text: Web Ratna LLP believes in integrating the latest technology to ensure maximum productivity. This cheat sheet covers 100s of functions that are critical to know as an Excel analyst that will calculate the linear trend line to the arrays of known y and known. As x increases by 1, g x 4 3x grows by a factor of 3, and h x 8 1 4 x decays by a factor of 1 4. $f(x) = k*2^x$ Use an initial value to get the value for k: $50 = k*2^1 → k = 25$ $f(x) = 25*2^x$ Finally, take the invers. the pattern created by a sequence or series. These parent functions illustrate that, as long as the exponent is positive, the graph of an exponential function whose base is greater than 1 increases as x increases — an example of exponential growth — whereas the graph of an exponential function whose base is between 0 and 1 decreases towards the x-axis as x increases — an example of. Exponential and Logarithmic Functions The first two sheets explore the graphs of exponential functions. First I run a linear regression, second is an exponential regression, and finally a polynomial regression. Some of the functions are also available in the Peak Analyzer tool, please refer to the Peak Analyzer Functions section also in Appendix 3. Download a free google spreadsheet portfolio tracker for stocks, mutual funds and fixed deposits. Principal branch of the logarithm ln(z). We stated in the section on exponential functions, that exponential functions were one-to-one. The Microsoft Excel GROWTH function returns the predicted exponential growth based on existing values provided. Resource includes a project checklist, planning sheets and preparation sheets so students can use Google Draw, Google Sheets and Google Docs to map their "paying it forward" idea, record. Open a spreadsheet with a chart where you want to add a trendline. This addition moves beyond the existing Microsoft Office JavaScript API. Graphing Exponential Functions Flip Book This flip book was created to be used as a stations activity to provide extra practice with graphing exponential functions and identifying the domain, range, y-intercept, asymptotes, and end behavior. The constant ais usually the starting value and bis the. We derive the derivatives of inverse exponential functions using implicit differentiation. See the complete profile on LinkedIn and discover Celeste’s connections and jobs at similar companies. where F-1 (u) is the inverse cumulative distribution function (ICDF) of the same distribution, and x is the resulting random number. 97)ᵗ, y = (1. Linear regression is a very powerful. 2)t/10, and classify them as representing exponential growth or decay. Stop wasting time in meetings and be more productive at work. Hi, yes I had looked at that formula earlier today. The steps below will walk through. "Cheat Sheet" 6 November 2019 Function Name Parent Function Graph Characteristics Algebra Constant ( T)= Exponential ( T)=10. To see the trend data and to create dynamic trend lines you can control with menus you need to use Excel’s TREND function. Click where you want to put the equation. See how to use the GROWTH function in Google Sheets below. Microsoft Word - 3. Refold to form a booklet. 9/24 Making Graphs in Google Sheets Nearpod 9/25 O3C2 part 1: Dot Plots and Stem and Leaf Plots Nearpod 9/26 O3C2 part 2: Histograms Nearpod 9/27 Quiz Review Notes 9/28 Quiz 10/1 Boxplot Nearpod 10/2 Interpreting Data from a Table 10/3 Boxplot Review 10/4 Group Poster Interpreting Data (pics to come) 10/10 Review for O3C3 Quiz 10/11 O3C3 Quiz. Choose from different chart types, like: line and bar charts, pie charts, scatter graphs, XY graph and pie charts. the pattern created by a sequence or series. GROWTH & DECAY (DAY 3) Linear Functions Exponential Functions General Equation Function Notation y ax b f(x) ax b General Equation Function Notation y abx f(x) abx (recall: variable is the exponent for an exponential function) a = b = a = b = x = Exponential Function are able to have both a_____ or _____ rate of change. Exponential+Growthand+DecayWord+Problems+!!! 4. Google Analytics is a powerful tool for the aspiring blogger. Exponential vs. A quiz and full answer keys are also provided. 6(3) x and y = 10(. Enter another add-on: autoCrat! This amazing tool automates the creation and sharing of personalized documents or PDFs from columns of data in a Google Sheet. Short overview video on how to run three different regressions in Google Sheets. Tracking your DGI Portfolio Last month, All About Interest posted a nice article about using Google Docs for your portfolio. Google today announced it has made dramatic improvements to copying and pasting between Gmail and Google Docs, Sheets, and Slides. To sum up, there are basically three different ways to pull real-time stock and dividend information into Google Sheets: Using Google's built-in GoogleFinance formulas Importing HTML and/or XML. Hunting for Spanish and Scandinavian special characters in this chart is so time-consuming as to be impractical. ; Basic Usage. Use Google Sheets to build your table and then use the embed function to place it on your page. 02)ᵗ, y = (0. Confusion comes from "Interval", "Minimum Interval", and "Maxim. Author summary The exponential growth of the number of available reference protein sequences has provided an opportunity to taxonomically annotate and quantify complex mixtures of organisms using bottom-up proteomics. decay function with a linear graph 2. Fetching contributors. This lesson has students examine the growth of Starbuck since 1971. - Create Functions using Google Sheets and analyze if linear or exponential - Exponential Functions Sheet Thurs 10/29/15 Learning Targets - students will solve problems involving percent increase and decrease 2 hour delay schedule Agenda - opening problem - Percent Increase Notes Wednesday 10/28/15 Learning Targets. This means that as the input increases by 1, the output value will be the product of the base and the previous output, regardless of the value of a. The two types of exponential functions are exponential growth and exponential decay. Budgeting Question: The national debt in the US is a continuing issue that has kept politicians and citizens worried for the past couple years. How? Thread starter Schuyler; There are two formulas in Excel that can model exponential growth: One is called LOGEST and the other is GROWTH. Student Answer Sheet and KEY are included. R ecently one of my readers asked me about the possibility of using dynamic ranges in Growth function in Google Sheets. Antonyms for exponentially. Practice: Exponential vs. And then using the "IRR" function, calculate an annual return number. Access stock quotes, historical financials, valuation and fundamental ratios, fair value targets and more. Exponential Growth and Decay Exponential decay refers to an amount of substance decreasing exponentially. This article describes the formula syntax and usage of the EXP function in Microsoft Excel. With the Dividend Meter Google Sheets Add-on tool and Data Feed, creating your own custom dividend tracking spreadsheet is easy. amount of the substance. A quality cheat sheet for time series, which I took time to re-create and decided to try an augment by adding code snippets for ARCH and GARH. The Windows version of Excel 2016 has a button that places an interactive forecast chart on your sheet. 