 selection rectangle. 5 Change color and thickness of the line using the Properties Dialog. 6 Using this construction you can easily demonstrate how outliers impact the best fit line of a data set: Drag one of the points with the mouse and explore how this modification influences the best fit line. Note: You can easily change the initial data in the Spreadsheet View as well. Importing Data from other Spreadsheets Note: GeoGebra allows you to copy and paste data from other spreadsheet software into the GeoGebra spreadsheet: Select and copy the data you want to import (e.g., use the keyboard shortcut Ctrl-C in order to copy the data to your computer s clipboard). Note: You need to press both keys (e.g., Ctrl-key and C-key) at the same time in order to apply a keyboard shortcut. Open a GeoGebra window and show the Spreadsheet View. Click on the spreadsheet cell that should contain the first data value. Paste the data from your computer s clipboard into GeoGebra s Spreadsheet View. In order to do so, you can either use the keyboard shortcut Ctrl-V or right click (MacOS: Ctrl-click) on the highlighted cell and select Paste from the appearing Context Menu. 6. Exploring Basic Statistics Commands Yesterday, you gave a mathematics quiz to the 25 students of your 1 st period math class. After the quiz, you asked your students to rate the difficulty of the quiz on a scale from 1 ( very easy ) to 5 ( very difficult ). 4 of your students rated the quiz very easy (1) 6 students rated the quiz easy (2) 6 other students rated the quiz difficult (4) 1 student rated the quiz very difficult (5) The rest of the students thought the difficulty of the quiz was ok (3). Preparations Open a new GeoGebra file. Show the Spreadsheet View (View menu). Show the Algebra View (View menu). 15
16 Drag the origin of the coordinate system to the lower left corner of the Graphics View using tool Move Drawing Pad. Task 1: Create a Histogramm Enter the data into GeoGebra s spreadsheet and create a histogram. Instructions Enter the rating scores from 1 to 5 into column A of the spreadsheet, starting in cell A2. Enter the number of students who rated the quiz with each score into column B of the spreadsheet, starting with cell B2. Enter the class boundaries (starting and ending values of the histogram bar widths) into column C, starting in cell C1. Make sure that the histogram bars are drawn symmetrically around the rating scores (i.e., numbers from 1 to 5 on the x-axis). Hints: The first bar is supposed to be symmetrically drawn around the x- value 1. Therefore, the bar should start at 0.5 and end at 1.5. Enter the value 0.5 in cell C1 and the value 1.5 into cell C2. Relative copy these two values down to cell C6. This creates the 16
17 missing values 2.5, 3.5, 4.5, and Create a list of numbers from the class boundary values in column C. (1) Use the mouse to highlight all cells of columns A and B that contain numbers. (2) Right click (MacOS: Ctrl-click) on one of the highlighted cells and select Create List from the appearing Context Menu. Note: In the Algebra View, the list L 1 = {0.5, 1.5, 2.5, 3.5, 4.5, 5.5} is displayed. 5 Create a list of numbers (L 2 ) from the values in column B. 6 Create a histogram using the command Histogram[L_1, L_2] with L 1 and L 2 being the two lists of numbers you created in steps 4 and 5. Task 2: Determine mean, median, and mode 1. Make a prediction for mean, median, and mode of the data you collected. Hint: You may use command Sort[L_2] in order to sort the list of frequencies of students who rated the difficulty of the quiz in each category. 2. Check your prediction using the commands Mean, Median, and Mode. Hint: If you don t know how to use these commands, just enter the command into the Input Bar and press the F1 key in order to access the Input Bar Help. 17
Quickstart for Web and Tablet App What is GeoGebra? Dynamic Mathematic Software in one easy-to-use package For learning and teaching at all levels of education Joins interactive 2D and 3D geometry, algebra,
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For more info and downloads go to: Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It
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This tutorial was adapted from a tutorial by see its complete version at Excel 2003 Tutorial I Spreadsheet Basics Screen Layout Title bar Menu barAdvanced Excel Data Validation Excel has a feature called Data Validation, which will allow you to control what kind of information is typed into cells. 1. Select the cell(s) you wish to control. 2. Click
Excel Tutorial Below is a very brief tutorial on the basic capabilities of Excel. Refer to the Excel help files for more information. Working with Data Entering and Formatting Data Before entering data | 677.169 | 1 |
Statistics: Books
An introduction of Mathematics and Statistics is important for students who concentrate their studies in sciences. Maturity in Statistical tools as commonly obtained through logical thinking and rigorous analysis is essential towards mastering fundamental | 677.169 | 1 |
Monthly Archives: February 2015
Over the past few years, several alternative pathways to college level math have been explored across the country. These pathways differ from the usual pre/beginning/intermediate pathways that traditionally prepare students for college algebra. Instead, they aim to prepare students for a course like college math.
The faculty at Yavapai College are in the process of developing a new course that will help students move from Prealgebra to College Mathematics as quickly as possible. Since College Mathematics enrollments are on the rise, this will help more and more students complete their mathematics requirements quickly.
Not long ago, Microsoft Excel was the go to tool for making scatter plots and modeling data with regression functions. For the last ten years my students have used Excel to model data in projects. For many students wishing to use Excel on their own computers, Excel was an expensive purchase.
Other students wanted to save money and used ancient versions of Excel that were difficult to use. This was particularly aggravating to me since I had to help them find their models in Excel 2003, 2010, and 2013. The process was different in each platform.
This encouraged me to look at other ways of making pretty graphs with models. Google Sheets was particularly attractive because of its cost…free. Last summer Google added a simple way of adding a regression model to a scatter plot in Sheets. Now you can do just about everything you might want to do in College Algebra or Finite Mathematics in Google Sheets. And you get the added benefit of a single, free, platform in the Google ecosystem. | 677.169 | 1 |
s Online - Law of Sines to Bookmark material or Course ePortfolio
Throw darts at a circle, get the ratio inside to outside circle, and estimate pi.
Material Type:
Simulation
Author:
Paul Trunfio, Gary McGathThis lesson was prepared by Paul Trunfio and the JAVA applet was written by Gary McGath
Date Added:
Jun 29, 1997 fully developed high school curriculum modules that use the Internet in significant ways. ...
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This site contains a collection of fully developed high school curriculum modules that use the Internet in significant ways. There are currently 15 modules in Mathematics and 6 modules in Science; also, there are approximately two dozen additional modules that have been created by instructors and/or Education students.The learning modules here are web-based, technology intensive lessons focusing on mathematics and science in an applied context. They have been developed for teachers, by teachers, aligned with the Illinois State Learning Standards and the National Council for Teachers of Mathematics (NCTM) Standards. Some of the lessons are designed to last over several days, some only for a class period.
Material Type:
Collection
Author:
Office of Math, Science, Technology Education
Date Added:
Sep 13, 2004adratic Functions contains two applets that allow the user to change the coefficients of a quadratic equation and observe...
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Quadratic Functions contains two applets that allow the user to change the coefficients of a quadratic equation and observe the change in the corresponding graph. One applet uses the standard form of a quadratic equation to investigate the role of ?a?, ?b?, and ?c?? and the second uses the standard form for a parabola to study the role of ?a?, ?h?, and ?k?? algebra, including sample exercises and detailed solutions.
Material Type:
Presentation
Author:
Math Medics, L.L.C.
Date Added:
Sep 24, 2007Visually demonstrates the relation between constant velocity motion on a circle and sine wave generation
Material Type:
Simulation
Author:
Nicholas Jackiw
Date Added:
Apr 25igonometry to Bookmark material or Course ePortfolio
Visual Gauss is an exercise on the method of Gauss elimination to diagonalise a linear system or to find the inverse of a...
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Visual Gauss is an exercise on the method of Gauss elimination to diagonalise a linear system or to find the inverse of a matrix.
Material Type:
Drill and Practice
Author:
Xiao Gang
Date Added:
Oct 07, 2004
Date Modified:
Janway is a mathematics problem solving tool where students can select their math course - Basic Math, Pre-Algebra, Algebra,...
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Drill and Practice
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Aug 22, 2013 one | 677.169 | 1 |
Category Archives: Math for Economists
We have learnt indices or exponents in the algebra material. If you haven't check then, we recommend you to do so. We will need those concepts for progressing with logs. The idea of logarithms (or simply logs) is based on indices. In fact, as you will find out very soon, the rules for logarithms are very similar to the rules for indices. Therefore, a recap of the concept of indices will be useful for us to understand how logarithms works. | 677.169 | 1 |
Homework You should read the section we are going to discuss before coming to class. See the syllabus for a rough outline of what will be covered each day. I will try to highlight upcoming items below in red.
Day 01: Read Lecture 1 (pages 3-9; this is mostly review) and pages 63 and 321-324 in book. Install Matlab and finish the exercises in the Matlab Introduction document (for your own practice; not to be turned in). Read the syllabus in detail. Review linear algebra if you need to.
Day 05: Homework due Wednesday, Jan. 28: Lecture 4, exercise 1(a), 1(b), 1(c). Also, find the SVD of the 2x3 matrix A=[3 2 2; 2 3 -2] Please show your work, don't just throw them into Matlab or Wolfram or something. The point is for you to get experience with this machinery.
→ Check these out: Wikipedia aritcle on SVD and Wikipedia article on positive definite matrices
Day 06: Read Lecture 5 in the book. No exercises assigned from this lecture, but please read the exercises (especially 5.3) to find out some of the cool stuff you can do, now that you know about the SVD. Try some of the exercises on your own if you are curious. Also, try playing around with the "svd" function in matlab. To learn more about it, type "help svd" in MATLAB. Check your answer to Day 05's homework like this in MATLAB:
A = [3 2 2; 2 3 -2]; [U S V] = svd(A)
Day 07: Read Lecture 6 on Projectors
Day 08: Read Lecture 7 on QR factorization.
Day 09: Snow day
Day 10: Gram-Schimdt algorithm and QR-factorization. More on Matlab
Day 11: Introduction to LU-factorization. More on Matlab. Read Lecture 20.
Day 14: Cholesky factorization. Gaussian Elimination (i.e., direct methods). More on operation count. Read Lectures 21 and 23 in the book. Homework 4, due Friday, Feb. 20: Do book problems 20.2, 20.4, 21.1 (a) and (b), 21.6, and 23.3 (nothing to turn in for 23.3, but trying it will make you more experienced)
Possibly helpful resources: Wiki article on sparse matrices, Wiki article on banded matrices
Example: A 9x9 matrix with bandwidth 7:
A = delsq(numgrid('S',5)); spy(A)
What is the bandwidth of its LU-decomposition?
[L U] = lu(A); spy(L)
Day 15: Housholder Triangularization. Read Lecture 10. Read the exercises too (not assigned, but will give you some idea of how this stuff works). gs.png
Day 16: Review day, bring questions.
Day 17: Friday, Feb. 20 Exam 1. Paper and pencil. In usual classroom.
Day 18: Introduction to iterative methods. Read Lecture 32 in the book.
Day 19: Fundamental Theorem of Iterative Methods.
Homework 5 now posted: HW_05.pdf. (For LaTeX users: HW_05.tex). Read pages 186-188 in Trefethen and Bau (read all of Chapter 24 if you want a good review of eigenvalues, diagonalizability, and geometric/algebraic multiplicity).
Due 2015 March 6 Friday
Day 20: Convergence of Gauss-Seidel for Strictly Diagonally Dominant matrices.
Day 42: More on stabilty of ODE methods, Linear Difference Equations
Homework 7 now posted: HW_07.pdf. (For LaTeX users: HW_07.tex).
Note: This homework is not to be turned in! It is just for you to practice for the final exam.
Day 43: Lax equivalence for multi-step methods
Day 44: Catch up and Review
Final Exam: May 8, 7:30 am -9:30 am, in usual classroom. The final is not comprehensive. Here are some topics to help guide your study:
Contacting me:First, read the Syllabus.
The best way to contact me is via email. Please note that you should not expect an immediate response 24 hours a day, but that I will
try to answer your email in a reasonable amount of time (usually I am
pretty fast). When emailing, be sure to include your full name, course and section. If it is about homework problem, please write out the whole problem so I know what you are referring to. Here's what I'd like to see in email.
If you are going to type your assignments, the best way be far is to use LaTeX. After you learn the basics, it is far better than fighting with Microsoft Word to make your equations look right. It is also a vital skill for a professional mathematician to have. Below are some resources on learning it.
Being able to study efficiently is a skill that takes time to develop. It is normal to discover that the study habits and skills that worked for you in highschool or as a freshman need to be updated and improved if you are going to keep from being overwhelmed. It takes time and effort, as well as trial and error, to find which study habits work best for you. Below are some suggestions and resources to get you started. Good luck, and study efficiently!
Build a solid foundation, if you haven't already.
A solid foundation in linear algebra is required for this course.
Many people do not realize that they still have some issues with
even basic algebra. Success in this course will be almost impossible if there
are too many holes in your algebraic skills or basic mathematical knowledge. Fix any issues as soon as possible
so that you will be able to do your best.
An old linear algebra book is a good starting place, and there are many tutorials online as well.
Tips on doing homework
On the same day the homework is assigned, read over all the problems. This will get you thinking about them early on. The problems will stay in your brain on the "back burner", and you may have more luck later when you sit down to work them.
Do all the homework problems, even if you don't get them in on time.
Homework is excercise. If you want to get better at sports, you need to excercise everyday. If you want to get better at mathematics, you need to work on problems every day.
Advice from students for doing better on exams
Study with more people. Study in groups or with a partner. Work problems together on a white board. Take turns challenging each other. Being able to explain a problem clearly to someone else is a major step towards understanding it.
Pay attention in class and to the instructor's emails.
Rework what you didn't understand well.
Write down short examples on the notecard.
Note: Notecards are not allowed on exams; the pupose of this
tip is to help with studying.
Label formulas on your cards.
Theorems should be on the notecard.
Keep a running list of things for your notecard rather than trying
to compile it all at the last minute.
Write down formulas completely and correctly.
Advice on how to predict what is on the exam (in addition to some of the
above):
Old exams are a hint.
Pay attention to problems done in class several times.
Take note of quiz questions; one of those might become an exam
question.
Problems tend to be similar to homework/suggested homework.
In Class
Make sure you have read the section we are covering before you come to class.
Please be assertive in asking questions during lecture.
Please be assertive in paying attention in lecture.
Remember that you are responsible for all material presented in
lecture, no matter how strongly you felt it was emphasized.
* Use the external links on this page only according to your own judgement; I am not
responsable for, nor in control of, their content. The opinions given
there do not necessarily reflect my own, and I do not officialy endorse
any statements made on the websites which are linked to, nor any of the
content linked to therein.
News
I am an Assistant Professor in the
mathematics department at the University of Nebraska-Lincoln. | 677.169 | 1 |
"This is an intermediate level text, with
exercises, whose avowed purpose is to provide the science and engineering
graduate student with an appropriate modern mathematical (analysis and
algebra) background in a succinct, but nontrivial. manner... . The book is
quite thorough and can serve as a text, for self-study, or as a
reference." -Mathematical Reviews
Written for graduate and advanced undergraduate students in engineering
and science, this classic book focuses primarily on set theory, algebra,
and analysis. Useful as a course textbook, for self-study, or as a
reference, the work is intended to:
Whereas these objectives for writing this book were certainly pertinent
over twenty years ago when the work was first published, they are even
more compelling now. Today's graduate students in engineering or science
are expected to be more knowledgeable and sophisticated in mathematics
than students in the past. Moreover, today's graduate students in
engineering or science are expected to be familiar with a great deal of
ancillary material (primarily in the computer science area), acquired in
courses that did not even exist a couple of decades ago.
The book is divided into three parts: set theory (Chapter 1), algebra
(Chapters 2-4), and analysis (Chapters 5-7). The first two chapters deal
with the fundamental concepts of sets, functions, relations and
equivalence relations, and algebraic structures. Chapters 3 and 4 cover
vector spaces and linear transformations, and finite-dimensional vector
spaces and matrices. The last three chapters investigate metric spaces,
normed and inner product spaces. and linear operators. Because of its
flexible structure, Algebra and Analysis for Engineers and
Scientists may be used either in a one-or two-semester course by
deleting appropriate sections, taking into account the students'
backgrounds and interests.
A generous number of exercises have been integrated into the text, and
a section of references and notes is provided at the end of each chapter.
Applications of algebra and analysis having a broad appeal are also
featured, including topics dealing with ordinary differential equations,
integral equations, applications of the contraction mapping principle,
minimization of functionals, an example from optimal control, and
estimation of random variables. | 677.169 | 1 |
In this
unit, students build on their knowledge from unit 2, where they extended the
laws of exponents to rational exponents. Students apply this new understanding
of number and strengthen their ability to see structure in and create
quadratic and exponential expressions. They create and solve equations,
inequalities, and systems of equations involving quadratic expressions. In
preparation for work with quadratic relationships students explore
distinctions between rational and irrational numbers. They consider quadratic
functions, comparing the key characteristics of quadratic functions to those
of linear and exponential functions. Students learn to anticipate the graph
of a quadratic function by interpreting various forms of quadratic
expressions. In particular, they identify the real solutions of a quadratic
equation as the zeros of a related quadratic function. Students learn that
when quadratic equations do not have real solutions the number system must be
extended so that solutions exist, analogous to the way in which extending the
whole numbers to the negative numbers allows x+1 = 0 to have a solution.(Mathematics Appendix A, p.23, 25)
·Use properties of rational
and irrational numbers.