19 February 2020. How It Contributes To Exponential Results And Growth:. Accessing Google Sheets & Slides from Apps Script. $f(x) = k*2^x$ Use an initial value to get the value for k: $50 = k*2^1 → k = 25$ $f(x) = 25*2^x$ Finally, take the invers. Exponential Growth and Decay Word Problems : In this section, we are going to see how to solve word problems on exponential growth and decay. The CCC sheet is an Excel sheet so we first want to copy that into the same Google Docs folder that contains your portfolio spreadsheet. I'm a very creative person but I also am utterly analytical when it comes to marketing. Instead I assigned 1 to the. Ask Question Asked 5 years, 7 months ago. 87 likes · 5 talking about this. - find the equation of a trend line for the data - analyze what this trend line is telling us. e or Euler's Number is the value that the following equation approaches as n grows large: (1 + (1/ n)) n. To apply for this position please go to https://careers. Lifetime access to the course. What exponential function best represents this species' population growth? Year Population 1 22 2 25 3 31 4 39 5 42. Use Google Sheets to build your table and then use the embed function to place it on your page. First, let me introduce to you the function syntax and then the formula example follows. But, this time we modeled both exponential growth and exponential decay. Protein synthesis, cell division, growth and differentiation all occur in the nucleus. Log In Product Premium Case Studies Curriculum. The real base b logarithm of a negative number is undefined. These values are used to curve fit an ideal exponential growth curve. Google Sheets gives you a variety of options for your graph, so if you want to show parts that make up a whole you can go for a pie chart, and if you want to compare statistics, a bar graph will. Add a trendline to a Chart. In three simple steps, you can create your own Dividend Meter or build a custom watch list spreadsheet with the new Dividend Meter Google Sheets Add-on and Data Feed. A spreadsheet formula for the correct calculation of an exponential moving average has been derived and successfully tested on Google Sheets. In fact, for any exponential function with the form $f\left(x\right)=a{b}^{x}$, b is the constant ratio of the function. The cell nucleus contains the hereditary information of the cell. When you want more than one value, you must enter it as an array or your must include it in an INDEX function. How to Create a Jeopardy Game to Use in you Geometry Classes!. Exponential Growth and DecayWorksheet 1. While online surveys are one of the more basic functions of Google Forms, they can be quite handy — especially for teachers looking to collect honest feedback from anonymous respondents. The GROWTH function is a built-in function in Excel that is categorized as a Statistical Function. Expect to find charts, graphs, pivot tables, functions and formulas, but you’ll also discover what makes Google Sheets unique, like file sharing, revision history, web-based scripts, macros, and much more. Principal branch of the logarithm ln(z). This is all real time – any change I make in my Google Sheet is immediately visible. To select formula click at picture next to formula. - go over example of fitting a linear function to data in Google Sheets (5 min) - Examine Tree Ring picture (15 min) - using Google Sheets create a table and scatterplot of the radius of the tree as a function of the year. A quiz and full answer keys are also provided. On the bottom of the dialog, click Options to expand the dialog. F7 Understand and use exponential growth and decay; use in modelling (examples may include the use of e in continuous compound interest, radioactive decay, drug concentration decay, exponential growth as a model for population growth); consideration of limitations and refinements of exponential models Exponential Functions. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Hey, it’s free as well!” – Dan Butler “Google Docs is by far the most versatile, powerful and fast method for doing this, in my personal experience. Google Chrome has hundreds of plug-ins to use, and the integration of social media websites into everyday life has opened up plenty of opportunities for micro-SaaS type products. For example, identify percent rate of change in functions such as y = (1. The two types of exponential functions are exponential growth and exponential decay. After you have selected all the formulas which you would like to include in cheat sheet, click the "Generate PDF" button. But, if you first paste the values into a Google Sheet, then copy that table into Desmos, it works fine. 11 Exponential and Logarithmic Functions Worksheet Concepts: • Rules of Exponents • Exponential Functions - Power Functions vs. Works in Mac and Windows. Then plot the points and either exponential growth or decay , and identify the growth or decay factor. Things to Remember About Exponential Function (EXP) in Excel. Plan your 60-minute lesson in Math or Algebra with helpful tips from James Bialasik. E xponential Growth. The function helps calculate predicted exponential growth by using existing data. The best part is that Google offers a way to seamlessly pull data from their Google Finance service into Sheets. Exponential growth is a specific way in which an amount of some quantity can increase over time. Sign in - Google Accounts. Write!an!exponential!function!to!model!each!situation. Model-Fitting with Linear Regression: Exponential Functions In class we have seen how least squares regression is used to approximate the linear mathematical function that describes the relationship between a dependent and an independent variable by minimizing the variation on the y axis. On June 2, 2017 June 2, 2017 By admin. In statistics, nonlinear regression is a form of regression analysis in which observational data are modeled by a function which is a nonlinear combination of the model parameters and depends on one or more independent variables. About Us Our Values Team Careers Press Blog. Later he revised that law to doubling every two years. - go over example of fitting a linear function to data in Google Sheets (5 min) - Examine Tree Ring picture (15 min) - using Google Sheets create a table and scatterplot of the radius of the tree as a function of the year. This video tutorial belongs to the Education category which is going to show you how to match exponential graphs. (Drawing the pictures is very helpful in learning about how to do all this, though, which is why we did it that way to start with. This example teaches you how to apply exponential smoothing to a time series in Excel. Google Sheets has been designed with the needs of agile organizations in mind. Lots of business people use Google Sheets as a web-based application to do financial calculations, and lots of people use it to handle their personal finances as well, because the cloud-based spreadsheet app includes many powerful financial functions and can be used to set up a wide variety of investment spreadsheets. Exponential functions and equations Student Activity Sheet 3; use with Exploring "Growth and decay. When you want more than one value, you must enter it as an array or your must include it in an INDEX function. Sheet1 Algebra II Honors ASSIGNMENT SHEET CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Date,Day,Topic,Assignment 12/20,Fri,6. Google Sheets: Excluding Weekends in Conditional Formatting. Built-in formulas, pivot tables and conditional formatting options save time and simplify common spreadsheet tasks. Smith's Life. – Alan Wells Dec 8 '15 at 17:18. l G iA wl wll 4r ci9g 1h6t hsi qr Feks 2e vrHv we3d9. 2 - Logarithmic Functions and Their Graphs Inverse of Exponential Functions. Exponential growth often causes this kind of surprising result, even when considering the vastly large orders of magnitude that come from space. If you'd like to follow along, you can download our example. When plotted on a chart, the curve would begin slowly, stay flat for some time, then increase exponentially until it becomes almost vertical. Don't let the above function names confuse you. ” Time to pause, raise more money, and build a. 9(C) – write exponential functions in the form f(x) = abx (where b is a rational number) to describe problems arising from mathematical and real-world situations, including growth and decay A. In real-world applications, we need to model the behavior of a function. It appreciates 6% per year. Calculate an exponential moving average from an array of numbers. This is the general Exponential Function (see below for e x):. We derive the derivatives of inverse exponential functions using implicit differentiation. This function applies the least squares method, so that instead of calculating the growth rate based only on the initial and ending values, it takes into account all the values to provide an annual rate of growth that best fits the historical trend. The base can be altered and the appropriate graphs shown. For that reason, we have many formulas and tools to aid us in that direction. 