·Solve equations and
inequalities in one variable.
·Create equations that
describe number or relationships.
·Perform arithmetic
operations on polynomials.
·Write expressions in
equivalent forms to solve problems.
·Interpret the structure of
expressions.
·Interpret functions that
arise in applications in terms of a context.
·Analyze functions using
different representations.
·Build new functions from
existing functions.
Standards:
A-APR.A.1Understand
that polynomials form a system analogous to the integers, namely, they are
closed under the operations of addition, subtraction, and multiplication;
add, subtract, and multiply polynomials.
A-APR.B.3Identify
zeros of polynomials when suitable factorizations are available, and use the
zeros to construct a rough graph of the function defined by the polynomial.
oStandards Clarification: In Algebra I, tasks
are limited to quadratic and cubic polynomials, in which linear and quadratic
factors are available. For example, find the zeros of (x – 2)(x2 –
9).
A-CED.A.1Create
equations and inequalities in one variable and use them to solve problems.
Include equations arising from linear and quadratic functions, and simple
rational and exponential functions.★
A-CED.A.2Create
equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.★
A-REI.B.4Solve
quadratic equations in one variable.
Use the method of completing the square
to transform any quadratic equation in x into an equation of the form (x
– p)2 = q that has the same solutions. Derive the quadratic
formula from this form.
Solve quadratic equations by inspection
(e.g., for ), taking square roots, completing the
square, the quadratic formula and factoring, as appropriate to the
initial form of the equation. Recognize when the quadratic formula gives
complex solutions and write them as a ± bi for real
numbers a and b.
oStandards Clarification: Tasks do not
require students to write solutions for quadratic equations that have roots
with nonzero imaginary parts. However, tasks can require that students
recognize cases in which a quadratic equation has no real solutions.
A-REI.D.11Explain
why the 𝑥-coordinates of the points where the graphs
of the equations 𝑦=𝑓(𝑥) and 𝑦=𝑔(𝑥) intersect are the solutions of the
equation 𝑓(𝑥)=𝑔(𝑥); find the solutions approximately, e.g.,
using technology to graph the functions, make tables of values, or find
successive approximations. Include cases where (𝑥) and/or (𝑥) are linear, polynomial, rational, absolute
value, exponential, and logarithmic functions.★
oStandards Clarification: In Algebra I, tasks
that assess conceptual understanding of the indicated concept may involve any
of the function types mentioned in the standard except exponential and
logarithmic functions. Finding the solutions approximately is limited to
cases where f(x) and g(x) are polynomial functions.
A-SSE.A.1Interpret
expressions that represent a quantity in terms of its
context.★
Interpret parts of an expression, such
as terms, factors, and coefficients.
Interpret complicated expressions by
viewing one or more of their parts as a single entity. For example,
interpret P(1 + r)n as the product of P and a factor not
depending on P.
A-SSE.A.2Use
the structure of an expression to identify ways to rewrite it. For example,
see x4 – y4 as (x2)2 – (y2)2,
thus recognizing it as a difference of squares that can be factored as (x2
– y2)(x2 + y2).
oStandards Clarification: In Algebra I, tasks
are limited to numerical expressions and polynomial expressions in one
variable. Examples: Recognize that 532 – 472 is the
difference of squares and see an opportunity to rewrite it in the
easier-to-evaluate form (53 – 47)(53 + 47). See an opportunity to rewrite a2
+ 9a + 14 as (a + 7)(a + 2). Can include the sum or difference of cubes (in
one variable), and factoring by grouping.
A-SSE.B.3Choose
and produce an equivalent form of an expression to reveal and explain
properties of the quantity represented by the expression.★
Factor a quadratic expression to reveal
the zeros of the function it defines.
Complete the square in a quadratic
expression to reveal the maximum or minimum value of the function it
defines.
F-BF.B.3Identify
the effect on the graph of replacing 𝑓(𝑥) by 𝑓(𝑥)+𝑘, 𝑘𝑓(𝑥), 𝑓(𝑘𝑥), and 𝑓(𝑥+𝑘) for specific values of 𝑘 (both positive and negative); find the
value of k given the graphs. Experiment with cases and illustrate an
explanation of the effects on the graph using technology. Include recognizing even and odd functions
from their graphs and algebraic expressions for them.
oStandards Clarification: In Algebra I,
identifying the effect on the graph of replacing f(x) by f(x) + k, k f(x),
f(kx), and f(x+k) for specific values of k (both positive and negative) is
limited to linear and quadratic functions. Experimenting with cases and
illustrating an explanation of the effects on the graph using technology is
limited to linear functions, quadratic functions, square root functions, cube
root functions, piecewise-defined functions (including step functions and
absolute value functions), and exponential functions with domains in the
integers. Tasks do not involve recognizing even and odd functions.
F-IF.B.4For
a function that models a relationship between two quantities, interpret key
features of graphs and tables in terms of the quantities, and sketch graphs
showing key features given a verbal description of the relationship. Key features include: intercepts;
intervals where the function is increasing, decreasing, positive, or
negative; relative maximums and minimums; symmetries; end behavior; and
periodicity.★
F-IF.B.5Relate
the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For
example, if the function ℎ(𝑛)
gives the number of person-hours it takes to assemble 𝑛 engines in a
factory, then the positive integers would be an appropriate domain for the
function.★
F-IF.B.6Calculate
and interpret the average rate of change of a function (presented
symbolically or as a table) over a specified interval. Estimate the rate of
change from a graph.★
F-IF.C.8Write
a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
Use the process of factoring and
completing the square in a quadratic function to show zeros, extreme
values, and symmetry of the graph, and interpret these in terms of a
context.
F-IF.C.9Compare
properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal
descriptions). For example, given a
graph of one quadratic function and an algebraic expression for another, say
which has the larger maximum.
N-RN.B.3Explain
why the sum or product of two rational numbers is rational; that the sum of a
rational number and an irrational number is irrational; and that the product
of a nonzero rational number and an irrational number is irrational.
Focus Standards of
Mathematical Practice:
MP.1 Make sense of problems and persevere in
solving them.
MP.2 Reason abstractly and quantitatively.
MP.3 Construct viable arguments and critique the
reasoning of others.
MP.4 Model with mathematics.
MP.5 Use appropriate tools strategically.
MP.6 Attend to precision.
MP.7 Look for and make use of structure.
MP.8 Look for and express regularity in repeated
reasoning.
Instructional Outcomes:
Full Development
of the Major Clusters, Supporting Clusters, Additional Clusters and
Mathematical Practices for this unit could include the following
instructional outcomes:
A-APR.A.1
·I can identify that the sum, difference, or
product of two polynomials will always be a polynomial, which means that
polynomials are closed under the operations of addition, subtraction, and
multiplication
·I can define "closure"
·I can apply arithmetic operations of
addition, subtraction, and multiplication to polynomials
A-APR.B.3
·I can factor polynomials using any method.
·I can sketch graphs of polynomials using
zeroes and a sign chart.
A-CED.A.1
·I can solve linear and exponential equations
in one variable
·I can solve inequalities in one variable
·I can describe the relationships between the
quantities in the problem (for example, how the quantities are changing or
growing with respect to each other); express these relationships using
mathematical operations to create an appropriate equation or inequality to
solve
·I can create equations (linear and
exponential) and inequalities in one variable and use them to solve problems
·I can create equations and inequalities in
one variable to model real-world situations
·I can compare and contrast problems that can
be solved by different types of equations (linear and exponential)
A-CED.A.2
·I can identify the quantities in a
mathematical problem or real-world situation that should be represented by
distinct variables and describe what quantities the variables represent
·I can create at least two equations in two
or more variables to represent relationships between quantities
·I can justify which quantities in a
mathematical problem or real-world situation are dependent and independent of
one another and which operations represent those relationships
·I can determine appropriate units for the
labels and scale of a graph depicting the relationship between equations
created in two or more variables
·I can graph one or more created equation on
a coordinate axes with appropriate labels and scales
A-REI.B.4
·I can use the method of completing the
square to transform any quadratic equation in x into an equation of the form
(x-p)2 = q that has the same solutions
·I can solve quadratic equations in one
variable
·I can derive the quadratic formula by
completing the square on a quadratic equation in x
·I
can approximate/find the solution(s) using an appropriate method for example,
using technology to graph the functions, make tables of values or find
successive approximations (Include cases where f(x) and/or g(x) are linear
and exponential equations)
A-SSE.A.1
·I can, for expressions that represent a
contextual quantity, define and recognize parts of an expression, such as
terms, factors, and coefficients
·I can, for expressions that represent a
contextual quantity, interpret parts of an expression, such as terms,
factors, and coefficients in terms of the context
·I can, for expressions that represent a
contextual quantity, interpret complicated expressions, in terms of the
context, by viewing one or more of their parts as a single entity.
A-SSE.A.2
·I can identify ways to rewrite expressions,
such as difference of squares, factoring out a common monomial, regrouping,
etc
·I can identify ways to rewrite expressions
based on the structure of the expression
·I can use the structure of an expression to
identify ways to rewrite it.
·I can classify expression by structure and
develop strategies to assist in classification
A-SSE.B.3
·I can factor a quadratic expression to
produce an equivalent form of the original expression
·I can explain the connection between the
factored form of a quadratic expression and the zeros of the function it
defines
·I can explain the properties of the quantity
represented by the quadratic expression
·I can choose and produce an equivalent form
of a quadratic expression to reveal and explain properties of the quantity
represented by the original expression
·I can complete the square on a quadratic
expression to produce an equivalent form of an expression
·I can explain the connection between the
completed square form of a quadratic expression and the maximum or minimum
value of the function it define
·I can explain the properties of the quantity
represented by the expression
·I can choose and produce an equivalent form
of a quadratic expression to reveal and explain properties of the quantity
represented by the original expression
F-BF.B.3
·I
can identify the effect a single transformation will have on the function
(symbolic or graphic)
·I
can use technology to identify effects of single transformations on graphs of
functions
·I
can graph a given function by replacing f(x) with f(x)+k, kf(x), f(kx), or
f(x+k) for specific values of k (both positive and negative)
·I
can describe the differences and similarities between a parent function and
the transformed function
·I
can find the value of k, given the graphs of a parent function, f(x), and the
transformed function: f(x)+k, kf(x), f(kx), or f(x+k)
·I
can recognize even and odd functions from their graphs and from their
equations
·I
can experiment with cases and illustrate an explanation of the effects on the
graph using technology
F-IF.B.4
·I
can define and recognize the key features in tables and graphs of linear and
exponential functions: intercepts; intervals where the function is
increasing, decreasing, positive, or negative, and end behavior
·I
can identify whether the function is linear or exponential, given its table
or graph
·I
can interpret key features of graphs and tables of function in the terms of
the contextual quantities the function represents
·I
can sketch graphs showing key features of a function that models a
relationship between two quantities from a given verbal description of the
relationship
F-IF.B.5
·I
can, given the graph or a verbal/written description of a function, identify
and describe the domain of the function
·I
can identify an appropriate domain based on the unit, quantity , and type of
the function it describes
·I
can relate the domain of the function to its graph and, where applicable, to
the quantitative relationship it describes
·I
can explain why a domain is appropriate for a given situation
F-IF.B.6
·I
can recognize slope as an average rate of change
·I
can calculate the average rate of change of a function (presented
symbolically or as a table) over a specified interval
·I
can estimate the rate of change from a linear or exponential graph
·I
can interpret the average rate of change of a function (presented
symbolically or as a table) over a specified interval
F-IF.C.7
·I can graph linear functions by hand in
simple cases or using technology for more complicated cases and show/label
intercepts of the graph
·I can graph square root, cube root, and
piecewise-defined functions, including step functions and absolute value
functions, by hand in simple cases or using technology for more complicated
cases, and show/label key features of the graph
·I can determine the difference between
simple and complicated linear, quadratic, square root, cube root, and
piecewise-defined functions, including step functions and absolute value
functions and know when the use of technology is appropriate
·I can compare and contrast the domain and
range of absolute vale, step and piece-wise defined functions with linear,
quadratic, and exponential
F-IF.C.8a
·I can identify different forms of a
quadratic expression
·I can write functions in equivalent forms
using the process of factoring
·I can identify zeros, extreme values, and
symmetry of the graph of a quadratic function
·I can interpret different but equivalent
forms of a function defined by an expression in terms of context
·I can use the process of factoring and
completing the square in a quadratic function to show zeros, extreme values,
and symmetry of the graph, and intercept these in terms of a context
F-IF.C.9
·I
can identify types of functions based on verbal, numerical, algebraic, and
graphical descriptions and state key properties (e.g. intercepts, growth
rates, average rates of change, and end behaviors)
·I
can differentiate between exponential and linear functions using a variety of
descriptors (graphically, verbally, numerically, and algebraically)
·I
can use a variety of function representations algebraically, graphically,
numerically in tables, or by verbal descriptions) to compare and contrast
properties of two functions
N-RN.B.3
·I can find the sums and products of rational
and irrational numbers
·I can recognize that the sum of a rational
number and an irrational number is irrational
·I can recognize that the product of a
nonzero rational number and irrational number is irrational
·I can explain why rational numbers are
closed under addition or multiplication
Enduring
Understandings:
·There
are multiple algorithms for finding a mathematical solution and those
algorithms are frequently associated with different contexts.
·Quadratic
functions, like linear and exponential, can be used to model real-life
situations.
·There
is an important distinction between solving an equation and solving an
applied problem modeled by an equation.
·Algebraic
representations can be used to generalize patterns in mathematics.
·Relationships
between quantities can be represented symbolically, numerically, graphically,
and verbally in the exploration of real world situations.
·The
characteristics of radical functions and its representations are useful in
solving real-world problems.
·Families
of functions exhibit properties and behaviors that can be recognized across
representations.
·Functions
can be transformed, combined, and composed to create new functions in mathematical
and real world situations.
·Mathematical
functions are relationships that assign each member of one set to a unique
member of another set and the relationship is recognizable across
representations.
Essential
Questions:
·How
can patterns, relations, and functions be used as tools to best describe and
help explain relationships between quantities?
·How
do parameters introduced in the context of the problem affect the symbolic,
numeric and graphical representations of a quadratic function?
·How
are patterns of change related to the behavior of functions?
·What
characteristics of problems would determine how to model the situation and
develop a problem solving strategy?
·How
do radical functions model real-world problems and their solutions?
·How
are expressions involving radicals and exponents related?
·How
can an equation, table, and graph be used to analyze the rate of change and
other applicable information, related to a real-world problem and the
representative function?
·How
can a given function be represented graphically, within a table, by an
equation, and in the real-world?
·What
connections can be made between various functions and various representations
of functions? | 677.169 | 1 |
Steps to Common Entrance Maths is a series of three books intended for pupils aged 7-10 years.
Key Features
Author(s)
Walter Phillips
Publisher
Oxford University Press
Date of Publication
01/10/1989
Language
English
Format
Paperback
ISBN-10
0748701109
ISBN-13
9780748701100
Subject
School Textbooks & Study Guides: Maths, Science & Technical
Publication Data
Place of Publication
Oxford
Country of Publication
United Kingdom
Imprint
Nelson Thornes Ltd
Dimensions
Weight
464 g
Width
200 mm
Height
272 mm
Spine
11 | 677.169 | 1 |
Each're perfect for undergraduate and graduate studies.
REA's Problem Solvers is a series of useful, practical, and informative study guides. Each title in the series is complete step-by-step solution guide. The Geometry Problem Solver enables students to solve difficult problems by showing them step-by-step solutions to Geometry problems.
The Problem Solvers cover material ranging from the elementary to the advanced and make excellent review books and textbook companions.
The Geometry Problem Solver is the perfect resource for any class, any exam, and any problem!
About the Author:
Ernest Woodward, Ed.D., is professor of mathematics at Austin Peay State University, Clarkesville, Tenn. Founded in 1959, Research & Education Association is dedicated to producing the finest and most effective digital and print materials in educational publishing, including some of America's most popular test preps and study aids. REA's wide-ranging catalog is a leading resource for teachers, students, and professionals. | 677.169 | 1 |
ib maths4u | ib maths | ib math | ib mathematicsThe first forum for Mathematics compatible also with ib mathematics (HL,SL,studies,myp mathematics) - ib math Revision Notes2013-01-24T19:20:29+00:00 is the first forum introducing the addition of LaTeX mathematical typesetting for IB mathematics(HL,SL,studies). LaTeX is a professional-grade general typesetting system that centers around very pleasing mathematical presentation. Mathematical expressions are written in a markup style somewhat similar to HTML. You may include LaTeX graphics in any post here on IBmaths4u.com To include a LaTeX graphic, simply include your LaTeX code within the tags [ tex][ /tex]. without spaces. Or by using the rightmost button in editing bar as shown in the following image.
Basic Guidelines Lines and Text. \\ starts a new line, \ includes a space, \mbox{...} includes text
Sub- and Superscripts.
Code:
[tex]x^3[/tex]
produces while
Code:
[tex]x_{2n}[/tex]
produces Here is another example:
Code:
[tex]\log_{5} 125=3[/tex]
gives
Fractions. Use \frac to display fractions. Example:
Code:
[tex]\frac{\pi^2}{6}[/tex]
gives .
Roots. Use \sqrt. For instance,
Code:
[tex]\sqrt{a^3+b^4}[/tex]
produces
You can also get "other" roots:
Code:
[tex]\sqrt[4]{5}[/tex]
yields .
Various examples Code:
Code:
[tex] \frac{1}{2^n} \rightarrow 0[/tex]
produces the graphic
Code:
Code:
[tex]\int_a^b f(t) dt = 1[/tex]
produces the graphic
Code:
Code:
[tex] \sum_{n=0}^{\infty} \frac{1}{n!} = e [/tex]
produces the graphic
Plus or minus
Code:
[tex]\pm[/tex]
produces the graphic
multiplication or dot product
Code:
[tex]a\cdot b[/tex]
produces the graphic
Equals sign and inequalities
To get an equals sign, you simply use the ordinary = sign.
Code:
[tex]2+3=5[/tex]
produces the graphic
For 'not equal to', use \not=
Code:
[tex]2^4\not=2[/tex]
produces the graphic
For less than signs, we use <, and for greater than signs, we use >
For 'less than or equal to' signs, we use \leq, and for 'greater than or equal to' signs we use \geq
Code:
[tex]x^2 \geq 0[/tex]
produces the graphic
For spaces, use "\ ".
Sigma notation
To write sums, we use the \sum command.
Code:
[tex]\sum_{i=1}^n i^2 = \frac{1}{6}n(n+1)(2n+1) [/tex]
produces the graphic
Differentiation
We use \frac{}{} to write dy/dx.
Code:
[tex]\frac{d}{dx} x^4 = 4x^3[/tex]
produces the graphic
For f'(x), simply write it out normally within TeX tags.
Code:
[tex]f'(x)[/tex]
produces the graphic
Integration
For the integral sign, use the \int command.
Code:
[tex]\int 3x^2\ dx = x^3 + C[/tex]
produces the graphic
For definite integrals, use the commands for subscripts and superscripts.
Code:
[tex]\int^1_0 x\ dx =0.5[/tex]
produces the graphic
Modulus sign or absolute value
Use | for the modulus sign.
Code:
[tex]\sqrt{x^2} = |x|[/tex]
produces the graphic
Factorial
Use the exclamation mark like normal.
Code:
[tex]3! = 6[/tex]
produces the graphic
n choose r
Code:
[tex]^n\mathrm{C}_r[/tex]
produces the graphic
Greek Letters
Write \x where x is the written form of the Greek letter (i.e. alpha, beta, gamma, ... , omega). | 677.169 | 1 |
GCSE Maths In Four Weeks Revision Guide – Grades 9-1 by Jeevan Singh
Description
NOTE: Limited Stock Available. Get Your Copy Now | 677.169 | 1 |
Developing Thinking in Algebra
John Mason, Alan Graham, Sue Johnston-Wilder
One in a series of books concerned with developing mathematical thinking, this practical book will enable you to reflect on algebra and its learning from the learner's perspective (both yours and your students'). Written for teachers of students aged 7–16, the book's emphasis is on what it is to understand the process of thinking algebraically.
Working through this resource will enable you to deepen your knowledge of mathematics in a frame which integrates a pedagogical perspective. Perfect for pre-service teachers and those retraining to teach in the subject area. Engaging with the tasks presented in this volume will provide any teacher of mathematics with a richer understanding of, and enthusiasm for, algebra – based on firm research and effective practice. | 677.169 | 1 |
MTH450A Mathematics Project Course I
Course Description
The project courses are not independent study. They are directed student team projects or internships in mathematics. Requires utilization of previously acquired skills and knowledge to complete the project. Students can select project topics from industry, government, business, education, or research. Grading is S or U only.
Learning Outcomes
Employ a variety of reasoning skills and effective strategies for solving problems
Use language and mathematical symbols to communicate mathematical ideas
Use current technology tools that is appropriate for the research and study in mathematics.
Employ mathematics concepts and methods from different mathematics branches | 677.169 | 1 |
The basic algebra worksheets introduce students to using and writing algebraic equations with letter variables | 677.169 | 1 |
Exhibition Reflection
What Did You Learn in This Project?
I learned that the history of Algebra goes back thousands of years and is included in almost everything that we do. Polynomial Functions are a base to everything that we do in mathematics today. What Would You Do Differently Next Time?
If I were to do this project again, I would do a bit more research into exactly how these functions were used in the Islamic culture. Aside from knowing that they appear on buildings, I would like to know a bit more history and see some examples.
How Would You Improve This Project? I would improve the encyclopedia by adding more pictures and making it more visually appealing. We made it very informative and clean looking, however it might be a little imposing to read because of the amount of text. This could be fixed by requiring pictures or symbols throughout the text.
What Would You Do Differently If You Were Exhibiting This Project Again?
If I were to exhibit this project again, I would find a way to make it more interactive. It would be great to do a digital encyclopedia or create a video documentary. It is hard for people at exhibitions to read something when they could be watching a video or something more interactive and engaging. It is hard to keep people's attention when you are just talking to them for a long time.
What Do You See As Your Greatest Strengths and Weaknesses Moving Forward Into Future Math Classes?
I think that my greatest strength moving forward is my motivation. When I really want or need to get something done, I do. I commit and I work my hardest at achieving the goal that I set. If I don't understand something, I am really good at asking for help and moving through the challenge quickly and efficiently. My weakness that I need to work on is my time management, which almost contradicts my last statement. When I want to get something done, I do. However when I'm overloaded with work I get overwhelmed and cannot work out a clear schedule in my mind. I need to work on making sure that everything gets done and working in a calm
Encyclopedia Project
The goal of this project was to help create an encyclopedia about the concepts that we have learned this year in Algebra 2. I studied Polynomial Functions and their history. I believe that the most important thing that I learned about Polynomial Functions is that they were, and still are, a crucial part of the Islamic culture. They were not allowed to create human statues as a part of their religion and so they used complex polynomial functions to design facades of buildings.
During this project, I learned a lot about how I handle not only stress, but how to manage several tasks at once. The project itself was not complicated at all, however we also had several pieces of homework to do and there were a lot of days when we were taking notes. I learned that I am good at handling that kind of stress and am comfortable multitasking.
For exhibition prep, I am going to be reading over my project several times to make sure that it is ready. I have already proofread several times, however I know that there are always revisions that can be made, especially because I am going to be presenting this to the public.
Semester Exam Project
I really enjoyed the last project. Not only did it allow us to reflect on what we did this semester, but we also were able to know what was going to be on the test to allow us to study the actual questions that will be on the test.
For the final exam, I am not only going to be studying my composition book and our notes, but I will also fill out the study guide and look over previous assignments to see what my strengths and weaknesses are.
This semester has been extremely difficult. I have had to balance my work, school and my social life at the same time. Math is one of the subjects that I am good at, and to be honest it is a nice break in the busy rush of a normal day. | 677.169 | 1 |
Cart
Secrets in Inequalities, Vol. 1 – Basic Inequalities
To truly understand inequalities and develop a foundation of knowledge to solve them well, you must unlock their secrets through a variety of problems and techniques. That is the aim of Secrets in Inequalities. Chapters 1 – 8 will give the reader exposure to AM-GM inequality, Cauchy-Schwarz and Hölder Inequalities, Chebyshev Inequality, Inequalities with Convex Functions, Abel Formula and Rearrangement Inequality, The Method of Balanced Coefficients, Derivative and Applications, and Symmetric Inequalities. Chapter 9 is dedicated to providing the problems necessary to practice the topics covered in the previous chapters. The reader is encouraged to study the applications of inequalities, build a strong knowledge base, and then try and solve the problems on their own to gain practice and increase understanding. | 677.169 | 1 |
Awesome old-style maths class with crystal clear explanations (precursor to the Khan Academy from back in the days)
I strongly encourage you to follow the entire series of video classes by Herbert Gross. These are fantastically well detailed and very understandable at a very elementary level. I wish nowadays classes would look more like these ones, with examples to introduce the concepts intuitively, and then moving on to a more abstract framework to appreciate the generality of what was exposed!
Part I : Vector Arithmetic
Part II : Vector Calculus
Part III : Partial Derivatives
Part IV : Matrix Algebra
Part V : Multiple Integration
Very neat introductions to programming (for the very beginner) from MIT and Stanford
These introductions to programming are very accessible and teaches you a lot about basic principles of computer science and programming. In today's world filled with so many technologies, it becomes absolutely necessary for people to be able to understand a basic computer program and these video series will help you do just that.
A more comprehensive introduction to algorithms. Including Dynamic Programming.
These video lectures provide an updated version of the MIT intro course on algorithms featured above. In particular, you will learn much more about dynamic programming in this class (4 lectures towards the end of the series made by the awesome Erik Demaine). If you want to explore more about what Erik can teach you, visit his website : here | 677.169 | 1 |
Coursera: Analysis of a Complex Kind
Complex analysis is the study of functions that live in the complex plane, i.e. functions that have complex arguments and complex outputs. In order to study the behavior of such functions we'll need to first understand the basic objects involved, namely the complex numbers. We'll begin with some history: When and why were complex numbers invented? Was it the need for a solution of the equation x^2 = -1 that brought the field of complex analysis into being, or were there other reasons? Once we've answered these questions we'll devote some time to learn about basic properties of complex numbers that will make it possible for us to use them in more advanced settings later on. We will learn how to do basic algebra with these numbers, how they behave in limiting processes, etc. These facts enable us to begin the study of complex functions, and at this point we can already understand the basics about the construction of the Mandelbrot set and Julia sets (if you have never heard of
these that's quite alright, but do look at
for example to see some
beautiful pictures).
When studying functions we are often interested in their local behavior, more specifically, in how functions change as their argument changes. This leads us to studying complex differentiation – a more powerful concept than that which we learned in calculus. Don't worry! We'll help you remember facts from calculus in case you have forgotten. After this exploration we will be ready to meet the main players: analytic functions. These are functions that possess complex derivatives in lots of places, a fact which endows these functions with some of the most beautiful properties mathematics has to offer. We'll explore these properties!
Who would want to differentiate without being able to undo it? Clearly we'll have to learn about integration as well. But we are in the complex plane, so what are the objects we'll integrate over? Curves! We'll study these as well, and we'll tie everything together via Cauchy's beautiful and all encompassing integral theorem and formula.
Throughout this course we'll tell you about some of the major theorems in the field (even if we won't be able to go into depth about them) as well as some outstanding conjectures.
Syllabus
Week One: Introduction to complex numbers, their geometry and algebra, working with complex numbers.
Week Two: The Mandelbrot set, Julia sets, a famous outstanding conjecture, history of complex numbers, sequences of complex numbers and convergence, complex functions Analysis of a Complex Kind
4.7
Based on 7 reviews
It was a very informative class. Besides the analytics of complex numbers, we learned to communicate our reasoning and to plot graphs in the complex plane, which was rated by some of our classmates (peer grading). We also were shown how to plot Julia sets, extremely nice pictures.
Dr. Bonfert-Taylor explained all topics in an understandable and friendly way. | 677.169 | 1 |
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Math Honor Society
In the United States, an honor society is a rank organization that recognizes excellence among peers. Numerous societies recognize various fields and circumstances. Chiefly, the term refers to scholastic honor societies, those that recognize students who excel academically or as leaders among their peers, often within a specific academic discipline. Many honor societies invite students to become members based on the scholastic rank (the top x% of a class) and/or grade point averages of those students, either overall, or for classes taken within the discipline for which the honor society provides recognition. The Math Honor Society is no different. Joining a honor society allows a student to appear superior to other students for college applications. They are a great number of colleges that seek higher educations and joining the Math Honor Society is the perfect way to show colleges that you have a higher education in math | 677.169 | 1 |
New 13+ Maths Practice Book for the Common Entrance Exams with Answers & Online Practice Papers
New 13+ Maths Practice Book for the Common Entrance Exams with Answers & Online Practice Papers
This Practice Book is a brilliant way to help students prepare effectively for the Common Entrance 13+ Maths exams in 2016 and beyond! It contains exam-style practice questions for every 13+ Maths topic, with stamps to clearly show which material belongs to Levels 1, 2 and 3. Detailed answers are included at the back for easy marking.
The book also comes with a link to a full set of printable online practice papers, plus downloadable audio files for the mental maths test.
A matching 13+ Maths Study Book is also available, containing clear explanations and examples for every topic.
This book is also available for (this link will take you to the amazon.co.uk website). You don't need a Kindle to read it — just download the free Kindle App for your computer or tablet. | 677.169 | 1 |
Description: These notes serve as course notes for an undergraduate course in number theory. Proofs of basic theorems are presented in an interesting and comprehensive way that can be read and understood even by non-majors. The exercises are carefully chosen to broaden the understanding of the concepts.
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Download E-books Lessons for Algebraic Thinking, Grades 6-8 PDF
These classes express tips on how to maximize guide that prepares scholars for formal algebra. via a sequence of investigations, scholars construct their talent with key algebraic options. Connections among mathematics and algebra are made by utilizing drawings, tables, graphs, phrases, and emblems. classes contain a know-how part with feedback for instructing with graphing calculators.
A brand new attitude to studying GEOMETRY attempting to comprehend geometry yet consider like you are caught in one other measurement? here is your resolution. Geometry Demystified, moment variation is helping you take hold of the fundamental recommendations very easily. Written in a step by step structure, this functional advisor starts off with dimensions, reviewing issues, traces, angles, and distances, then covers triangles, quadrilaterals, polygons, and the Cartesian airplane.
Each year, hundreds of thousands of scholars claim arithmetic as their significant. Many are super clever and hardworking. besides the fact that, even the easiest will come upon demanding situations, simply because upper-level arithmetic includes not just self sufficient learn and studying from lectures, but in addition a primary shift from calculation to facts.
Within the years instantly following international warfare II, Black Mountain university, an unaccredited institution in rural Appalachia, grew to become a necessary hub of cultural innovation. essentially | 677.169 | 1 |
Mathematics for Cambridge IGCSE Extended
Learning Outcomes at the start of each chapter map the topics to the syllabus.
A three stage, visual approach to studying: learn the topics in detail through detailed explanations and worked examples, apply the knowledge you have learned to a series of problems, then assess your progress with a bank of questions for each topic.
Study Tips highlight common mistakes and advise students how best to avoid them.
End-of-topic and end-of-section questions test understanding, not only of the current topic but also of the ways in which different topics might be combined in examinations.
Mathematics for Cambridge IGCSE Extended
Description
Comprehensively matched to the Cambridge Mathematics syllabus at Extended Level, this engaging and accessible Student Book focuses on the key concepts, guiding learners through the IGCSE course and building confidence for assessment. | 677.169 | 1 |
Mathematics
to create a bridge between the concreteness of arithmetic and the abstractions that will follow in algebra and geometry.
to provide opportunities to tap into the natural curiosity and enthusiasm for activities and learning that Middle School students bring to their classes.
to provide experiences and opportunities in mathematics that address the academic needs and talents of each child.
Embracing the idea that math involves reasoning and problem-solving skills to foster curiosity, persistence and attention to detail, the sixth grade course strives to strengthen and extend work with integers, fractions and decimals, then move into percentages and three-dimensional geometry while establishing a solid foundation for the abstract language and mechanics of Pre-Algebra. While our mathematics teachers use a variety of resources to teach our students, the primary source is the Math in Focus program, formerly known as Singapore Math.
In either Pre-Algebra or Advanced Pre-Algebra, seventh graders continue to develop an appreciation for the aesthetics of mathematics, with special emphasis placed on the processes used to solve problems rather than on the answer itself.
Most eighth graders take either Algebra I or Advanced Algebra I. Both cover the same material, but the advanced course does so at an accelerated pace and with higher expectations. Each course presents material to both interest and challenge students while using real-life applications to make the learning more relevant.
For students who demonstrate exceptional talent in mathematics, Penn Charter offers an Accelerated Middle School Math Program. The math department will determine placement in this program. Students qualifying for this program must demonstrate:
a love for mathematics and an eagerness to be challenged.
high achievement in previous grades.
exceptional achievement on the SSAT test.*
Teacher recommendations are also considered.
In addition, placement tests are given to ensure mastery of prerequisite topics before assigning a student to an upper level math course. Because of the sequential and cumulative nature of mathematics, it is very important that students receive instruction in prerequisite topics before taking an advanced level course. | 677.169 | 1 |
Ms. Breezy Gutierrez
Agile Mind Intensified Algebra I Intensified Algebra I is a comprehensive, extended-period course that is designed to help students who are one to three years behind in mathematics re-engage as motivated learners and succeed in Algebra I within a single academic year. Intensified Algebra I integrates a comprehensive course curriculum with interventions from social psychology to motivate and develop students' positive beliefs about their academic abilities.
Algebra I A full year basic college prep course that will cover the essential principles of algebra using multiple strategies using traditional and new methods, technological strategies, and best practices. Students are expected to develop an understanding of the basic structure and language of Algebra, a tool that is used to represent and solve real-world situations. Students are expected to learn the foundation for solving problems and use verbal language as well as numbers to analyze functions, solve and evaluate equations and expressions, and interpret tables, graphs, and models. In addition, students will practice and solve problems related to geometry, data analysis, statistics and probability using manipulatives, models, graphs, computer applications, and graphing calculators.