2015-04-30T12:06:00Z The letter F. r is the growth rate when r>0 or decay rate when r<0, in percent. The Forecast function can forecast any number of periods into the future. In the process you will be looking at very large numbers and the effects of exponential growth on something we now live with all the time. Find local businesses, view maps and get driving directions in Google Maps. Domain and Range of Exponential and Logarithmic Functions Recall that the domain of a function is the set of input or x -values for which the function is defined, while the range is the set of all the output or y -values that the function takes. Sign in - Google Accounts. This lesson has students examine the growth of Starbuck since 1971. For forecasting of linear data you can use the Forecast function. 9(D) – graph exponential functions that model growth and decay and identify key features, including y-intercept and asymptote, in mathematical and real-world problems. (To be precise, it's the line. In exponential functions the variable is in the exponent, like y=3ˣ. This tool enumerates possible states and calculates overall system reliability (probability of success). ARCHIVED: In Microsoft Excel, how can I make a log-log graph? This content has been archived , and is no longer maintained by Indiana University. The Microsoft Excel GROWTH function returns the predicted exponential growth based on existing values provided. You can add trendlines to scatter plot, line, column, and bar charts. Enter the items you want to be able to select from the dropdown. A fact not so surprising from a company whose initial investors were from Google and Facebook. Essentially, today’s EMA is a function of today’s closing price and yesterday’s EMA. Introduction. Fetching contributors. Trendlines are often helpful for financial or scientific data. Add a trendline to a Chart. initial amount of the substance b. MuPAD® notebooks will be removed in a future release. Your decisions make you. 91 \times 10^{12} 2 n = 3. How to Create a Jeopardy Game to Use in you Geometry Classes!. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. ; If you work with exponential data you can use the Growth function to forecast and an exponential trends can also be visualized in a Excel chart. The rate of growth/decay is. Google Sheets supports autocomplete for custom functions much like for built-in functions. 6 Exponential and Logarithmic Functions. Heyo Hacker, very excited to announce that Hacker Noon has joined the Google Cloud for Startups program, which provides high growth companies cloud credits (up to $100K), technical support, and access to an engaging startup community. To accomplish both of these I created a function that can be placed in the header and returns an array of results, including the header itself, of a moving average of the value from another column. It would be great if data entered on contact forms could automatically sync with a google sheet. Learn line graphs at your own pace. Exponential Growth Worksheet 1. Antonyms for exponentially. Exponential Equations Brain Genie Create Writing a Exponential Growth Function given a table of Values (ask Mr. Get daily or periodic email alerts on your portfolio growth. It is good for an introduction to exponential functions in general. A linear trendline is the straight line that most closely approximates the data in the chart. To sum up, there are basically three different ways to pull real-time stock and dividend information into Google Sheets: Using Google's built-in GoogleFinance formulas Importing HTML and/or XML. 87 likes · 5 talking about this. Add a trendline to a Chart. It forces students to create a t-table (with x-values already supplied), and then a graph for an exponential growth function. Here is my dividend growth portfolio that is housed in my 401k plan. The method we're using works with all versions of Excel, and it also works with other spreadsheet applications like Google Sheets. Go to Data –> Data Validation. Google Images. Just follow these instructions to find the slope of any line graph in Google Sheets. Learn line graphs at your own pace. e or Euler's Number is the value that the following equation approaches as n grows large: (1 + (1/ n)) n. ASSESSSMENT TASK OVERVIEW & PURPOSE: In the Population Project, students will deepen their understanding of exponential growth and decay by researching and analyzing a country's population growth or decay every decade over a 100-year period. EMA Function In Google Spreadsheet. notebook January 30, 2015 Writing Exponential Growth and Decay Functions in Real Life!!! Remember this from yesterday? A(t) = a(1 + r)t Sometimes you have to use given values to find r! 15. But sometimes, you will need to do exponential calculation to a range of cells, how can you apply exponential calculation in Excel?. 4 Properties of Exponents & Rati. Over 50+ different web development, SEO, design, analytics, reporting, and productivity tools are used by us. linear growth. Google Sheets: Excluding Weekends in Conditional Formatting. An envelope. Harold's Exponential Growth and Decay Cheat Sheet 16 May 2016 Discrete Continuous 𝐴=𝑃(1+ N J) 𝑛𝑡 𝐴=𝑃 𝑟𝑡 Simple Interest: 𝐴=𝑃+ =𝑃+𝑃=𝑃(1+ N P) A = Amount after time t P = Original amount, such as principle e = The natural number (~2. Exponential Growth and Decay: Teacher Guidance This teacher guidance contains information including what prior student knowledge is required, suggested approaches and possible extension activities. It indicates the ability to send an email. Lots of business people use Google Sheets as a web-based application to do financial calculations, and lots of people use it to handle their personal finances as well, because the cloud-based spreadsheet app includes many powerful financial functions and can be used to set up a wide variety of investment spreadsheets. Most folks who use spreadsheets are used to this convention. Use this interactive quiz and printable worksheet to check your understanding of the definition of a logarithmic function and how to rewrite it as. Use Google Sheets to build your table and then use the embed function to place it on your page. Learn line graphs at your own pace. The mathematical model of exponential growth is used to describe real-world situations in population biology, finance and other fields. Take advantage of the QUERY( ) function in Google Sheets to start treating your tables like databases where you can retrieve any figure based on SQL code. The ease of use and out of the box reporting is great but I hope this post has shown you how to get even more value from your traffic data and build a powerful content analytics dashboard with minimum effort. The Windows version of Excel 2016 has a button that places an interactive forecast chart on your sheet. Seen in population growth, logistic function is defined by two rates: birth and death rate in the case of population. There is extensive help for each of the functions, including examples of how the formula can be used. High School 6. Exponential Growth and Decay Exponential decay refers to an amount of substance decreasing exponentially. where x is a variable, and a is a constant called the base of the function. You can start by taking the two dates and calculating the number of months that have elapsed using the DATEDIF() function like this: =DATEDIF(A2,C$1,"M") But you'll have to adjust the two dates because a simple DATEDIFF between today's date and cells A3 and A4 will both return 2 which isn't really right. Compound Interest; 11 Graph Theory: Introduction and Euler Paths and Circuits. Say you’re doing an audit on your content strategy… And you want to figure out if the founder interviews that you’ve been posting on your company’s blog have been performing well. Functions can be used to create formulas that manipulate data and calculate