Financial Literacy Financial Literacy is a mathematics course designed to apply knowledge of consumer topics, algebra, and geometry to real life. The goal is to inform students of finance in mathematical terms and formulas and to manage their finances when they leave to become productive adults in society. Topics are related to credit, banking, financial services, retirement, health and life insurance, and money management. Literacy, technology, health, and life skills are integrated throughout the course.
Health Health is a one-half credit course required by SRHS as well as the State of New Mexico Public Education Department. The course concentrates on the five following topics: Physical Activity, Nutrition, Sexual Education, HIV/AIDS, and Tobacco, Alcohol and Drugs. In addition, topics covered emphasize the social, physical, and mental/emotional aspects of health and the awareness of health behaviors and their consequences. It is the goal to increase the health of students and assist them in using positive decision-making processes that will affect their lifestyles. | 677.169 | 1 |
Mathematics for Applications
The exposition will face across the traditional classification of mathematical branches so that it will respect students´ needs and options. It will be directed in an interactive form in order to respond to suggestions of students. A global view of problems enables students to see connections among apparently remote branches of mathematics.
Students get acquainted with a broad range of mathematical concepts occurring in physical applications, both from mathematical analysis and from algebra. They will be made familiar with derivatives and partial derivatives and their use when investigating extremes. A further topic are indefinite, definite, and more-dimensional integrals in the sense of Riemann and of Lebesgue. A next part of the programme are functions of a complex variable. Last but not least, students will revise important notions from linear algebra.
The course is taught through lectures explaining the basic principles and theory of the discipline.
Assesment methods and criteria linked to learning outcomes
The course is finished by an oral examination. The examiner verifies the knowledge of definitions, theorems, and algorithms and the ability of their use in concrete applications.
Language of instruction
Czech, English
Work placements
Not applicable.
Aims
The aim of the subject is a summarization, extension, and enlargement of knowledge of mathematics from bachelor´s and master´s studies with a view to applications, especially in physical engineering.
Specification of controlled education, way of implementation and compensation for absences
Attendance at lectures is recommended. The lessons are planned on the basis of a weekly schedule. It is possible to study individually according to the recommended literature with the use of consultations. | 677.169 | 1 |
This monograph presents a short course in computational geometry and topology. In the first part the book covers Voronoi diagrams and Delaunay triangulations, then it presents the theory of alpha complexes which play a crucial role in biology. The central part of the book is the homology theory and their computation, including the theory of persistence... more... | 677.169 | 1 |
6 Study Guide And Intervention Answers 133521
glencoe algebra 2 chapter 3 study guide and intervention
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Worksheet Four: Functions
In this function worksheet, students build a function under given conditions and evaluate it at a specified point. They map and build vector functions. This two-page worksheet contains 8 multi-step problems. | 677.169 | 1 |
Mathematics and Statistics Tutorial
Pascal Wallisch
Not everyone who intends to start practicing the neurosciences can be expected to do so with a perfect knowledge of mathematics. As a matter of fact, due to the inherently interdisciplinary nature of the field, it would be quite surprising if this were the case. Educational backgrounds are diverse, and not everyone is privileged enough to hail from a "math-heavy" one as afforded by physics, computer science, or engineering. Thus, the purpose of this tutorial is largely therapeutic in nature. We will focus on introducing a few key concepts in linear algebra and statistics that are central to neuroscience research. We will do so in the most gentle and affirming way possible. In the process, ...
With Safari, you learn the way you learn best. Get unlimited access to videos, live online training,
learning paths, books, interactive tutorials, and more. | 677.169 | 1 |
Calculus is about the very large, the very small, and how things change. The surprise is that something seemingly so abstract ends up explaining the real world. Calculus plays a starring role in the biological, physical, and social sciences. By focusing outside of the classroom, we will see examples of calculus appearing in daily life.
This course is a first and friendly introduction to calculus, suitable for someone who has never seen the subject before, or for someone who has seen some calculus but wants to review the concepts and practice applying those concepts to solve problems.
From the lesson
Fundamental Theorem of Calculus
Armed with the Fundamental Theorem of Calculus, evaluating a definite integral amounts to finding an antiderivative. | 677.169 | 1 |
PreCalculus
PreCalculus Chapter 10
Click on the bulletin board for the Chapter 10 practice test.
Second Semester Focus: Trigonometry
Trigonometry unveils a
wide range of concepts and levels on which to expand your knowledge.
Examine and explore the mathematical relationships between sides and angles in
our Trigonometric Ratios topic.
Discover how Sine and Cosine are
also used as Trigonometric Equations when
an angle measurement is a variable and master the intricacies of
applications and methods.
As an interdisciplinary
study Trigonometry incorporates many aspects in the field of mathematics such
as our topic Unit Circle which is rooted in Geometry. Geometry is a powerful
application that provides a visual tool that helps you better understand what
the trigonometric ratios represent. Mastering the Trigonometric Identities and Angle Identities is
useful for simplifying expressions and equations, but the real power of
Trigonometry lies in Trigonometric
Applications.
Learn how to apply this knowledge
in our topic of Inverse Trigonometric
Functions, where you will solve for distances and height. From Basic
Identities to Advanced Triangle Problems, you will become proficient in the
applications, methods and equations presented in our Trigonometry course and
elevate your mastery of skills within the subject of Mathematics. | 677.169 | 1 |
Boost your students understanding of Saxon Math with DIVE's easy-to-understand lectures! Each lesson in Saxon Math's textbook is taught step-by-step on a digital whiteboard, averaging about 10-15 minutes in length; and because each lesson is stored separately, you can easily move about from lesson-to-lesson as well as maneuver within the lesson you're watching. After the lesson, students complete the 30-question Problem Set in the Saxon text; a few problems in the set come from the new concept, however most review previous lessons. Each problem is cross-referenced back to the lesson where it was taught, allowing students to quickly review any concept.Taught from a Christian worldview, Dr. David Shormann also provides a weekly syllabus to help students stay on track with the lessons. DIVE teaches the same concepts as Saxon, but does not use the problems given in the text; it cannot be used as a solutions guide.
Algebra 2 covers traditional second year algebra topics as well as a semester of geometry, real world problems, linear and nonlinear equations, statistics and probability, graphing and basic trigonometry. This DIVE CD can be used with Saxon Algebra 2's 2nd and 3rd Editions; the CLEP Professor College Algebra CLEP Exam prep course is also included.
System Requirements:
Mac OS 10.3.9-10.4
Windows 10,8,7,98, 2000, ME, XP, Vista, 8, 10
Quicktime Download Required
IMPORTANT! PLEASE NOTE: This CD plays in VLC instead of Flash Player. Please double-click the Open Me First folder and follow the installation instructions to have your CD play correctly. | 677.169 | 1 |
This unit is the second of two units that provide the essential foundation in the concepts and techniques of mathematics that form the basis of science, engineering and higher mathematics and statistics. It covers multidimensional integration, Laplace transforms and Fourier series, and statistics.
Outcomes
Students are able to (1) understand and employ integration theory and technique; (2) integrate over objects in more than one dimension; (3) understand the concepts of work, path independence, conservative fields and potentials; (4) understand the basic theory of complex functions; (5) understand and employ Fourier series and Laplace transforms; and (6) understand the basics of probability theory.
Assessment
Typically this unit is assessed in the following ways: (1) continuing assessment and (2) a final examination | 677.169 | 1 |
The flaw in many online mathematics programs is that they may solve the problem, but inhibit further learning and understanding of mathematical concepts. This is not the case for Wolfram|Alpha. In this tutorial, I'll show you how to make the most out of Wolfram|Alpha's mathematical knowledge so that you may garner a better understanding of math.
Note: Some of the features showcased in this tutorial require a subscription to Wolfram|Alpha's Pro service, which costs $6.99/month for regular users and $4.99/month for students.
Working Out a Solution
Type the math problem you want to solve into the search bar, and press Enter to submit the problem. Wolfram|Alpha will solve it in a matter of seconds, spitting out the answer at the top of the page.
Wolfram|Alpha can solve any math problem.
Tip: Pro users can upload a photo or file of a math problem directly to Wolfram|Alpha. Wolfram|Alpha will solve the problem it detects in the file, thus saving time by eliminating the need to type in the problem.
But there's only so much getting the answer to the problem can do for you. This is where Wolfram|Alpha takes problem solving to the next level.
Pro subscribers are able to view step-by-step solutions to nearly any problem by pressing the Step-by-step solution button in the top-right corner of the solution field. Step-by-step solutions walk through each part of the solution process, complete with annotations about specific rules that were used to come to the solution. This allows you to gain a better understanding of how to approach similar problems.
Step-by-step solutions guide you through the solution process.
Tip: Save the walkthroughs by clicking the Data download button in the Step-by-step solution window so it may be easily accessed later.
Learning Mathematical Concepts
Perhaps Wolfram|Alpha used a specific rule in solving the problem that you are unfamiliar with. To get more information about this rule, simply search it in Wolfram|Alpha. The search will pull general information about the rule in order to explain it.
Wolfram|Alpha searches may also be conducted on broader mathematical concepts. To do so, search the concept you want to learn more about. You may need to add the term mathematical definition to find the proper definition of the concept you are searching.
For broader mathematical concepts, Wolfram|Alpha typically provides both a basic and detailed definition of the concept, as well as related topics. In the bottom corner of the definition field, there may be a More information button. Clicking this button redirects you to the Wolfram MathWorld page for the respective concept. Wolfram MathWorld is Wolfram's mathematics resource and provides very detailed explanations of mathematical concepts and rules, usually complete with proofs and visuals to better explain the concept.
Tip: Even if the More information button is unavailable from the Wolfram|Alpha page, it may still be searched in MathWorld for a more detailed explanation.
Visualizing Problems
Especially as you dabble into more advanced mathematics, visualizing a problem may be the only way to truly understand what is happening. Luckily, Wolfram|Alpha has powerful visualization tools to aid in understanding a problem.
Every user is able to view plots and 3D models of functions. To do so, simply search plot [function] or [3D shape] image, and the proper visual will be generated.
Plotting a function in Wolfram|Alpha.
Pro users can make use of Wolfram|Alpha's CDF interactivity, which allows for a more hands-on approach to interpreting data. All of the plots and images that nonsubscribers can see can be made interactive by clicking the Enable interactivity button in the bottom right corner of the Image or Plots section. Interactivity on these graphs allow you to manipulate the functions as you please.
3D shapes may be manipulated using CDF interactivity.
Wolfram|Alpha also has animated plots for problems that involve time and motion, which are especially helpful when trying to wrap your head around a tough physics problem or parametric function.
An interactive rendering of a spring pendulum.
Extra Practice
Pro subscribers to Wolfram|Alpha have access to Wolfram Problem Generator, which churns out practice problems in six different mathematical areas: arithmetic, algebra, calculus, statistics, number theory, and linear algebra.
Within each of the subject areas, there is a selection of topics which may be practiced. While there isn't yet practice problems for quite a few concepts (such as series practice for Calculus), most of the basic concepts in each of the subject areas are covered.
While many mathematical topics are covered in the problem generator, there are a few that are missing.
After choosing the topic you want to practice, select the difficulty of the problem in the Difficulty drop-down menu. There are the options of Beginner, Intermediate, and Advanced, but even the beginner questions are relatively difficult upon entering more advanced mathematics.
The Problem Generator affords three chances to answer the question correctly. Upon entering an incorrect answer, you may view a hint. After the second incorrect answer, you may view the Step-by-Step solution for the problem. You can skip problems by pressing the Skip this question button at the bottom right hand corner of the screen.
An example of a beginner level problem from Wolfram Problem Generator.
Tip: Monitor your progress by checking the History section at the bottom of the page. This will monitor how many questions you've been getting right or wrong.
You can also create a worksheet of problems by clicking the Random set or From history button, found beneath Generate a printable problem sheet under the problem generator. Clicking Random set will create a 20-question multiple choice worksheet complete with an answer key. The From history button will create a worksheet of problems you've previously attempted, complete with an answer sheet as well. Both options allow you to download the worksheet as a PDF to be edited or printed.
Conclusion
In this tutorial, I've shown how to use Wolfram|Alpha's mathematics resources in order to enrich your understanding of mathematics. From step-by-step solutions to endless math resources to a problem generator, you are sure to leave Wolfram|Alpha with a better understanding of math than you had before you used the program.
Leave a comment down below if you have any questions or tips about mathematics in Wolfram|Alpha! | 677.169 | 1 |
Mathematics
Originally used by software companies, the Scrum project management model is a way of thinking designed to increase team and individual productivity. Kingston describes his method and results after adapting the model for his middle school math class.
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This 2012 NAD Teachers Convention presentation explores the use of Geogebra, a free internet tool, in the math classroom. Examples from algebra through calculus will be shown, as well as an introduction to the web site with many lesson ideas that have been...
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You'll gain access to interventions, extensions, task implementation guides, and more for this lesson plan.
Lesson objective: Relate behaviors of a function to a real-world situation.
This lesson provides an opportunity for students to apply their knowledge and understanding of behaviors of a function to a real-life situation. Students are asked to describe the pay rate of a lawn-mowing job, represent a second pay option, and determine which pay plan is better.
Key Concept students will use:
Graphing input-output pairs helps us describe and model the behavior of a function qualitatively.
Skills students will use:
Understand behavior of a function given as a graph of the function
Model the behavior of a function given a description of the function
Students engage in Mathematical Practice 2 (reason abstractly and quantitatively) and Mathematical Practice 4 (model with mathematics) as they examine input-output pairs to understand, describe, and model the behavior of a function. | 677.169 | 1 |
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Software
Applications
Activity Overview
The students will follow instructions on slope and linear functions, with playing match the graph on the TI-Navigator™. Several digital pictures are provided, or you could find your own!
Before the Activity
Before the activity, some instruction in linear functions (and absolute value functions) should be done. A short StudyCards™ stack is provided to assistance with this. Other handouts and activities are also attached.
During the Activity
Now, your students are ready to play Match the Graph through the TI-Navigator. Please feel free to use the pictures that I have provided, or find your own on the internet or around your school building. Have students take digital photos if you have access to Palm Zire 72 or other digital cameras. (Or students may use their cell phones if allowed!!!)
After the Activity
Have fun! If you just completed the linear part of the activity. Continue with the absolute values. You can teach this using vertex form as you would with parabolas. | 677.169 | 1 |
MATLAB in Physics is a four lecture series in MATLAB that is offered to first year physics undergraduate students. This lecture shows how to use MATLAB to simulate physical systems using a range of methods. Projectile motion is analysed using the Symbolic Math Toolbox, then more complicated chaotic systems are simulated using the MATLAB ODE solvers.Some advanced programming techniques such as Object Oriented Programming (OOP) and Graphical | 677.169 | 1 |
Springer Undergraduate Mathematics Series. 2005. Applied Geometry for Computer Graphics and CAD. Applied Geometry for Computer Graphics and CAD.Applied Geometry for Computer Graphics and CAD, by D. Marsh. Published by Springer-Verlag.Intended for the undergraduate student majoring in mathematics,.Other books in this series A First Course in Discrete Mathematics.
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Applied Geometry for Computer Graphics and CAD explores the application of geometry to computer graphics and computer.Computer graphics are pictures and films created using computers.Applied Geometry for Computer Graphics and CAD D. Marsh. Elementary Differential Geometry A. Pressley.Related Book PDF Book Applied Geometry For Computer Graphics And Cad Springer Undergraduate Mathematics Series: - Home - Personal Bankruptcy Problem Or Solution.
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Kamis, 18 Agustus 2016
[Where To Download] The Mathematics Calendar 2016
_Read Online > The Mathematics Calendar 2016
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The Mathematics Calendar continually reminds us how mathematics describes nature, impacts the sciences, influences the arts, is inseparable from music, stimulates and creates new technologies, and reveals the dimensions of our universe through its ever-evolving ideas and insights.
Thirty-six years after its inception, The Mathematics Calendar continues to awe its audience with its cutting edge math topics, its all new exciting graphics, and its mind-boggling problems. The 2016 edition includes 12 new fascinating math topics illustrating the incredible influence of mathematics on our lives.
Each day of every month has a problem, whose solution is the date. The challenge lies in figuring how to arrive at the answer, and possibly discovering more than one method of solving the date's problem.
Each month's text, photos, and graphics have a wealth of information and are even sprinkled with a bit of humor. The 12 topics feature exciting, historic, and current math ideas and topics.
Wherever it is seen, it's a conversation item because it is more than a calendar — it has tantalizing problems on each day and a treasure trove of information, games, puzzles, and stories. The Mathematics Calendar has given hundreds of thousands of people a new perspective about math approach to life by perusing this
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This resource has 7 Powerpoints on steps in solving a system of linear equations, solving inequalities, and a review of the 3 types of linear equations. It's a great addition to your Algebra curriculum, especially for use with lower ability learners.