strings and numbers. Salary Circa $60k -$70kReporting to Head of Finance The Company Autopilot is an Australian founded global marketing software automation business. In this case, the slope is +1251. The GROWTH function will help you fit an ideal exponential growth trend, and give you data about said growth trend. The tutorial shows how to do time series forecasting in Excel with exponential smoothing and linear regression. where my words occur. These functions occur frequently in a wide variety of applications, such as biology, chemistry, economics, and psychology. The forecast for the next value in the time series is 74. Curious and eager to learn. Or a book containing different plans and instructions for a particular purpose, such as weddings, school, fitness, and many more. Exponential Growth and Decay Word Problems : In this section, we are going to see how to solve word problems on exponential growth and decay. But sometimes, you will need to do exponential calculation to a range of cells, how can you apply exponential calculation in Excel?. Custom functions will appear in this list if their script includes a JsDoc @customfunction tag, as in the DOUBLE() example below. In the function, 6 represents the: a. A Point-of-Sale (POS) is a key component for businesses focusing on retail and service. The Senior Accountant will work with various members of the Finance team for systems & process improvements that will optimize the Finance function of the business. Example: An ant colony is growing according to an exponential growth model with P0 = 80 and P1 = 92 where n is the of days. Plan your 60-minute lesson in Math or Algebra with helpful tips from James Bialasik. To sum up, there are basically three different ways to pull real-time stock and dividend information into Google Sheets: Using Google's built-in GoogleFinance formulas Importing HTML and/or XML. Exponential Functions In this chapter, a will always be a positive number. Choose from different chart types, like: line and bar charts, pie charts, scatter graphs, XY graph and pie charts. For example, identify percent rate of change in functions such as y = (1. " That will add the equation that Google Sheets used to calculate the trendline, and the slope of our line is the part to the left of the "*x" term. Both Google Sheets and Microsoft Excel refer to cells by their addresses using a system called 'A1 notation'. ©v K2u0y1 r23 XKtu Ntla q vSSo4f VtUweaMrneW yLYLpCF. This week's tutorial covers a number functions in Google's Spreadsheet app, Google Sheets. In the example below, we'll demonstrate how Google Sheets solves a complex formula using the order of operations. It gives us the value below which the specified percentage of observations in a group fall. r is the growth rate when r>0 or decay rate when r<0, in percent. e or Euler's Number is the value that the following equation approaches as n grows large: (1 + (1/ n)) n. Google Images. Operators’ energy costs keep rising, but efficiency measures and organizational change can lower them by 15 to 20 percent in a year, benefiting company profits and the environment. It starts off with matching the basic function, y = 2 to the power x. Web Ratna LLP believes in integrating the latest technology to ensure maximum productivity. Download a free google spreadsheet portfolio tracker for stocks, mutual funds and fixed deposits. Access Google Sheets with a free Google account (for personal use) or G Suite account (for business use). Built-in formulas, pivot tables and conditional formatting options save time and simplify common spreadsheet tasks. Google Sheets is a great free tool for creating and editing spreadsheets. com rather than a standalone portal. Match each exponential function with its graph. Exponential vs. But, if you first paste the values into a Google Sheet, then copy that table into Desmos, it works fine. For example, identify percent rate of change in functions such as y = (1. 77 percent in five years, it's cause for celebration. On thinglink. Consider a function of the form y = ba x. It forces students to create a t-table (with x-values already supplied), and then a graph for an exponential growth function. If you lack the cohort analysis capabilities you need in Google Analytics and you don’t use other analytics systems, you can build a cohort report manually in Google Sheets. Google G Suite Enterprise, which is software as a service, is more of an alternative to Microsoft Office 365. Note: We will be using Google Sheets to learn Excel basics. On the bottom of the dialog, click Options to expand the dialog. 0 packages. The rate of growth/decay is. To learn more we recommend this article. " "Here are a couple of examples, y =. The templates are ready to use with all of the basic formulas and functions pre-loaded. transform the data mathematically, so that the graph looks like a straight line (Fig 2). Example: If a population of rabbits doubles every month, we would have 2, then 4, then 8, 16, 32, 64, 128, 256, etc!. It set us the challenge of delivering an analytics function suitable for all, that adds real value to the Google Sheets user experience. Though there are many other powerful tools available, we’re already using Sheets extensively, so familiarity was an important factor. Apparently Google doesn´t support foreign language input in Sheets. Re: exponential interpolation 4 different types of curves Shg I don't think there's a natural curve that looks like this. NumXL wraps common mundane calculations such as auto-correlation, log-likelihood, model fitting/calibration, residuals diagnosis, forecasting and much more, and into a simple extension of Excel. This video tutorial belongs to the Education category which is going to show you how to match exponential graphs. A linear trendline is the straight line that most closely approximates the data in the chart. An exponential function where a > 0 and 0 < b < 1 represents an exponential decay and the graph of an exponential decay function falls from left to right. This student will infect one other student (randomly), those two students will infect one more each. Perfect for short term forecasts that predict production, sales, demand, website traffic and much more! Linear forecasting is very simple and easy to do - it’s always a great place to start. The two types of exponential functions are exponential growth and exponential decay. Population Growth Rate Calculator. It allows you to easily work in real time and collaborate with others. The GROWTH function is most useful when data values rise/fall at increasingly higher rates. Instructions can be found here. Algebra 1 IXL offers hundreds of Algebra 1 skills to explore and learn! Not sure where to start? Go to your personalized Recommendations wall and choose a skill that looks interesting!. First I run a linear regression, second is an exponential regression, and finally a polynomial regression. The AS alleles encode a key “gatekeeper” mutation designed to preserve catalytic function while enabling the kinase to be inhibited by addition of a cell-permeable ATP analog. Here we take a practical look at using weekdays in calculations, introduce "if" statements and learn how to refer dynamically to the contents of a current cell. ©v K2u0y1 r23 XKtu Ntla q vSSo4f VtUweaMrneW yLYLpCF. Specifically, the natural logarithm ln(x) is the inverse of the exponential function e x. -1-Sketch the graph of each function. There's almost an identical usage in Excel and Sheets for all of these functions, including some product, brand, brand between, forecast, growth, the EXP function, the correlation function, and range names. Calculating exponential growth to a known point. For any positive number a>0, there is a function f : R ! (0,1)called an exponential function that is deﬁned as f(x)=ax. The unrestricted growth of bacteria is an example of exponential population growth. Resource includes a project checklist, planning sheets and preparation sheets so students can use Google Draw, Google Sheets and Google Docs to map their "paying it forward" idea, record. Google Sheets only.