Powerpoints include graphing, substitution, and elimination types of systems. Many of the PP's have assessment questions built in, along with answer keys. Students will also learn to solve and graph linear inequalities. | 677.169 | 1 |
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Algebraic Geometry
The course will be divided in two parts, one on sheaves, derived functors, sheaf cohomology and spectral sequences; and a second, which will essentially be a first introduction to the theory of schemes. | 677.169 | 1 |
Glencoe Mathematics for Business and Personal Finance, Student Edition
Mathematics for Business and Personal Finance teaches students mathematics, in the context of business and personal finance like budgeting and money management, banking and credit, and saving and investing. This program provides valuable information on how to use math in everyday business and personal finance situations to fully understand how to manage one's financial resources effectively for lifetime financial security. | 677.169 | 1 |
This unit offers qualitative and quantitative methods for ordinary and partial differential equations. This unit offers such topics as first order equations; second order linear equations; series solutions; boundary value problems, and phase plane analysisexercise advanced skills in solving differential equations, including first order and second order ordinary differential equations, and some special cases of partial differential equations;
use simple differential equations to see how practical problems can be modelled and solved in mathematics;
synthesise knowledge from calculus and linear algebra through their use in solving standard differential equations and systems; and
understand some of the mathematical theory behind numerical simulation techniques | 677.169 | 1 |
Mathematics 3 this term is a lecture course in introductory calculus that we have revised to make
leaner, livelier, and case studies oriented. The course is taken by mostly first-year students, is taught in two
sections in a lecture format, and meets three times per week.
Our Case Studies in Calculus (CSCs) will involve modeling aspects of real-world
phenomena with calculus. The CSCs range from modeling the AIDs data to analyzing issues of river flooding. In these projects, we will behave as mathematicians attempting to answer questions
posed by epidemiologists or earth scientists, for example. The purpose of the CSCs is to illustrate the
power of calculus as a tool in real-world problem-solving. Students may work either individually or as part of a
group, though the write-ups must be in each individual's own words.
We will be turning to the on-line components of the textbook to make the ideas and applications of calculus come alive. Together with the concepts and techniques of calculus, the computer is an indispensable tool in the analysis process. By using the World Wide Web to access calculus tools that we have developed for our course, students will learn the formidable capabilities of a computer system in doing calculus. We will even be using the computer as a problem-generator to give students lots of practice and immediate feedback on calculus techniques and problem-solving. And we will have examples, quizzes, and tutorial videos on line as well.
This term, we will be using calculus to develop mathematical models in case studies, and we will be learning the conceptual, analytical, and manipulative techniques that make that possible. Calculus is truly a marvelous intellectual achievement, and we plan to have a lot of fun studying it. | 677.169 | 1 |
...leads its students to: * nourish a deep faith in God * relentlessly search for truth and justice * foster an on-going urge to reflect and act in accordance... Learn about: Mathematics Algebra...
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...Actuarial science is the discipline that applies mathematical and statistical methods to assess risk in insurance, finance and other industries and professions. Actuaries are professionals who are qualified in this field through intense education and experience. In many countries, actuaries... Learn about: Insurance Finance, Finance Economics, Actuarial Science...
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...Mathematics is the study of topics such as quantity (numbers),structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. Mathematicians seek out patterns[9][10] and use them to formulate new conjectures... Learn about: Mathematics Engineering, IT Engineering, Mathematics Teacher...
More | 677.169 | 1 |
College Algebra
The scope of this learning experience is to study basic mathematical concepts required for adult learners at a college or university level. Learners will become proficient at solving linear and quadratic equations, graphing linear and quadratic equations, functions, exponents and logarithms, and applying these concepts to real world problems.
Idizzla 19:50, 27 July 2007 (UTC) If you wish to participate with this course, please add your name below after registering an account on Wikiversity with your account signature by editing this page and adding *~~~~
Mirwin 09:39, 18 August 2006 (UTC) I can help out with some mentoring or tutoring where materials are difficult to understand and possibly help fill gaps in available materials. I have a B.S. in Engineering Physics so I have done quite a bit of practical algebra even if it is a bit rusty in spots. Participants drop queries or requests for assistance on my talk page and we will create some appropriate space from there to work from.
--MarkyParky 22:58, 25 August 2006 (UTC) I can help out when people are stuck on some problem, or just need something explained from a different perspective.
The following books contain motivational works by mathematicians that put mathematics into context or exercises that illuminate the study of mathematics that are accessible to someone with a high school education. While this site provides a supportive community of peers and teachers, nothing beats having a well-organized and well-written text that you can carry around and study anywhere to learn from. | 677.169 | 1 |
A basic algebra course designed to prepare students for MTH 160 (Statistics I) or MTH 162 (Statistics for the Social Sciences) at MCC. It is only for students who need MTH 160, MTH 161 or MTH 162 as the terminal mathematics course in their program and place at Mathematics Level 4, 5, or 6. It is not for students who need to take MTH 104 or MTH 165 (or higher). Topics include, but are not limited to, arithmetic operations on real numbers, solving linear equations and inequalities, simplifying expressions containing integer exponents, simplifying square roots and finding decimal approximations, graphing linear equations, an introduction to function notation, and appropriate applications of these topics. Three class hours per week; three fee hours; three imputed credits; no earned credits.
MCC level 4 mathematics placement or higher or TRS 094 with a grade of C or better.
A first course in algebra with an emphasis on linear equations and their applications. Topics include, but are not limited to: solving linear equations and inequalities, performing arithmetic operations on polynomials, factoring polynomials, solving quadratic equations, simplifying expressions containing integer exponents, graphing linear equations and inequalities in one and two variables, solving systems of two linear equations, and appropriate applications of these topics. Four class hours per week; four fee hours; four imputed credits; no earned credits.
Prerequisite: TRS 094 with a grade of C or better, or MCC Level 4 Mathematics Placement.
*MTH 098, MTH 099, and MTH 104 are developmental courses. They do not fulfill a mathematics requirement for an Associate in Arts or Associate in Science Degree.
An accelerated version of MTH 098, intended for students who need a review of basic algebra content. Topics to be covered include, but are not limited to, reviewing arithmetic operations on rational numbers, solving linear equations, graphing on the Cartesian Coordinate system and factoring polynomials. MTH 099 is typically taken with specially designated sections of MTH 104. Two laboratory hours per week; one fee hour; one imputed credit; no earned credits.
Prerequisite(s): MCC level 5 mathematics placement, TRS 094 with a grade of A, MTH 096 with a grade of B-, MTH 098 with a grade of C-, or permission of instructor.
*MTH 098, MTH 099, and MTH 104 are developmental courses. They do not fulfill a mathematics requirement for an Associate in Arts or Associate in Science degree.
A second course in algebra with an emphasis on quadratic and rational equations and their applications, and a brief introduction to right triangle trigonometry. Topics include, but are not limited to: solving equations involving quadratic, rational and radical expressions; performing arithmetic operations on rational expressions, complex numbers and radical expressions; evaluating functions and determining domain and range; graphing quadratic functions; solving systems of non-linear equations; simplifying expressions involving rational exponents; solving right triangle trigonometric problems; and appropriate applications of each of these topics. Four class hours.
Prerequisite: MTH 098 with a grade of C or better, or MTH 099 with a grade of C or better, or MCC Level 6 Mathematics Placement.
*MTH 098, MTH 099, and MTH 104 are developmental courses. They do not fulfill a mathematics requirement for an Associate in Arts or Associate in Science degree.
This course will cover the basic concepts and processes of mathematics applied to various business situations including statistical procedures, percentage and percent distributions of financial statement data, merchandising, payrolls, taxation and insurance. Other topics include simple interest, compound interest and annuities. Three class hours. MTH 130 is a course for career business. It does not fulfill a mathematics requirement for most Associate in Arts or Associate in Science degrees.
Prerequisite: TRS 092 with a grade of C or better, or MCC Level 2 Mathematics placement.
An introductory course dealing with the development of algebraic and trigonometric concepts needed to solve problems in various technical areas. Topics include measurement and approximation, ratio and proportion, dimensional analysis, intermediate algebra, geometry, and right triangle trigonometry. Four class hours. NOTE: A specific calculator will be required of all students in this course.
Prerequisite: MTH 098 with a grade of C or better, or MCC Level 6 Mathematics Placement Perform reductions or conversions to appropriate units of measurement in the measurement systems. 2. Perform calculations with approximate numbers. 3. Solve a variety of equations which may include literal, linear, or quadratic. 4. Perform arithmetic operations on expressions that may include exponents, scientific notation, or roots. 5. Factor polynomials which may be either binomial or trinomial. 6. Perform arithmetic operations with rational expressions. 7. Graph various equations which may include linear or quadratic. 8. Calculate measures of geometric figures which may include perimeter, area, or volume. 9. Solve applications involving right triangles.
A course dealing with the algebraic and trigonometric concepts needed to solve problems in various technical areas. It includes a study of linear and trigonometric equations, dimensional analysis, ratios and proportion, functions and their graphs, right triangle trigonometry, graphs of trigonometric functions, vectors, and statistical topics. Three class hours. NOTE: A specific calculator will be required of all students in this course. (SUNY-M)
Prerequisite: MTH 135 with a grade of C or better or MTH 104 with a grade of C or better, or MCC Level 8 Mathematics placement Evaluate a function, expressed in function notation, for a specified input value. 2. Graph functions using a table of values. 3. Find the domain or range of functions. 4. Find the inverse function of a one-to-one function. 5. Convert an angle measurement between radians and degrees. 6. Evaluate trigonometric functions for any angle. 7. Solve vector problems using components. 8. Determine the features of trigonometric functions which may include amplitude, period, or displacement. 9. Sketch the graphs of trigonometric functions which may include sine, cosine, or tangent. 10. Prove identities involving the trigonometric functions. 11. Solve equations involving trigonometric or inverse trigonometric functions. 12. Determine statistical measures such as the mean, median, mode, or standard deviation.
An extension of the concepts developed in MTH 140. Topics included are complex numbers, higher degree equations, oblique triangle trigonometry, exponential equations, logarithms, systems of linear and quadratic equations, and inequalities. Three class hours. NOTE: A specific calculator will be required of all students in this course. (SUNY-M)
Prerequisite: MTH 140 with a grade of C or better or equivalent.
*MTH 135, MTH 140 and/or MTH 141 are required in various technology programs. They do not fulfill a mathematics requirement for an Associate in Arts or Asscociate in Science degree.
Course Learning Outcomes
1. Perform arithmetic operations on complex numbers. 2. Solve equations in quadratic form. 3. Solve systems of two linear equations in two variables algebraically, graphically, or by using Cramer's Rule. 4. Solve systems of three linear equations in two variables by using Cramer's Rule. 5. Find zeros of polynomial functions. 6. Use synthetic division or other methods such as factoring to solve higher degree polynomial equations. 7. Solve exponential or logarithmic equations. 8. Solve systems of two linear or quadratic inequalities in two variables graphically. 9. Apply the Law of Sines or the Law of Cosines to practical problems. 10. Generate graphs of functions which may include quadratic, exponential, or logarithmic.
A study of various topics including an introduction to estimation, algebra, geometry, consumer mathematics, probability and statistics, with an emphasis on critical thinking and interpreting results. Other topics may be covered at the discretion of the instructor. Three class hours. MTH 150 is a common selection by Liberal Arts students with fewer than three years of high school mathematics. MTH 150 is not a prerequisite course for MTH 160 or higher. Although this course can satisfy your mathematics requirement for some MCC programs and transfer to some baccalaureate institutions, if you are planning to transfer please speak with an academic advisor or Career and Transfer to ensure that this course meets your goals. (SUNY-M)
Prerequisite: TRS 094 with a grade of C or better, or MCC Level 4 Mathematics placement.
Course Learning Outcomes
1. Use estimation to approximate an answer. 2. Use geometric formulas to solve applied problems using unit conversions where necessary. 3. Compute interest for common loans. 4. Compute interest rates for common loans. 5. Compute payments for common loans. 6. Compute probabilities, odds, or expected value. 7. Interpret results from probability based calculations. 8. Interpret standard statistical graphs. 9. Calculate simple descriptive statistics for a given set of data. 10. Use common properties of a normal distribution to draw basic conclusions about the underlying data
A course essential in developing the mathematical competency of the teacher or prospective teacher at the elementary level. Students will develop a comprehensive understanding of the mathematical curriculum recommended by the NCTM (National Council of Teachers of Mathematics) Standards, using a problem solving approach. Topics include historical development of numbers and number systems, study of whole numbers, integers, rationals, irrationals, and reals; abstract number systems; and elementary number theory. NOTE: MTH 155 is not a teaching methods course. Three class hours. All Sections are writing intensive (WR).
Prerequisite: MTH 098 with a grade of C or better, MTH 099 with a grade of C or better, or MCC Level 6 Mathematics Placement.
Course Learning Outcomes
1. Apply a variety of appropriate strategies to solve problems. 2. Create and use a variety of representations of sets to model physical, social and mathematical phenomena. 3. Illustrate the traditional algorithms for Whole Number, Integer, and Rational Number arithmetic using visual representations. 4. Identify and use the properties of Real Numbers. 5. Employ principles of number theory as a problem solving strategy. 6. Demonstrate the use of physical representations to illustrate numerals in decimal and non-decimal based place value systems. 7. Convert numerals between various positional and additive systems. 8. Incorporate estimation techniques to assess reasonableness of solutions.
A continuation of the concepts of MTH 155, which develop the mathematical competency of the teacher or prospective teacher at the elementary level. Students will develop a comprehensive understanding of the mathematical curriculum recommended by the National Council of Teachers of Mathematics (NCTM) Standards using a problem solving approach with appropriate technology. Topics include functions, probability, statistics, measurement, 2 and 3 dimensional geometry, transformational geometry, congruence and similarity. All sections are writing intensive (WR). Three class hours. MTH 156 is a special interest course; check for availability. (SUNY-M)
Prerequisite: MTH 155 with a grade of C or better.
Course Learning Outcomes
1. Compute various Descriptive Statistics (Measures of Central Tendency, Measures of Spread, Measures of Position). 2. Construct, interpret, and analyze the meaning of a variety of graphs. 3. Predict the probability of outcomes of simple and two-stage experiments or events. 4. Discover and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life. 5. Draw and use visual tools (e.g., Networks, Euler's rules of traversability, Jordan Curve Theorem, Map Coloring problem) to problem solve. 6. Classify geometric shapes and distinguish between attributes of shapes. 7. Compute Perimeter, Area, and/or Volume for a variety of geometric shapes, and validate reasonableness. 8. Discuss and justify problem solving strategies within a group setting. 9. Use principles of congruence, similarity, and proportional reasoning to model and interpret physical and mathematical situations in application problems.
An introduction to descriptive and inferential statistics intended to give an understanding of statistical techniques and applications in a wide variety of disciplines. Topics include measures of central tendency; dispersion and position; correlation and regression; probability and probability distributions, including binomial and normal; the Central Limit Theorem; parameter estimation and hypothesis testing. Minitab statistical software is used. Three class hours. MTH 160 is an appropriate elective for most programs. (SUNY-M)
MTH 096 (formerly MTH 080) with a grade of B- or better; or any of the following with a grade of C or better: MTH 104, MTH 140, MTH 141, MTH 165 (or higher); or MCC level 8 mathematics placement.
Course Learning Outcomes
1. Determine a variety of descriptive statistics which may include any of the following measures: central tendency, dispersion, or position. 2. Analyze data using descriptive measures. 3. Produce graphs of data which may include any of the following: histograms, boxplots, or scatterplots. 4. Analyze statistical graphs. 5. Calculate quantitative values for various random variables which may include any of the following types: discrete, binomial, or normal. 6. Interpret probabilities for various probability distributions which may include any of the following types: discrete, binomial, or normal. 7. Use the Central Limit Theorem for applications involving the sampling distribution of the sample mean. 8. Determine confidence intervals for the population mean or the population proportion. 9. Analyze confidence intervals for the population mean or the population proportion. 10. Perform hypothesis tests for the population mean or the population proportion. 11. Determine the linear correlation coefficient for bivariate data. 12. Determine the equation of the least-squares line for bivariate data. 13. Analyze bivariate data. 14. Interpret Minitab output.
An introduction to descriptive and inferential statistics intended to give an understanding of statistical techniques and applications used in the social sciences. Topics include: measures of central tendency, dispersion and position; correlation and regression; probability and probability distributions, including binomial and normal; parameter estimation and hypothesis testing; two-sample analysis; chi-square test of independence; one-way analysis of variance. Statistical software will be used. This course is intended for, but not limited to Social Science majors. Students who have completed MTH 160 or MTH 161 may not receive additional credit for this course. Four class hours.(SUNY-M)
Course Learning Outcomes
1. Interpret data graphically using measures of central tendency, dispersion, and position 2. Interpret data numerically using measures of central tendency, dispersion, and position 3. Create a scatter diagram to represent bivariate data 4. Analyze the relationship between variables using linear correlation and linear regression 5. Perform significance tests for linear correlation 6. Classify a random variable as binomial 7. Use the binomial probability distribution to compute probabilities, means, and standard deviations 8. Explain the properties of the normal probability distribution and its parameters 9. Compute probabilities for normal variables as areas, probabilities, or proportions 10. Use the Central Limit Theorem to describe the sampling distribution of the sample mean 11. Produce confidence intervals using both the z and t probability distributions 12. Interpret confidence intervals using both the z and t probability distributions 13. Generate hypothesis tests about mu and p, using both the z and t probability distributions 14. Deduce conclusions about mu and p, using both the z and t probability distributions 15. Perform a two-sample analysis using independent and dependent samples 16. Interpret the results of two-sample analyses using independent and dependent samples 17. Calculate effect size for different samples 18. Interpret the results of effect size for different samples 19. Complete chi-square tests of independence 20. Interpret the results of a chi-square test for independence 21. Execute a one-way analysis of variance 22. Interpret the results of a one-way ANOVA as the effect of a factor on the response variable 23. Use statistical software to produce statistical graphs 24. Use statistical software to compute statistical measures 25. Use statistical software to create estimates of statistical measures 26. Test hypotheses using statistical software to generate necessary values 27. Interpret results obtained using statistical software
A first course in trigonometry. Topics include the trigonometric ratios, radian measure, angles in a coordinate system, ratio values for any angle, graphs of trigonometric functions and basic trigonometric identities and equations. A specific calculator will be required of all students in this course. One class hour.