text:
Context

Identifying neuromuscular screening factors for anterior cruciate ligament (ACL) injury is a critical step toward large-scale deployment of effective ACL injury-prevention programs. The Landing Error Scoring System (LESS) is a valid and reliable clinical assessment of jump-landing biomechanics.

Objective

To investigate the ability of the LESS to identify individuals at risk for ACL injury in an elite-youth soccer population.

Design

Cohort study.

Setting

Field-based functional movement screening performed at soccer practice facilities.

Patients or Other Participants

A total of 829 elite-youth soccer athletes (348 boys, 481 girls; age = 13.9 ± 1.8 years, age range = 11 to 18 years), of whom 25% (n = 207) were less than 13 years of age.

Intervention(s)

Baseline preseason testing for all participants consisted of a jump-landing task (3 trials). Participants were followed prospectively throughout their soccer seasons for diagnosis of ACL injuries (1217 athlete-seasons of follow-up).

Main Outcome Measure(s)

Landings were scored for “errors” in technique using the LESS. We used receiver operator characteristic curves to determine a cutpoint on the LESS. Sensitivity and specificity of the LESS in predicting ACL injury were assessed.

Results

Seven participants sustained ACL injuries during the follow-up period; the mechanism of injury was noncontact or indirect contact for all injuries. Uninjured participants had lower LESS scores (4.43 ± 1.71) than injured participants (6.24 ± 1.75; t1215 = −2.784, P = .005). The receiver operator characteristic curve analyses suggested that 5 was the optimal cutpoint for the LESS, generating a sensitivity of 86% and a specificity of 64%.

Conclusions

Despite sample-size limitations, the LESS showed potential as a screening tool to determine ACL injury risk in elite-youth soccer athletes.

Key Points
• The Landing Error Scoring System (LESS) score may effectively identify elite-youth soccer athletes at higher risk of sustaining anterior cruciate ligament (ACL) injuries.

• Elite-youth soccer athletes with LESS scores of 5 or more were at higher risk of sustaining ACL injuries than athletes with LESS scores less than 5.

• Individuals with LESS scores of 5 or more may be targeted for ACL injury-prevention exercise programs.

Soccer is the most popular sport in the world, with more than 265 million participants. It is also one of the sports most commonly associated with athletic injury, such as to the anterior cruciate ligament (ACL). More than 25% of individuals with ACL injuries do not return to previous activity levels even after successful surgery and rehabilitation.1  Within 7 years after an ACL injury, 65% of individuals no longer play soccer.1  More than one-third of skeletally immature children also face challenges returning to sports such as soccer within 2 years of an ACL injury.2  Regardless of treatment, patients also have reported moderate to severe disability with walking (31%) and activities of daily living (44%).3,4  Adolescent female soccer players appear to be at greatest risk for negative long-term consequences, with more than 22% of these athletes undergoing either surgery to the contralateral knee or revision surgery during the 5-year period after the index surgery.5  Given that surgery and rehabilitation do not prevent long-term morbidity, the need to prevent ACL injuries in youth soccer athletes is great.

For effective injury prevention, prospective risk factors for injury should be established before preventive measures are introduced.6  Most theorized risk factors for ACL injury, such as sex, hormonal changes, notch width, and static postural alignment, are not modifiable through prevention efforts. Abnormal lower extremity biomechanics, however, are modifiable and are critical factors for study because they produce abnormal knee loading. Specifically, 3-dimensional knee loading, including knee-extension moment, proximal anterior tibial shear force, knee valgus-varus moment, and knee internal-external–rotation moment, often is implicated in ACL injury7,8  and imposes the greatest strain on the ACL.9,10  Data regarding ACL loading and injury mechanisms help clinicians understand what occurs at the time of injury.11  Prospective risk factors, however, provide information for identifying individuals at risk for sustaining injuries, potentially years before injury.11

Only a small body of evidence exists on prospective biomechanical risk factors for ACL injury. A limitation of these studies is the use of laboratory-based instrumentation and testing procedures.12,13  Whereas these instruments are the criterion standard for biomechanical measures, they are impractical in time and cost for large-scale mass screenings on the athletic field. To be feasible, a field-assessment tool for high-risk biomechanics should be brief; use minimal or inexpensive equipment; facilitate large-scale, field-based screening; and provide a valid and reliable measure of the biomechanics that predict injury.

The Landing Error Scoring System (LESS) is a field-assessment tool for identifying potentially high-risk movement patterns (“errors”) during a jump-landing maneuver. Padua et al14  demonstrated that the LESS has concurrent validity using 3-dimensional motion analysis and that good interrater and intrarater reliability can be obtained. However, few researchers15  have investigated the LESS as a prospective screening tool. Therefore, the purpose of our study was to examine the validity of the LESS (total score and individual items) in identifying individuals at risk for ACL injury in elite-youth soccer athletes. We hypothesized that higher LESS scores, representing a greater number of movement errors, would predict ACL injury in this population.

### Design and Participants

We used a prospective cohort design to evaluate the LESS as a predictor of ACL injury in elite-youth soccer athletes. Two soccer leagues (1 from North Carolina, 1 from Maryland) with under-11 to under-18 age divisions agreed to participate in the study. The 2 leagues competed in similar levels of competition and always on natural grass. All teams were simultaneously performing an injury-prevention program warm-up during this study.

A total of 829 participants (348 [42%] boys, 481 [58%] girls; age = 13.9 ± 1.8 years) were enrolled in this study: 565 (68%) from North Carolina (261 [46%] boys, 304 [54%] girls) and 264 (32%) from Maryland (87 [33%] boys, 177 [67%] girls). Of these participants, 207 (24.9%) were from 11 to 12 years of age. The group accumulated 1217 athlete-seasons of observation. At the initial test session, all participants were free from any injury or illness that prohibited competitive soccer activity. Participants and their legal guardians provided written informed assent and consent, respectively, before the initial test session. All procedures were approved by the Biomedical Institutional Review Board of the University of North Carolina at Chapel Hill.

### Procedures

Each participant completed a baseline questionnaire and movement assessment at the beginning of each soccer season between August 2006 and January 2009. A brief baseline questionnaire addressed demographics and sport-related injury history. The movement assessment consisted of a jump-landing task that was graded using the LESS by video review at a later date. Returning athletes were retested at the start of their returning seasons using identical procedures.

All participants were monitored prospectively for ACL injury from the date of their enrollment (August 2006 through May 2009). A member of the research team visited each soccer team weekly during the monitoring period to record injuries that participants sustained. Coaches and participants were instructed to identify any participant who missed a soccer-related activity during the previous week due to an injury or an unknown reason. The research team member followed up with all participants with suspected injuries. All participants with reported ACL injuries completed a specific questionnaire to verify the injury and obtain information about the circumstances of injury. All reported ACL injuries were verified during surgical reconstruction and indicated on this self-reported questionnaire. Noncontact and indirect-contact ACL injuries were defined operationally as injuries that occurred without direct contact to the lower extremity from an external source at the time of injury. We defined noncontact mechanism of injury as an injury that did not involve contact with the participant. We defined indirect-contact mechanism of injury as an injury due to contact with a body part other than the knee (ie, trunk).

Participants performed 3 trials of a standardized jump-landing task during each test session on a soccer field before practice (Figure 1). The participant began the task standing on a 30-cm-high box placed at a distance of half the body height away from a landing area, which was marked by a line on the ground. Participants were instructed to jump forward so that both limbs left the box simultaneously, to land just past the line, and to jump for maximal height immediately after landing. They practiced until they were comfortable with the task and performed it correctly. Trials were excluded and repeated if the participants jumped vertically from the box or if they did not jump for maximal height upon landing. Two digital video cameras (model DCR-HC30; Sony Corporation of America, Park Ridge, NJ) were placed 10 ft (3 m) in front of and to the right of the participants to capture frontal and sagittal images of all jump-landing trials.14,16

Figure 1.

The standardized jump-landing task consists of 2 segments: A, participant jumps down from box and lands on ground; B, participant immediately jumps vertically as high as possible.

Figure 1.

The standardized jump-landing task consists of 2 segments: A, participant jumps down from box and lands on ground; B, participant immediately jumps vertically as high as possible.

Close modal

### Data Reduction

Two research assistants (L.J.D., M.J.D.), who were blinded to injury status, graded the digital videos of all participants using the LESS and free computer software (QuickTime; Apple, Inc, Cupertino, CA). The LESS is a valid and reliable (interrater reliability: intraclass correlation coefficient [2,1] = 0.84, standard error of the mean = 0.71) clinical movement-analysis tool that evaluates specific jump-landing characteristics.14  Movements were analyzed at the initial contact frame, which was defined as the frame immediately before the foot was flat on the ground, and between initial contact and maximal knee flexion (Table 1). The LESS primarily uses a dichotomous scoring rubric to identify obvious movement errors, such as limited knee flexion or excessive medial knee displacement. Therefore, a 1-point differential in the total LESS score can be associated with moderate to large differences in certain biomechanical variables.14  A higher LESS score indicates a greater number of landing errors and consequently poorer jump-landing technique. The average LESS score from the 3 trials at each testing session was used for data analyses.

Table 1.

Operational Definitions for Individual Landing Error Scoring System Items

### Statistical Analyses

Standard statistics for screening tests were used and included receiver operator characteristic curve analyses to select a test cutpoint, followed by computation of sensitivity and specificity. We compared mean LESS scores in the injured and uninjured participants using t tests. The 1-season risk of ACL injury in athletes whose screening was positive was divided by the 1-season risk of ACL injury in athletes whose screening was negative to compute the injury risk ratio. We used SPSS software (version 16.0; SPSS Inc, Chicago, IL) and SAS software (version 9.2; SAS Institute, Cary, NC) to analyze the data. Returning athletes contributed more than 1 season to the analysis, and multiple seasons were treated as statistically independent. Exact methods, such as the Fisher test, were used for analyses and confidence intervals (CIs). The α level was set at .05.

Seven participants sustained ACL injuries during follow-up (1 boy, 6 girls; age = 14.9 ± 0.7 years; age range, 13 to 17 years; Table 2). The mechanism of injury was indirect contact for 3 injuries and noncontact for 4 injuries. All injuries occurred during games. No participant who sustained an ACL injury had reported a previous lower extremity injury on the preseason baseline questionnaire.

Table 2.

Descriptive Characteristics of Participants Sustaining Anterior Cruciate Ligament Injury After Baseline Landing Error Scoring System Testing

The receiver operator characteristic analysis identified a LESS cutpoint of 5.17 as having optimal screening properties: 86% sensitivity and 71% specificity (Figure 2). The area under the curve was 0.78 (95% CI = 0.61, 0.95). The LESS is scored as a continuous variable; however, we do not consider that the LESS can be reliably and reproducibly scored to 1 decimal place in a clinical setting. Therefore, we selected the nearest whole number (5.00) as a clinically meaningful cutpoint. Use of the 5.00 cutpoint yielded the same sensitivity (86%; 95% CI = 42%, 99%) but reduced the specificity to 64% (95% CI = 62%, 67%). Given the very low incidence of ACL injury, the positive predictive value (PPV) was low (1.4%; 95% CI = 0.6%, 3.1%), whereas the negative predictive value (NPV) was high (99.8%; 95% CI = 99.1%, 99.9%).

Figure 2.

Receiver operator characteristic curve derived for the total Landing Error Scoring System (LESS) score and anterior cruciate ligament injury using different cutpoints. Using a LESS cutpoint of 5.17 revealed a sensitivity of 86% and a specificity of 71%. The area under the receiver operator characteristic curve was 0.78 (95% confidence interval = 0.61, 0.95).

Figure 2.

Receiver operator characteristic curve derived for the total Landing Error Scoring System (LESS) score and anterior cruciate ligament injury using different cutpoints. Using a LESS cutpoint of 5.17 revealed a sensitivity of 86% and a specificity of 71%. The area under the receiver operator characteristic curve was 0.78 (95% confidence interval = 0.61, 0.95).

Close modal

### The ACL Injury Risk

The 1-season risk of ACL injury in the overall sample was 0.58% (95% CI = 0.23%, 1.18%). The 1-season risk was 1.37% (95% CI = 0.50%, 2.96%) in athletes with preseason LESS scores of 5 or more and 0.13% (95% CI = 0.01%, 0.71%) in athletes with preseason LESS scores less than 5. Thus, the 1-season risk difference between a LESS score of 5 or more and less than 5 was 1.24% (95% CI = 0.12%, 2.36%; P = .01). The risk ratio for a LESS score of 5 or more compared with a score of less than 5 was 10.7. A greater LESS score was associated monotonically with a greater 1-season risk of ACL injury in elite-youth soccer (Table 3).

Table 3.

Preseason Landing Error Scoring System Score by In-Season Incidence of Anterior Cruciate Ligament Injurya

### The LESS Score by Injury Status

The mean LESS score was higher in athletes who sustained ACL injuries (6.24 ± 1.75; 95% CI = 4.62, 7.86) than in athletes who did not sustain ACL injuries (4.43 ± 1.71; 95% CI = 4.34, 4.53) (t1215 = −2.784, P < .005). We compared the item errors on the LESS scale between injured and uninjured participants to determine the individual items driving the difference in mean LESS scores. The most predictive items were trunk-flexion displacement, hip-flexion displacement, joint displacement, trunk flexion: initial contact, foot position: external rotation, and knee-flexion displacement (Table 4).

Table 4.

Preseason Landing Error Scoring System Errors by In-Season Incidence of Anterior Cruciate Ligament Injury

Validating field screening tools for identifying high-risk individuals for noncontact and indirect-contact ACL injuries is an important step in preventing these injuries in youth soccer players. We examined the predictive ability of a field-expedient assessment of movement quality during a jump landing. Our most important finding was that elite-level youth soccer athletes with LESS scores of 5 or more were at greater risk (1.2% risk difference) of sustaining noncontact or indirect-contact ACL injuries than their counterparts with LESS scores below 5.