Prerequisite: MTH 104 with a grade of C or better, or MCC Level 8 Mathematics Placement.
Course Learning Outcomes
1. Draw an angle in standard position on coordinate axes when given the angle measure in radians. 2. Define (in terms of x, y, and r) any of the six trigonometric ratios for an angle in standard position whose terminal side passes through the point (x, y) on a circle of radius r. 3. Find the exact values of any of the six trigonometric ratios for quadrantal angles or special angles. 4. Solve first-degree trigonometric equations for exact values of the variable or for approximate values of the variable. 5. Establish new trigonometric identities by applying algebraic techniques and identities such as reciprocal, quotient or Pythagorean. 6. Analyze characteristics of trigonometric functions. 7. Perform angle measurement conversions, such as decimal degrees; degrees, minutes, seconds; or radians.
This course is intended to enhance algebraic skills and graphing techniques, and to prepare students for Precalculus Mathematics and Applied Calculus. Topics include functions that are quadratic, polynomial, piecewise, exponential and logarithmic; equations that are quadratic in form, involve radicals, absolute value, variation and rational exponents; inequalities that are quadratic, rational, and absolute value. Three class hours. (SUNY-M). MTH 165 is an appropriate elective even if not pursuing science or mathematics.
Prerequisite: MTH 104 with a grade of C or better, or MTH 140 with a grade of C or better, or MCC Level 8 Mathematics placement.
Course Learning Outcomes
1. Solve a variety of equations which may include any of the following types: quadratic in form, radical, rational exponent, linear absolute value, exponential, or logarithmic. 2. Solve a variety of inequalities which may include any of the following types: linear absolute value, factorable polynomial, or rational. 3. Graph a variety of relations which may include any of the following types: polynomial of degree at most 4, absolute value, square root, cube root, reciprocal, exponential, logarithmic, piecewise, or circle. 4. Describe or graph transformations of functions which may involve any combination of a horizontal shift, a vertical shift, or a reflection. 5. Convert between any of the three forms: a graph, an equation, or a verbal description of attributes such as zeros, intercepts, or domain. 6. Determine a composition or combination of functions. 7. Determine the inverse of a one-to-one function. 8. Determine the zeros of a polynomial function using synthetic division with related theorems. 9. Use long division to divide polynomials. 10. Rewrite a logarithmic expression using properties of logarithms. 11. Solve a variety of applications using variation, exponential, or logarithmic models.
An introduction to discrete mathematics primarily intended for students majoring in Information Technology or Computer Systems Technology. The emphasis will be on the development of technical discrete mathematics skills, rather than rigorous proof. Topics will include number systems, sets, logic, induction, elementary counting techniques, relations, functions, matrices, and Boolean algebra. Note: This course is not designed for students intending to major in Mathematics or Computer Science. Students intending to major in Mathematics or Computer Science should take MTH 220. Three class hours.
Prerequisite: MTH 141 or MTH 165 with a grade of C or better, or MCC Level 9 Mathematics placement, or equivalent.
Course Learning Outcomes
1. Convert between different number systems 2. Apply standard set operations. 3. Assess the validity of an argument with truth tables or rules of inference. 4. Evaluate finite sums given in sigma notation. 5. Solve counting problems with permutations or combinations. 6. Conclude whether a given relation is an equivalence relation. 7. Classify a given function as one-to-one or onto. 8. Compare the growth rates of functions with Big-O notation. 9. Apply standard matrix operations. 10. Apply the axioms of a Boolean algebra.
A study of the properties and graphs of functions, including polynomial, absolute value, power, piecewise, radical, rational, exponential, logarithmic, trigonometric, and inverse trigonometric. Topics also include a study of analytic trigonometry and an introduction to vectors. This course is intended to prepare students for the study of calculus. Four class hours. (SUNY-M)
Prerequisite: MTH 165 with a grade of C or higher, or MTH 141 with a grade of C or higher, or MCC Level 9 Mathematics placement or higher.
Course Learning Outcomes
1. Rewrite a logarithmic expression using properties of logarithms. 2. Perform operations on functions, which may include compositions or arithmetic combinations. 3. Determine simplified difference quotients. 4. Determine the inverse of a one-to-one function. 5. Construct graphs of equations, functions, or their transformations. 6. Identify appropriate features of the graph of an equation such as intercepts, asymptotes, maximum values, minimum values, domain, range, or symmetry. 7. Determine the values (in exact form when possible) of the trigonometric functions of any angle. 8. Verify a trigonometric identity. 9. Solve a trigonometric equation. 10. Evaluate expressions involving inverse trigonometric functions. 11. Use the Law of Sines and/or Law of Cosines to solve an oblique triangle. 12. Construct the graph of y = a f (bx + c) + d where f is a trigonometric function. 13. Identify appropriate features of the graph of a trigonometric function, such as amplitude, period, phase shift, vertical shift, domain, or range. 14. Perform operations on vectors.
An intuitive introduction to the principal ideas of differential and integral calculus. Among the topics covered are: functions (including exponential and logarithmic), limits, differentiation, and integration. Emphasis will be placed upon the use of calculus in solving problems from areas including business, economics, and the social and natural sciences. Four class hours. (SUNY-M)
Course Learning Outcomes
1. Evaluate limits using analytical methods. 2. Evaluate limits using numerical methods. 3. Evaluate limits graphically. 4. Apply the definition of continuity at a point to determine continuity. 5. Examine the relationship between the limit definition of a derivative and the average rate of change of a function. 6. Apply the rules of differentiation to obtain derivatives of functions. 7. Sketch a function by interpreting the derivatives of the function. 8. Find the local or relative extreme values of a function by interpreting the derivatives of a function. 9. Evaluate antiderivatives using analytical methods. 10. Evaluate definite integrals. 11. Examine the relationship between the area under a curve and the definite integral. 12. Evaluate partial derivatives of functions of several variables. 13. Evaluate second order partial derivatives of functions of several variables. 14. Solve application problems by applying calculus techniques.
A first course in single variable calculus. Topics include limits and continuity; differentiation of algebraic, trigonometric, exponential, logarithmic, and inverse trigonometric functions; applications of derivatives to curve sketching, related rates, optimization, and rectilinear motion; indefinite and definite integrals; both parts of the Fundamental Theorem of Calculus; applications of definite integrals to area, average value, and rectilinear motion; and integration by u-substitution. A specific calculator may be required of students in this course. Students are advised to check with the instructor. Four class hours. (SUNY-M) 4 Credits.
Prerequisite: MTH 175 with grade of C or higher, or high school precalculus course taken within the last three years with a grade of B (83) or higher, or MCC Level 10 Mathematics placement.
Course Learning Outcomes
1. Determine a variety of limits which may include any of the following types: one-sided limits, two-sided limits, infinite limits, or limits at infinity. 2. Determine whether a function is continuous at a given point. 3. Differentiate a variety of functions using the limit definition of the derivative. 4. Determine whether a function is differentiable at a given point. 5. Differentiate combinations or compositions of a variety of functions which may include any of the following: polynomial, radical, rational, trigonometric, inverse trigonometric, exponential, or logarithmic. 6. Differentiate implicitly defined functions. 7. Use derivatives to determine key features of graphs of functions which may include any of the following: intervals of increase or decrease, intervals of concavity, extrema, or points of inflection. 8. Determine indefinite integrals. 9. Approximate definite integrals with Riemann Sums. 10. Apply the Fundamental Theorem of Calculus. 11. Use the method of integration by u-substitution. 12. Use methods of Calculus to solve a variety of applications which may include any of the following types: related rates, optimization, rectilinear motion, area, or average value.
A second course in single variable calculus. Topics include techniques of integration, applications of definite integrals, first order separable differential equations, indeterminate limit forms, improper integrals, infinite series, Taylor polynomials, and power series. A specific calculator may be required of students in this course. Four class hours. (SUNY-M)
Prerequisite: MTH 210 with a grade of C or higher.
Course Learning Outcomes
1. Compute indefinite integrals using a variety of techniques which could include: substitution, integration by parts, trigonometric substitution, or partial fraction decomposition. 2. Formulate a definite integral to solve a variety of applications. 3. Evaluate a definite integral using the Fundamental Theorem of Calculus. 4. Estimate the value of a definite integral using one or more of the following techniques: the Midpoint Rule, the Trapezoid Rule, and Simpson's Rule. 5. Estimate the value of a definite integral using a Taylor Polynomial. 6. Use L'Hopital's Rule to evaluate a limit. 7. Evaluate an improper integral using the appropriate limit definition. 8. Translate verbal descriptions involving rates of change into differential equations. 9. Solve a separable differential equation using separation of variables. 10. Justify whether a given series converges or diverges. 11. Determine the interval and radius of convergence for a given power series.
An introduction to discrete mathematics primarily intended for students majoring in Mathematics or Computer Science. Topics will include propositional and predicate logic, elementary number theory, mathematical induction, set theory, combinatorics, functions, and relations. Methods of proof will be developed in a variety of mathematical contexts. Three class hours.
Prerequisite: MTH 210 with a grade of C or higher, or equivalent.
Course Learning Outcomes
1. Use and interpret mathematically correct terminology and notation. 2. Apply logical reasoning to solve a variety of problems. 3. Formulate a correct proof of a universally quantified statement. 4. Propose a counterexample to demonstrate that a statement is false. 5. Solve a variety of combinatorial problems. 6. Classify a function using concepts such as one-to-one, onto, and bijection. 7. Analyze a given relation to determine whether or not it is reflexive, symmetric, transitive, or an equivalence relation.
An introduction to ordinary differential equations and their applications. Analytical methods include: separation of variables, linear first order equations, substitution methods, second order linear equations with constant coefficients, undetermined coefficients, variation of parameters, autonomous systems of two first order equations, series solutions about ordinary points, and the Laplace Transform. In addition to analytical methods, quantitative and qualitative analysis will be employed through the use of Euler's Method, phase lines, phase planes, and slope fields. Four class hours.
Prerequisite: MTH 211 with a grade of C or better.
Course Learning Outcomes
1. Classify a differential equation using appropriate mathematical terminology. 2. Solve a variety of differential equations using analytical methods. 3. Describe the qualitative behavior of the solutions of a differential equation. 4. Estimate the solutions of a differential equation using numerical and graphical methods. 5. Solve a differential equation using power series. 6. Solve an autonomous system of two first order differential equations. 7. Examine the qualitative behavior of the solutions of an autonomous system of two first order differential equations. 8. Solve initial value problems using the Laplace Transform. 9. Solve applied problems using differential equations.
Prerequisite: MTH 212 with a grade of C or better, or both MTH 211 and MTH 220 with a grade of C or better in each, or permission of instructor.
Course Learning Outcomes
1. Solve systems of linear equations with Gaussian or Gauss-Jordan elimination. 2. Apply standard matrix operations. 3. Formulate the inverse of a square matrix or determine that the inverse does not exist. 4. Evaluate determinants of square matrices. 5. Classify sets of vectors in a vector space as linearly independent or dependent. 6. Identify sets of vectors in an inner product space as orthogonal or orthonormal. 7. Calculate the eigenvalues of a square matrix. 8. Produce a basis for the eigenspace of a square matrix associated with a given eigenvalue. 9. Construct a basis for the kernel of a linear transformation. 10. Construct a basis for the range of a linear transformation. | 677.169 | 1 |
focus of this unit is on developing and using simple formulae to describe and represent relationships between variables in real life contexts and on using simple maths problem solving techniques to interpret and solve straight forward problems.
Pre-requisite Courses and Assumed Knowledge and Capabilities
None
National Competency Codes and Titles
National Element Code & Title:
VBQU154 Investigate and use simple mathematical formulae for everyday purposes
Elements:
1. Investigate and use simple mathematical formulae in familiar contexts
They contribute directly to the achievement of ACSF indicators of competence at Level Four Numeracy (4.10, 4.11, 4.12, 4.13)
Overview of Assessment
This course is delivered as part of an integrated program.
You must demonstrate an understanding of all elements of the competencies in order to be deemed competent. Assessment methods have been designed to measure achievement of each competency in a flexible manner over a range of assessment tasks.
Feedback: assessment charter ( summarises your responsibilities as an RMIT student as well as those of your teachers.
Your courses assessment conforms to RMIT assessment principles, regulations, policies and procedures which are described and referenced in a single document: | 677.169 | 1 |
Elementary Mathematics for Engineers explains mathematical jargon and develops the formulae used by engineers from first...
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Elementary Mathematics for Engineers explains mathematical jargon and develops the formulae used by engineers from first principles. The first chapter is a summary so the reader can quickly see where further study is needed. The book is in two parts, pure and applied. Pure mathematics covers the following subjects;positive, negative, and fractional indiceslogarithmspartial fractionsirrational and complex functionstrigonometrical and hyperbolic functionsproperties of triangles and polygonsareas and volumes rules for differentiationmethods for integrationdifferential equationsFourier analysis Applied mathematics covers the following subjects;forces in equilibrium equations of motion and Newton's lawspotential and kinetic energyconservation of energymomentum and angular momentumsimple harmonic motionhanging chainsgyroscopes
Material Type:
Open (Access) Textbook
Author:
W. J. R. H. Pooler Mathematics for Engineers to Bookmark material or Course ePortfolio
This book provides a refreshing introduction to the three fascinating and fundamental subjects of communication, public...
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.
Material Type:
Open (Access) Textbook
Author:
Georgios P. Piperopoulos
Date Added:
Jan 09
The present book describes the state-of-art in the middle of the 20th century, concerning first order differential equations...
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The present book describes the state-of-art in the middle of the 20th century, concerning first order differential equations of known solution formulæ. Among the topics can be found exact differential forms, homogeneous differential forms, integrating factors, separation of the variables, and linear differential equations, Bernoulli's equation, and Riccati's equation. All topics are illustrated by examples.
Material Type:
Open (Access) Textbook
Author:
Leif Mejlbro Ordinary differential equations of first order to Bookmark material or Course ePortfolio
The book will introduce the reader to the following topics:The necessity for evaluation of investment and finance...
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The book will introduce the reader to the following topics:The necessity for evaluation of investment and finance opportunitiesMost common motives behind investment Basic assumptions behind investment and finance evaluationsMost common methods and techniques used for evaluation of investment and finance opportunitiesPitfalls in methods and techniques for evaluation of investment and finance opportunitiesPlanning of investment financing considering flexibility, cost, and timing of liquidityThe book contains numerous exercises, which the reader can solve to train the understanding of the topics presented above.
Material Type:
Open (Access) Textbook
Author:
Lars Wøldike Petersen
Date Added:
Jan 09 Investment and Finance to Bookmark material or Course ePortfolio
This book is the fourth in a series of books on software development in Java. This book has a primary focus on Java and...
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This book is the fourth in a series of books on software development in Java. This book has a primary focus on Java and numerous of details regarding Java as an object-oriented programming language. The book presents techniques that can help to ensure the development of robust and maintenance-friendly programs, but also techniques needed to know in order to develop robust programs in a modern programming language.
Material Type:
Open (Access) Textbook
Author:
Poul Klausen 4: Java's type system and collection classes Software Development to Bookmark material or Course ePortfolio
textbook is not to make you an expert. True expertise in any field is a years-long endeavor. Here I will...
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Material Type:
Open (Access) Textbook
Author:
Pueblo to Bookmark material or Course ePortfolio
Adult Learning Academy - Pre-Algebra is a competency-based, contextualized and accelerated basic academic skills course aligned to the competencies in the credit developmental education Pre-Algebra math course. Students develop their basic arithmetic skills, improve their ability to solve problems, and successfully transition toward college coursework in pursuit of an academic credential. Successful completion of Adult Learning Academy - Pre-Algebra is an essential step on a pathway to healthcare, IT, machining, and other STEM-related careers.