Almost no literature exists regarding prospective biomechanical screening for ACL injury. Increased external-knee abduction (valgus) moment during a drop landing has been reported as a prospective risk factor for ACL injury, with a sensitivity and specificity of 78% and 67%, respectively.13  Our observations agreed with this research by demonstrating that movement quality is also a biomechanical risk factor predictive of noncontact or indirect-contact ACL injury, with very similar levels of sensitivity (86%) and specificity (64%). Confirmation in subsequent reports that the field-based LESS has a predictive capacity similar to that of laboratory instrumentation would be a major step forward in ACL injury-prevention screening. Using laboratory equipment is expensive, demands technical expertise, and is time consuming. In contrast, the LESS requires approximately $300 to$400 to be spent on equipment, a few hours of administration training, 1 to 2 minutes of athlete time per individual screened, and fewer than 5 minutes per athlete to grade the LESS. This time may be reduced by scoring the LESS in real time, which has been shown to be reliable.16  However, no research exists regarding the injury-prediction validity of the LESS in real time.

Our preliminary study showed that the LESS is not a perfect screening tool. Its PPV of 1.4% at first glance seems unacceptable. However, the PPV of any ACL screening test will always be low because the rate of ACL injury is very low, even in reportedly high-risk populations. For instance, assuming an ACL incidence of 1 per 1000 athletes, a screening test with 99% sensitivity and 99% specificity would have a PPV of only 10% simply due to the low incidence of the injury. Similarly, the NPV for a screening test of a low-incidence condition will always be mathematically very favorable. For instance, in our population of elite-youth soccer athletes, a LESS score below 5.0 had an NPV of 99.9%. Given that PPV and NPV are highly dependent on incidence, the best measures of screening efficacy are sensitivity and specificity, which are not influenced by injury incidence.

Most researchers have shown that ACL injury-prevention programs pose no risk to participants. In relation to our results, these programs have also been shown to specifically reduce LESS scores below the cutpoint threshold that we detected for injury-risk susceptibility.17,18  However, these programs require large investments of athlete time, and most successful programs involve professional supervision, which results in substantial logistical constraints and monetary cost. The LESS can be performed quickly with minimal expense and minimal technical expertise. Given that ACL injury-prevention programs are not harmful, one would always want to maximize sensitivity over specificity in designing a tool for screening youth athletes to focus limited injury-prevention resources on the portion of the population (roughly one-third in our study) at high risk for ACL injury. The ability to focus injury-prevention resources would represent an important step forward for injury-prevention efforts in large populations with substantial time and resource constraints, such as youth athletic leagues, public school systems, and military training populations.

The LESS score is quantified as the sum of 17 possible individual movement errors during a jump-landing task. A limitation of the LESS score is that all identified movement errors are grouped into a single overall score, which could comprise any combination of individual specific movement errors. In our study, the most predictive individual items for ACL injury were limited trunk-flexion displacement and average lower extremity (hip, knee, and ankle) joint displacement. However, these results should be interpreted with caution because they are based on a small number of injured athletes in a specific population. Furthermore, the LESS score is affected by age and sex. We caution that use of a single cutpoint (ie, 5) may not be appropriate for all ages and sexes. Additional studies with larger sample sizes are needed to explore whether age-specific or sex-specific cutpoints would be appropriate and would enhance the predictive capacity of the LESS.

Our observations are not consistent with those reported in a well-designed case-control study conducted by Smith et al,15  who screened high school (n = 2910) and collegiate (n = 966) athletes with the LESS. Twenty-eight of these individuals sustained noncontact ACL injuries (9 males, 19 females), and 64 matched control participants were included in the final data analysis, which showed no relationship between ACL injury risk and LESS score. Whereas these researchers did not report any associations, 2 observations suggested a relationship. Athletes with LESS scores of 6 or more tended to be at higher risk than athletes with LESS scores of 4 or below (males: P = .08, females: P = .06).

Some differences between our study and the study of Smith et al15  may account for the lack of agreement between studies. We focused on youth soccer athletes (age = 13.9 ± 1.8 years; 25% from 11 to 12 years of age), whereas Smith et al15  studied a diverse multisport group that included both high school and college-aged athletes (age = 18.3 ± 2 years). The LESS was not predictive of ACL injury when the sample was limited to high school athletes; however, the average age of the high-school athletes was 16.9 ± 1.1 years,15  which was greater than the average age of athletes in our study. The natural decline of LESS scores in older athletes was likely an effect of maturation or “selection effect” in competitive sports and might have limited the ability of the LESS to predict ACL injury in an older population. Smith et al15  also removed individuals with histories of ACL injury from the analysis. Previously, ACL-injured individuals were known to be at high risk for ACL injury1921  and to have faulty movement patterns.2224  Removing these athletes may have further limited the predictive ability of the LESS in this population because many individuals who could have been detected by the LESS might have been filtered from the source population.

Another factor to consider when comparing our observations and those of Smith et al15  is differences in sport or athlete type between studies. We examined only soccer athletes, whereas Smith et al15  included basketball, field hockey, football, gymnastics, lacrosse, rugby, soccer, and volleyball athletes. The ACL injury rates are higher in soccer than in sports such as field hockey, gymnastics, lacrosse, and volleyball.25  In addition, direct-contact ACL injuries are more common in sports such as rugby and American football than in soccer. Predicting ACL-injury risk in sports with lower injury rates or more direct-contact injuries may influence the ability to predict ACL injury using the LESS. By focusing on a high-risk group for ACL injury, we eliminated the potential confounding factors of age and sport type. Finally, the differences in the 2 datasets may simply reflect random variation due to the small numbers of ACL injuries (28, Smith et al15 ; and 7, current study) observed in these 2 studies.

Our results suggest that the LESS may have screening potential for ACL injury in elite-youth soccer athletes. These observations may have implications for ACL injury-prevention efforts in youth soccer because the LESS is a field assessment of movement quality that requires no laboratory-based instrumentation. Ideally, one would want to use movement screening programs that identify individuals at risk for ACL injury as young as possible to allow time for preventive training to correct high-risk movement patterns before the peak ages for ACL injury risk (age range, 15 to 18 years). Whereas a high LESS score may lack precision in identifying which athlete will sustain an ACL injury, the LESS score may be effective in separating athletes into high-risk and low-risk subgroups. These data should be interpreted with caution due to the small number of ACL injury cases in this analysis. However, given the devastating consequences of ACL injury, the paucity of other ACL screening tests, the favorable feasibility profile of the LESS, and the costs associated with large-scale ACL injury-prevention efforts, the screening value of the LESS needs further investigation in youth-sport populations.

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text:
# 4.5: Fourier Series Approximation of Signals

Learning Objectives

• Fourier series approximations.

It is interesting to consider the sequence of signals that we obtain as we incorporate more terms into the Fourier series approximation of the half-wave rectified sine wave. Define sK(t) to be the signal containing K+1 Fourier terms.