Material Type:
Open (Access) Textbook
Author:
MoWINs first | 677.169 | 1 |
Synopsis: New Mathematics Counts for Secondary 1 to 5 Normal (Academic) is a comprehensively revised version of the current recommended Mathematics Counts for Secondary 1 to 5 Normal (Academic). This edition meets the requirements of the new syllabuses for the Singapore-Cambridge GCE 'O' and 'N' Level Examinations in Mathematics. It promotes the infusion of the Ministry of Education's three initiatives: Thinking Skills, Information Technology and National Education. The aim of this series of five coursebooks, written specially for Normal (Academic) students, is twofold: to help students excel in the Singapore-Cambridge GCE 'N' level Mathematics examination and to enable students in Secondary Five to take the GCE 'O' level Mathematics examination successfully. This series adopts an approach emphasizing investigation, practice, information handling and problem solving to help students develop skills of long-term value. In each book, mathematical concepts are clearly presented in simple language and with extensive use of worked examples and a wealth of graded questions. Other salient features of the books in the series are:
An interesting introduction in each chapter to stimulate discussion of the topic. An overview at the start of each chapter outlines the key points that will be covered. Pedagogically sound treatment of topics in which concrete examples precede abstract reasoning. Extensive use of worked examples with explanatory notes in side columns to enhance understanding Well-graded exercises in each chapter to enable students to master fundamental concepts Explication of thinking skills at relevant sections in the text Activities that employ computer programs as an investigative tool Activities in the Investigating Maths and Fun Maths sections which enable students to explore and discuss mathematical ideas, use mathematical tools to analyse data related to national issues as well as to develop thinking skills A topical revision paper at the end of each chapter and ample assessment papers at appropriate intervals to provide students with necessary practice and reinforcement A summary at the end of each chapter which highlights the main points and concepts covered Written by highly qualified and experienced Mathematics teachers, this series can help students to achieve a high level of competence in the subject | 677.169 | 1 |
Purple Math Purple Math is a good online resource for math help for the following subjects: basic
math; beginning, intermediate, and advanced (college) algebra; word problems. While
watching video explanations, the user may do practice problems, immediately learning
whether the answers were correct or not. Worksheets and preparation for standardized
tests are also available.
Paul's Online This Lamar University web site is helpful for students wanting information on college
algebra and all levels of calculus (AHC's Math 181, 182, 183, and 184). It contains
class notes and explanations, downloadable handouts, practice problems, and reviews.
West Texas The West Texas A & M University web site offers assistance in the beginning, intermediate,
and advanced (college) algebra. Go to this site to view tutorials, work sample problems,
or prepare for the START (ACCUPLACER) or GED tests.
Achieve Success at the Math Center | 677.169 | 1 |
Math 4
Delivery
Online Asynchronous with print resources
Course Description
Math is a required area of study in Saskatchewan's Core Curriculum. The aim of the curriculum is to prepare learners to value Math and appreciate its role in their life and society. The curriculum is intended to prepare learners to confidently use Mathematics in everyday situations by thinking, reasoning, and inquiring. Lesson plans include written instructions, instructional videos, handouts and assessment tools. Learners will be assessed in a variety of ways that include ongoing home assessment and regular submissions to the DLC teacher | 677.169 | 1 |
Linear Algebra Writing Service
Linear Algebra Writing Service
Introduction
Linear Algebra is a stylish and lovely branch of mathematics with a wealth of applications in the Engineering field. Its approaches are perfect and pure, however at the exact same time, they provide students with troubles, due to its abstract nature.
Linear Algebra Writing Service
Linear algebra is the research of linear sets of formulas and their improvement homes. Linear algebra permits the analysis of rotations in area, least squares fitting, option of combined differential formulas, decision of a circle going through 3 provided points, in addition to numerous other issues in mathematics, physics, and engineering. Confusingly, linear algebra is not in fact an algebra in the technical sense of the word "algebra".
Linear algebra is the easiest method to take a look at functions of numerous variables, which typically occur in engineering by the discretization of a principle specified in regards to a continuum, e.g. the law governing the relation in between tensions and pressures in a structure.
Linear Algebra is utilized rather greatly in Structural Engineering. Typically these formulas are linear, even when product contortion (i.e. flexing) is thought about.
It is branch of mathematics that is interested in mathematical structures closed under the operations of addition and scalar reproduction which consists of the theory of systems of linear formulas, matrices, factors, vector areas, and linear improvements.
The set of points with collaborates that please a linear formula forms a hyperplane in an n-dimensional area. The conditions under which a set of n hyperplanes converge in a single point is an essential focus of research study in linear algebra.
Linear algebra is important to both pure and used mathematics. Algebra occurs by unwinding the axioms of a vector area, leading to a number of generalizations. Functional analysis research studies the infinite-dimensional variation of the theory of vector areas. Integrated with calculus, linear algebra assists in the option of linear systems of differential formulas.
Strategies from linear algebra are likewise utilized in analytic geometry, engineering, physics, lives sciences, computer technology, computer system animation, and the social sciences (especially in economics). Nonlinear mathematical vehicles are often estimated by linear designs due to the fact that linear algebra is such a strong theory.
As a field, it's beneficial to you due to the fact that you can explain (as well as carry out with the best libraries) complicated operations made use of in artificial intelligence utilizing the notation and formalisms from linear algebra.
Linear algebra is a sub department of algebra and falls under the wider topic of mathematics. If you do not have any interest in mathematics then you need to not have actually selected it up in the very first location.
It is an algebraic system like a ring where components can be increased by complicated or genuine number consisting of other components too. A unique system of notation is adjusted to study unique system of relationship which is called as algebra of classes.
If you are a Linear Algebra student and looking for Linear Algebra Writing Help, then here is most dependable, exact and 100 % initial option at Academicpaperwriter.com. Our well certified and skilled specialists of Linear Algebra will offer you Linear Algebra writing assistance within due date.
We see that there are lots of writing aid service providers who ruin the quality in order to achieve the Linear Algebra writing Help in a specific time frame while overlooking the quality work, we supply you the finest of the finest service in Linear Algebra writing Help, focusing on time and quality level on very same level.
We provide substantial services for resolving linear algebra research for students of all ages and scholastic levels. There is no requirement for you to experience confusion or tension when fixing linear algebra.
Linear algebra is the kind of mathematics that is worried about the research of vectors (linear (vector) areas) in addition to with functions inputting one vector and outputting another in accordance with the guidelines. Our service can be a dazzling linear algebra composing aid for you and your mathematics issues. Given that our solvers in linear algebra options are offered 24hrs a day, they can aid you with linear algebra composing assistance whenever you require. | 677.169 | 1 |
This Includes motivated This discrete mathematics, which involves processes that consist of sequences of individual steps (as compared to calculus, which describes processes that change in a continuous manner). The principal topics presented in this course are logic and proof, induction and recursion, discrete probability, and finite state machines. This free course may be completed online at any time. See course site for detailed overview and learning outcomes. (Computer Science 202) exercises | 677.169 | 1 |
MAT199 Math Alive
Course Instructors
How is life different from 25 or even 10 years ago? Mathematics has profoundly changed our world, from the way we communicate with each other and listen to music, to banking and computers. This course is designed for those who haven't had college mathematics but would like to understand some of the mathematical concepts behind important modern applications. It will consist of largely independent two-week units covering topics such as: Voting and social choice; Probability and statistics; Error correction and data compression; Cryptography; and Birth, growth, death and chaos.
You can navigate through the units as we cover them using the navigation bar on the left.
Each unit is divided into two parts. For each part you can download
lecture notes in pdf format.
There will also be online labs that you can use to help with understanding.
The problem sets are made available on the web one by one, according to
the schedule. Every problem set will be available (at least) one week
before it is due.
Problem sets: You need to look at the online labs and at the problem
sets (clickable on the left). The online labs
review some of the material seen in class, and give you interactive
windows to try out various things. The
problem sets contain questions and assignments for you to answer or
complete. To answer some of these
questions you'll need the interactive pages from the online labs. On
the due dates for the problem sets (see Calendar) you hand in the completed homework only. (No need to
hand in a printout of the corresponding online lab.) | 677.169 | 1 |
Description
This undergraduate textbook is intended primarily for a transition course into higher mathematics, although it is written with a broader audience in mind. The heart and soul of this book is problem solving, where each problem is carefully chosen to clarify a concept, demonstrate a technique, or to enthuse. The exercises require relatively extensive arguments, creative approaches, or both, thus providing motivation for the reader. With a unified approach to a diverse collection of topics, this text points out connections, similarities, and differences among subjects whenever possible. This book shows students that mathematics is a vibrant and dynamic human enterprise by including historical perspectives and notes on the giants of mathematics, by mentioning current activity in the mathematical community, and by discussing many famous and less well-known questions that remain open for future mathematicians.Ideally, this text should be used for a two semester course, where the first course has no prerequisites and the second is a more challenging course for math majors; yet, the flexible structure of the book allows it to be used in a variety of settings, including as a source of various independent-study and research projects.
About Author
Bela Bajnok is a native of Hungary, and he remains greatly influenced by the excellent education he received there. He currently holds the endowed position of Alumni Chair Professor at Gettysburg College. He is also the recipient of the Mathematical Association of America's 2012 James P. Crawford EPADEL Teaching Award.
Contents
Preface to Instructors.- Preface to Students.- Acknowledgments.- I What's Mathematics.- 1 Let's Play a Game!.- 2 What's the Name of the Game?.- 3 How to Make a Statement?- 4 What's True in Mathematics?.- 5 Famous Classical Theorems.- 6 Recent Progress in Mathematics.- II How to Solve It?.- 7 Let's be Logical!.- 8 Setting Examples.- 9 Quantifier Mechanics.- 10 Mathematical Structures.- 11 Working in the Fields (and Other Structures).- 12 Universal Proofs.- 13 The Domino Effect.- 14 More Domino Games.- 15 Existential Proofs.- 16 A Cornucopia of Famous Problems.- III Advanced Math for Beginners.- 17 Good Relations.- 18 Order, Please!.- 19 Let's be Functional!.- 20 Now That's the Limit!.- 21 Sizing It Up.- 22 Infinite Delights.- 23 Number Systems Systematically.- 24 Games Are Valuable!.- IV. Appendices.- A. Famous Conjectures in Mathematics.- B The Foundations of Set Theory.- C All Games Considered.- D Top 40 List of Math Theorems.? - Index. | 677.169 | 1 |
Vector Fields in Mathematica
Study Section 8.8 carefully. Then do Project D on page 100. Create all of the graphs in a Mathematica notebook. Use the functions within Mathematica (such as Text, etc.) and write a paper about your findings. | 677.169 | 1 |
Great stuff! As I was doing an Internet search on effective math software, I came across the Algebra Helper Web site. I decided to purchase the software after seeing the online demo and never regretted it. Thanks! T.P., Wyoming
The best part of The Algebra Helper is its approach to mathematics. Not only it guides you on the solution but also tells you how to reach that solution12:
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Note: Citations are based on reference standards. However, formatting rules can vary widely between applications and fields of interest or study. The specific requirements or preferences of your reviewing publisher, classroom teacher, institution or organization should be applied.
Antieigenvalue analysis, with its operator trigonometry, is a unifying language which enables new and deeper geometrical understanding of essentially every result in operator theory and matrix theory, together with their applications. This title will open up its methods to a wide range of specialists.
Abstract:
Karl Gustafson is the creator of the theory of antieigenvalue analysis. Its applications spread through fields as diverse as numerical analysis, wavelets, statistics, quantum mechanics, and finance. This book opens up various methods to a range of specialists.Read more... | 677.169 | 1 |
Product Description: This exciting book contains 18 instructional lessons that guide students step-by-step through sequences of activities that help them learn the mathematical concepts using the TI-89 graphing calculator. Keying sequences are provided for each lesson. Students are given lots of exercises for practice of the concepts and complete solutions are given for most of the exercises.$16.95
96 pages20 Explorations develop the fundamental concepts of differential & integral calculus using the TI-89 or the TI-92. Topics included are: | 677.169 | 1 |
I'm having great difficulty understanding the logic behind the problem regarding software that will solve sentence algebra problems. Can anybody please assist me to know how to come up with a detailed answer and explanation regarding software that will solve sentence algebra problems specifically in topic of adding matrices? I was told how to solve this before but now I forgot and confused how to solve it. I find it difficult to understand it on my own so I think I need assistance since I think I can't do this on my own . If someone is familiar with software that will solve sentence algebra problems can you please help me? Thank you !
Hi, I believe that I can to help you out. Have you ever used a program to assist you with your algebra assignments? Some time ago I was also stuck on a similar problems like you, but then I came across Algebrator. It helped me a lot with software that will solve sentence algebra problems and other algebra problems, so since then I always count on its help! My math grades got better thanks to the help of Algebrator.
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The Math Question Center
The Math Question Center (MQC) is a community where students in Math M, 1a, 1b, 21a, and 21b
come together to work on their problem sets (or PSETs).
The center is staffed by undergraduate course assistants (CAs) who are there to help you
productively approach your PSETS. The goal of the MQC is for you to not only complete
your homework correctly, but also synthesize the main ideas from the PSETS so you can apply them more broadly.
The MQC is open Sundays - Thursdays (excluding holidays) from 7:30pm to 11pm, and is staffed by CAs
from 8:30pm to 10:30pm. As a general rule, if no class takes place the following day, then the
MQC is closed. Please refer to the schedule below for the exact location of the MQC for each course this semester:
Getting the Most Out of the MQC
In past semesters, the MQC has been used by many students. We are delighted that it has been so popular!
However, this means that MQC staffers are often able to spend only a few minutes
helping any one student. Here are some tips to help you make use of this time effectively:
Before you visit the MQC, work on your homework, and try each problem carefully.
Only ask for help on problems you've thought through on your own first.
At the MQC, discuss your questions with other students in addition to asking for help from a staffer.
Use the MQC as a place to find other students with whom to work on your homework. Your classmates can be your best resource!
Keep in mind that the MQC staffers are not there to do your homework for you; their job is to guide you to
understanding the material well enough so that you can do the homework yourself. Try to work in your group as
autonomously as possible, but do ask for help when your group gets stuck.
Remember that the MQC is not the only available source for help! Each course has problem sessions, which can better
prepare you for the homework, as well as office hours, which may be more suitable for one-on-one assistance. | 677.169 | 1 |
Mathematics Explained for Primary Teachers free download
1848601972 This much-loved textbook has been fully revised and updated to take account of the new Primary Curriculum, which was made statutory in 2010 and will be taught in primary schools beginning September 2011. The new edition will be a valuable resource for new primary teachers as they prepare to teach this curriculum.
Some of the changes in the new edition include the following:
New chapters on key ideas and key processes in primary mathematics
Reordering of the chapters to give more prominence to using and applying mathematics
Further material on graphs in the chapter on coordinates and linear relationships
References throughout to the new Primary Curriculum
Research focus in every chapter
Updating of suggestions for further reading
More discursive answers to some of the self-assessment questions
A companion website providing a comprehensive glossary and additional material to enable primary trainees to prepare with confidence for the ITT Numeracy test
A companion Student Workbook available for purchase, providing further self-assessment examples for checking understanding, for using and applying mathematics, and for teaching and learning
Extensively used on primary PGCE courses and undergraduate courses leading to QTS, this book is an essential resource for all new primary teachers. | 677.169 | 1 |
Looking for an inspection copy?
In this stimulating book, aimed at researchers both established and budding, Peter Elliott demonstrates a method and a motivating philosophy that combine to cohere a large part of analytic number theory, including the hitherto nebulous study of arithmetic functions. Besides its application, the book also illustrates a way of thinking mathematically: historical background is woven into the narrative, variant proofs illustrate obstructions, false steps and the development of insight, in a manner reminiscent of Euler. It is shown how to formulate theorems as well as how to construct their proofs. Elementary notions from functional analysis, Fourier analysis, functional equations and stability in mechanics are controlled by a geometric view and synthesized to provide an arithmetical analogue of classical harmonic analysis that is powerful enough to establish arithmetic propositions until now beyond reach. Connections with other branches of analysis are illustrated by over 250 exercises, structured in chains about individual topics | 677.169 | 1 |
Study Guide For 8th Grade Math
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Your new release is so much more intuitive! You team was quick to respond. Great job! I will recommend the new version to other students. R.B., New Mexico
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The course includes interval notation; absolute value, rational, radical and quadratic equations; absolute value and polynomial inequalities in one variable; linear inequalities in two variables; compound inequalities in one and two variables; operations on radical expressions and expressions containing rational exponents; complex number system introduction; and applications and modeling.
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going step by step with balancing equations where i type in the problem | 677.169 | 1 |
MATH 2700 - Euclidean and Non-Euclidean Geometry
University of Virginia, January Term 2018 Prof. David Sherman
Course description:
Euclid's Elements was the standard reference in geometry for about 2,000 years. In this course we will examine the assumptions and methods in the original text of Book I, then see how both of these have evolved over the last 200 years. Apart from Euclid, the main topics include the following: symmetries, spherical geometry, curvature, the dissection theory of area, constructible numbers, and the discovery of non-Euclidean geometry. Mathematical concepts will not always be presented in their most general and technical form, but there will be at least enough detail to facilitate simple computations.
Students will be challenged to think critically, to make conjectures, and to compose rigorous arguments. For geometries other than the plane, we will use various physical models: rubber bands on spheres, strings wrapped around cylinders, paper approximations to the hyperbolic plane, etc. Significant class time will be devoted to experimentation and group work on problem sets.
Required materials:
-- Experiencing Geometry: Euclidean and Non-Euclidean with History (third edition), D. W. Henderson and D. Taimina
-- Flatland, E. Abbott
-- Straightedge and compass
-- Tennis ball (or equivalent) and rubber bands
-- (optional) The Thirteen Books of the Elements: Vol. 1 (Books I-II), Euclid (ed. Heath). I have in mind a Dover paperback edition which includes lots of scholarly commentary and can be had for $10-15. (The remaining 11 books of The Elements come in two separate volumes and will not be considered in the course.) Of course any text of Book I is acceptable. Since all of The Elements is available for free on the internet (and linked to the course webpage), the purchase of a book form is optional. | 677.169 | 1 |
Plane Geometry (step-by-step)
Neste livro, disponibilizam-se construções passo-a-passo de Geometria Plana com que os utilizadores podem interagir, acompanhando a construção de acordo com o ritmo pretendido.
In this book are shown several constructions on Plane Geometry, with the intent to help students to follow them step-by-step and support their schoolwork, according to their own needs. | 677.169 | 1 |
Description: In this book the trigonometrical ratios are introduced as functions of the angle. The trigonometrical properties of the single angle are treated fully in the early chapters, and from the beginning the examples apply Trigonometry wherever it may be useful, to Geometry, Mensuration, Analytical Geometry, Physical formulae, etc11 views)
Open Trigonometry: Basic to Advanced - Wikibooks Book 1 is pre-calculus trigonometry. We assume the student is relatively new to algebra and do algebra step by step. Many of the pages have closely related free/YouTube videos at the Khan Academy. Many students find the video presentation helpful. (2001 views)
Plane and Spherical Trigonometry in three parts by Henry Bedingfield Goodwin - Longmans, Green, and Co. This book was intended to serve as an introduction to the study of Navigation and Nautical Astronomy for the junior officers under training in H.M. Fleet. The text plane trigonometry, spherical trigonometry, and the examples in the use of logarithms. (8004 views) | 677.169 | 1 |
Discrete mathematics achieves fascinating results using relatively simple means, such as counting. Explore this modern mathematical realm in Discrete Mathematics, 24 mind-expanding lectures delivered by popular Teaching Company Professor Arthur T. Benjamin. Covering combinatorics, number theory, and graph theory, this intriguing course is an ideal point of entry for you to start thinking more mathematically. While certainly rigorous, this entertaining course is accessible to anyone with a basic knowledge of high-school algebra and a willingness to learn logical thinking skills essential to living everyday life—as well as success in any future math courses you may take.
This book discusses examples of discrete mathematics in school curricula, including in the areas of graph theory, recursion and discrete dynamical systems, combinatorics, logic, game theory, and the mathematics of fairness. In addition, it describes current discrete mathematics curriculum initiatives in several countries, and presents ongoing research, especially in the areas of combinatorial reasoning and the affective dimension of learning discrete mathematics. | 677.169 | 1 |
Calculators can perform math functions quickly and easily. The most common functions are addition, subtraction,...
see more
Calculators can perform math functions quickly and easily. The most common functions are addition, subtraction, multiplication, and division. This course will use the Touch Method, which means using the calculator without looking at the keys. Using this method will help develop competency. After building competency, students will be able to use 10-key calculators to enter numeric data and perform calculations efficiently.
Material Type:
Online Course
Author:
Velda Arnaud
Date Added:
Aug 21, 2011 | 677.169 | 1 |
introduces computational and mathematical techniques for modeling, simulating, and analyzing the performance of various systems. Helps readers gain a better understanding of how systems operate and respond to change by: 1) helping them begin to model, simulate, and analyze simple-but-representative systems as soon as possible; and 2) whenever possible, encouraging the experimental exploration and self-discovery of theoretical results before their formal presentation. Features an approachable writing style that emphasizes concepts and insight without sacrificing rigor. Provides C software as source code for running simulations developed in the book, eliminating the need for readers to do all their programming from scratch. Emphasizes an algorithmic approach throughout. A useful reference for industrial engineers.
This textbook provided me with a solid introduction to a complex but interesting field of study. As a computer science student, it provided me with a much-needed understanding of mathematical topics (random variate generation and statistics) while keeping me interested with challenging programming problems.
I have picked up a copy of the popular simulation textbook "Simulation Modeling and Analysis" since taking my "Discrete Event Simulation" course. While the two books cover similar material, I think that "Discrete Event Simulation" moves through the material faster, focuses on some of the more important (and challenging) issues, and keeps the reader more engaged along the way.
Considering how widely modeling and simulation are used in science, engineering, and business today, I think that a course based on this book would be a valuable addition to many undergraduate and graduate curricula.
This is good book\,while I did not really understand them all. I bought this book for Modeling and simulation class. Because it was online class. I realy read this book. I could not read the whole contents but I read most of it. Random number theory was very interesting. | 677.169 | 1 |
Author: Publisher: ISBN: Size: 56.17 MB Format: PDF, Kindle View: 3980 DownloadRead Online
Reading Level This series flows nicely, beginning by using pictures alone for
explanation. Later levels combine pictures with text, and progress by using
increasingly complex mathematical language. The texts highlight definitions and
important points. Vocabulary lists start each chapter in the teacher's edition.
Tables of measures and symbols. Indexes, glossaries, and content area activities
in the student texts provide additional support and development of mathematics
terminology. | 677.169 | 1 |
Accessibility links
Navigation
A2 Level : Mathematics (Mei) at English Martyrs Catholic School
Course description
Year 13 - we do C3, C4, S2 - follow a similar pattern as in year 12 / students also have the option to do M2 and D2 if they want to do an applied module other than S2.
We also do Further Maths - a further 6 modules in addition to their core mathematics qualifications
Modules include D1, S1, S2, D2, M1, M2, FP1, FP2, Numerical methods.
Entry requirements
Grade B in either Methods or Applications GCSE
Assessment
Year 13 / A2 = 3 x 1.5 hours external exam
All students undertake mock exams at least 2 times a year.
Homework set in every week (sometimes x 2 a week)
Modules completed on MyMaths / OCR MEI website. Students given their own passwords. Students have access to past papers / marks scheme + teacher notes.
Exam Board: OCR MEI
Future opportunities
You could take this course with other advanced level courses to prepare for higher education in areas of Mathematics, related areas such as Engineering, Economics, Physics or more general higher education courses. With further training, you could go into
a job related to Mathematics such as an Accountant, Engineer, Financial Adviser or Teacher. The course also helps you develop the skills, understanding and knowledge that many employers across lots of industries are looking for, especially in the economic
and financial sectors.
How to apply
You can apply for this course through UCAS Progress. Add this course to your favourites so you can start making an application. | 677.169 | 1 |
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Oxidizing and reducing agents | Redox Reactions | Khan Academy0 to plus 1 is an increase in oxidation state,; so therefore, sodium, by definition, is
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your own digital archive and overlay graphs and other elements.
Explore surfaces by rotating them, investigate intersections of a
surface with a plane and create animations using sliders to
understand how changing different parameters affect objects with
TI-Nspires 3D Graphing View
Create, edit, save and share work in documents within a familiar
folder structure.
Vernier DataQuest supports data logging for collection and analysis
of real-life data in the maths classroom and in science experiments.
Ideal for all mathematical topics, such as graphing, geometry,
statistics, data logging and more.
USB port supports handheld to handheld communication and connection
to a computer or other presentation technology.
Buy with Confidence from a Texas Instruments | 677.169 | 1 |
About this title:
Synopsis: This revised and expanded edition is for the reader lacking a strong mathematical background. It makes statistics interesting and accessible by using realistic examples and offering clear, step-by-step explanations, sound pedagogy and quality exercise sets.
From the Publisher:
This revised and expanded edition is designed to motivate and challenge students without a strong mathematics background. Filled with real-world examples, step-by-step explanations, quality exercise sets that link applications to concepts and procedures, and sound pedagogy, this text makes statistics interesting and accessible. New features include problems designed for use with the graphing calculator, directions and screen shots for the Windows version of Minitab throughout the text and an increased focus on technology as a teaching tool. | 677.169 | 1 |
Geometry Standardized Test Practice
This Geometry Standardized Test Practice worksheet also includes:
In this figures and graphs review worksheet, 10th graders solve and complete 23 various types of problems that include identifying various figures and graphs. First, they name the transformation that maps the unshaded figure or preimage onto a shaded one. Then, students use the graph of the transformation to name the image and preimage of given line segments. | 677.169 | 1 |
Mathematics is a tool for describing the world around us. Linear equations give some of the simplest descriptions, and systems of linear equations are made by combining several descriptions.
In this unit we write systems of linear equations in the matrix form Ax = b. We explore how the properties of A and b determine the solutions x (if any exist) and pay particular attention to the solutions to Ax = 0. For a given matrix A we ask which b can be written in the form Ax | 677.169 | 1 |
Students in Algebra II may check out a textbook from the library to keep at home for homework help. Students also may use on-line textbook help-videos, textbooks,etc. I have given the students sign-in information.
Notes and copies of homework are available on Google Classroom for all students.
Algebra III- During the 5th six weeks, students will be working with relations and functions. Students will begin working on polynomials and rational functions..
College Prep Algebra- During the 5th six weeks, students will be working on radical and rational functions. Students will also be learning about conics.
Algebra II- During the 5th six weeks, students will be working on relations and functions. They will also be .learning to factor and quadratic functons-graphing and solving.
Algebra II Honors- During the 5th six weeks, students will be working on relations and functions. Students will also be learning about polynomial and radical functions. | 677.169 | 1 |
08218285bability Theory (Courant Lecture Notes)
This volume presents topics in probability theory covered during a first-year graduate course given at the Courant Institute of Mathematical Sciences, USA. The necessary background material in measure theory is developed, including the standard topics, such as extension theorem, construction of measures, integration, product spaces, Radon-Nikodym theorem, and conditional expectation | 677.169 | 1 |
Description
Contents
The point The right line Examples on the right line The right line--Abridged notation Right lines The circle Examples on circle Properties of two or more circles The circle--Abridged notation General equation of second degree Central equations The parabola Examples on conics Abridged notation Reciprocal polars Harmonic and anharmonic properties The method of projection Invariants and covariants The method of infinitesimals Notes. | 677.169 | 1 |
Course Objectives and Goals: Math 131 is a calculus course primarily intended for students in the life or social sciences, such as Biology, Pharmacy, and Economics. It is different (but not easier) than the four-credit calculus course, Math 141, designed for students who intend to take more advanced math, such as engineering, computer science, and mathemactics majors. The main emphasis will on the practical interpretation of calculus in numerical, graphical, and algebraic terms, although important theoretical concepts will also be covered. The main topics of the course are functions, differentiation, integration and applications.
Expectations: We expect that you will give this course 7-9 hours a week of your undivided attention, in addition to your class time. No kidding! This is an approximate figure of course, but don't assume that you can spend less time than this and still get a grade you'll like. We also expect that you will ATTEND YOUR CLASS
Weather phone number: (401) 874-snow or 874-7669 | 677.169 | 1 |
Description: The difference between advanced calculus and calculus is that all the theorems are proved completely and the role of plane geometry is minimized. Instead, the notion of completeness is of preeminent importance. This is a subject which places calculus as part of mathematics and involves proofs and definitions, not algorithms and busy work.
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Calculus II by Paul Dawkins - Lamar University These lecture notes should be accessible to anyone wanting to learn Calculus II or needing a refresher in some of the topics from the class. The notes assume a good knowledge of Calculus I topics including limits, derivatives and basic integration. (10660 views)
Graphical Calculus by Arthur Henry Barker - Longmans, Green, and Co. All teachers of engineering and applied sciences generally now recognize the vast superiority of graphical over purely mathematical methods of imparting instruction of almost every description. The former are much more convincing to the student. (825800 | 677.169 | 1 |
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Curves and Surfaces: Topology
On the basis of the concepts introduced in Chapter 6, this chapter discusses curves and surfaces in topologic spaces, with emphasis on the digital case.
7.1 Curves in the Euclidean Topology
The definition of a curve has an interesting history in mathematics. Jordan curves are defined by parameterization; see the "parameterized paths " in Section 6.3.3. Urysohn-Menger curves are defined using a topologic approach. The two definitions are equivalent, and they both define separations of the plane.
7.1.1 Jordan curves
Let φ be a parameterized continuous path φ :[a, b] → R2 such that a ≠ b, φ (a) = φ(b), and let φ(S) ≠φ(i) for all s,t (a ≤ s<t<b). The following set was defined by C. Jordan in 1893 to be ...
With Safari, you learn the way you learn best. Get unlimited access to videos, live online training,
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Compiling the best online articles about mathematics so there will be more in the future.
Friday, December 21, 2012
Advanced Mathematics Books
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Needless to say, the advanced mathematics books of math teachers exists has further created the advanced mathematics books for bright new mathematics tutors. Today as a high degree of faith. Yet we see evidence of this article and explain how basic mathematics was the advanced mathematics books, the advanced mathematics books about adding numbers. I carefully selected this story which explains some simple rules of math to see actually see that 3 objects subtracted from 4 objects is 1 object. From there, it is applied to reality. In the advanced mathematics books of the advanced mathematics books if the advanced mathematics books in accordance with the advanced mathematics books and looking forward to high school level math classes that are offered to students that don't have time for math, so they opt for other majors. Has a future Einstein been lost to this chain of events. | 677.169 | 1 |
Introduction to the methodology and subject matter of modern mathematics. Sets, functions, relations, cardinality, and induction. The axiomatic method: definitions and examples of graphs, metric spaces, groups and rings. Roots of complex polynomials. Selected further topics.
II. Rationale (see instructions):
We are expanding two required four-credit courses, Pure Math I and Pure Math II, to three required four-credit courses, Mathematical Perspectives, Abstract Algebra, and Analysis. The new structure divides the material more naturally and is more in accord with national standards.
Mathematical Perspectives requires Calculus I, but will otherwise be a stand-alone course. Besides being required of our majors, it will be attractive to minors and mathematically talented students from across the campus. Pure Math I did not serve the latter two constituencies well.
III. Other Course Information
First semester to be offered:
Spring 2002
Principal Faculty Sponsor(s):
David Roberts and Michael O'Reilly
Is course repeatable? (if yes, give max cr)
No
To what quarter course(s) does this correspond?
3250f Foundations of Mathematics
Course Type (choose one from instruction sheet):
Lecture
Course requires:(check all that apply):
Laboratory sections
Recitation/discussion sections
IV. Assessment Information
Please state clearly and briefly the GOALS of this course.
The goal of this course is help students make the transition from computational mathematics to conceptual mathematics. The students will develop technical skill in proof-writing, and familiarity with some of the basic objects of advanced mathematics. The students will emerge with a better perspective on mathematics.
How will you ASSESS the success of this course in achieving the GOALS described above?
Student projects and presentations. Quizzes, Tests, and Final Examination.
Regular Approval Process:
For long-term approval, curricular change forms must go through the following regular approval process (put check in box and date when approved): | 677.169 | 1 |
Easily master the fundamental concepts of mathematical analysis with ADVANCED CALCULUS. Presented in a clear and simple way, this advanced caluclus text leads you to a precise understanding of the subject by providing you with the tools you need to succeed. A wide variety of exercises helps you gain a genuine understanding of the material and examples demonstrate the significance of what you learn. Emphasizing the unity of the subject, the text shows that mathematical analysis is not a collection of isolated facts and techniques, but rather a coherent body of knowledge.
"synopsis" may belong to another edition of this title.
About the Author:
Patrick M. Fitzpatrick is Professor and Chair of Mathematics at the University of Maryland, College Park. | 677.169 | 1 |
MM1_Chapter_10 - Chapter 10 Matrices In Section 10.1 we...
Chapter 10MatricesIn Section 10.1 we introducem×nmatriceswhich are rectangular arrays of realnumbers consisting ofmrows andncolumns. For example,A=p213131Pis a 2×3 matrix with two rows and three columns. Vectors which we have encoun-tered in Chapter 9, are a special case of matrices. In Section 10.2, we learn howtoaddandsubtract matricesof the same size and how tomultiply matriceswith scalars. We also learn how tomultiply matriceswith each other. In Sec-tion 10.3, we introduce thedeterminantof a square matrix, which is a scalar. InSection 10.4, we learn how to compute theinverse matrixof a matrix with deter-minant di±erent from zero. Finally we will use matrices and their inverse matricestosolve linear systems of equationsin Section 10.5.10.1IntroductionWe start by introducing matrices as rectangular arrays of real numbers. In theprevious chapter we have already encountered vectors which are special cases ofmatrices.263
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10. Matrices265Defnition 10.3 (equal matrices)Two matrices are said to beequalif they have thesame size, that is, the samenumbers of rows and the same number of columns, and ifall correspondingentries are the same. In formulas, anm×nmatrixA= (ai,j)and anℓ×kmatrixB= (bi,j)areequalifm=ℓandn=kandai,j=bi,jfor alli= 1,2, . . ., mand allj= 1,2, . . ., n.If two matricesAandBareequalwe may writeA=B, and if two matrices arenot equalwe may writeAn=B.Example 10.4 (equal and unequal matrices)(a) Consider the matricesA=p2 3 1−42 1 05PandB=p2 3 12 1 0P.The matrixAis a 2×4 matrix, andBis an 2×3 matrix. Since the matrices havenot the same dimension, we haveAn=B.
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