$s_{K}(t)=a_{0}+\sum_{k=1}^{K}a_{k}\cos \left ( \frac{2\pi kt}{T} \right )+ \sum_{k=1}^{K}b_{k}\sin \left ( \frac{2\pi kt}{T} \right )$

Fig. 4.5.1 below shows how this sequence of signals portrays the signal more accurately as more terms are added.

Fig. 4.5.1 The Fourier series spectrum of a half-wave rectified sinusoid is shown in the upper portion.

The index indicates the multiple of the fundamental frequency at which the signal has energy. The cumulative effect of adding terms to the Fourier series for the half-wave rectified sine wave is shown in the bottom portion. The dashed line is the actual signal, with the solid line showing the finite series approximation to the indicated number of terms, K+1.

We need to assess quantitatively the accuracy of the Fourier series approximation so that we can judge how rapidly the series approaches the signal. When we use a K+1 -term series, the error—the difference between the signal and the K+1 -term series—corresponds to the unused terms from the series.

$\varepsilon _{K}(t)=\sum_{k=K+1}^{\infty }a_{k}\cos \left ( \frac{2\pi kt}{T} \right )+ \sum_{k=K+1}^{\infty }b_{k}\sin \left ( \frac{2\pi kt}{T} \right )$

To find the rms error, we must square this expression and integrate it over a period. Again, the integral of most cross-terms is zero, leaving

$rms(\varepsilon _{K})=\sqrt{\frac{1}{2}\sum_{k=K+1}^{\infty }a_{k}^{2}+b_{k}^{2}}$

Fig. 4.5.2 shows how the error in the Fourier series for the half-wave rectified sinusoid decreases as more terms are incorporated. In particular, the use of four terms, as shown in the bottom plot of Fig. 4.5.1 has a rms error (relative to the rms value of the signal) of about 3%. The Fourier series in this case converges quickly to the signal.

Fig.4.5.2 Approximation error for a half-wave rectified sinusoid

The rms error calculated according to  the equation above is shown as a function of the number of terms in the series for the half-wave rectified sinusoid. The error has been normalized by the rms value of the signal.

We can look at Fig. 4.5.3 to see the power spectrum and the rms approximation error for the square wave.

Fig. 4.5.3 The upper plot shows the power spectrum of the square wave, and the lower plot the rms error of the finite-length Fourier series approximation to the square wave. The asterisk denotes the rms error when the number of terms K in the Fourier series equals 99.

Because the Fourier coefficients decay more slowly here than for the half-wave rectified sinusoid, the rms error is not decreasing quickly. Said another way, the square-wave's spectrum contains more power at higher frequencies than does the half-wave-rectified sinusoid. This difference between the two Fourier series results because the half-wave rectified sinusoid's Fourier coefficients are proportional to 1/k2 while those of the square wave are proportional to 1/k. If fact, after 99 terms of the square wave's approximation, the error is bigger than 10 terms of the approximation for the half-wave rectified sinusoid. Mathematicians have shown that no signal has an rms approximation error that decays more slowly than it does for the square wave.

Exercise $$\PageIndex{1}$$

Calculate the harmonic distortion for the square wave.

Solution

Total harmonic distortion in the square wave is:

$1-\frac{1}{2}(\frac{4}{\pi })^{2}=20\%$

More than just decaying slowly, Fourier series approximation shown in Fig. 4.5.4 exhibits interesting behavior.

Fig. 4.5.4 Fourier series approximation to sq(t). The number of terms in the Fourier sum is indicated in each plot, and the square wave is shown as a dashed line over two periods.

Although the square wave's Fourier series requires more terms for a given representation accuracy, when comparing plots it is not clear that the two are equal. Does the Fourier series really equal the square wave at all values of $tt$ ? In particular, at each step-change in the square wave, the Fourier series exhibits a peak followed by rapid oscillations. As more terms are added to the series, the oscillations seem to become more rapid and smaller, but the peaks are not decreasing. For the Fourier series approximation for the half-wave rectified sinusoid above, no such behavior occurs. What is happening?

Consider this mathematical question intuitively: Can a discontinuous function, like the square wave, be expressed as a sum, even an infinite one, of continuous signals? One should at least be suspicious, and in fact, it can't be thus expressed. This issue brought Fourier much criticism from the French Academy of Science (Laplace, Lagrange, Monge and LaCroix comprised the review committee) for several years after its presentation on 1807. It was not resolved for almost a century, and its resolution is interesting and important to understand from a practical viewpoint.

The extraneous peaks in the square wave's Fourier series never disappear; they are termed Gibb's phenomenon after the American physicist Josiah Willard Gibbs. They occur whenever the signal is discontinuous, and will always be present whenever the signal has jumps.

Let's return to the question of equality; how can the equal sign in the definition of the Fourier series be justified? The partial answer is that pointwise—each and every value of t—equality is not guaranteed. However, mathematicians later in the nineteenth century showed that the rms error of the Fourier series was always zero.

$\lim_{k \to\infty }rms(\varepsilon _{K})=0$

What this means is that the error between a signal and its Fourier series approximation may not be zero, but that its rms value will be zero! It is through the eyes of the rms value that we redefine equality: The usual definition of equality is called pointwise equality: Two signals s1(t), s2(t) are said to be equal pointwise if:

$s_{1}(t)=s_{2}(t)$

for all values of t. A new definition of equality is mean-square equality: Two signals are said to be equal in the mean square if:

$rms(s_{1}-s_{2})=0$

For Fourier series, Gibb's phenomenon peaks have finite height and zero width. The error differs from zero only at isolated points—whenever the periodic signal contains discontinuities—and equals about 9% of the size of the discontinuity. The value of a function at a finite set of points does not affect its integral. This effect underlies the reason why defining the value of a discontinuous function, like we refrained from doing in defining the step function, at its discontinuity is meaningless. Whatever you pick for a value has no practical relevance for either the signal's spectrum or for how a system responds to the signal. The Fourier series value "at" the discontinuity is the average of the values on either side of the jump.

### Contributor

• ContribEEOpenStax


text:
# Help! Can you get my puzzle back?

Puzzling has a lot of really complicated stuff these days, so I just wanted to post a simple Boggle grid.

You know that game, right? You form words by starting from one letter, then taking adjacent letters moving in any of the eight directions, but you can't use the same letter twice to form a single word.

Here are the words you are looking for, they should be familiar to most of you:

And here is the grid:

Um... that was not what the grid was supposed to look like at all! Have I got one of those newfangled viruses or something?

Since we're already here, can you help me recover my puzzle? I didn't take any backups while I was making this, and I'm really desperate to remember what it was like!

Luckily, the virus or whatever it was seems to have picked colorblind-friendly colors for the image, but if that somehow is still an issue here is a text transcription, in which different letters represent different colors:

ABCBB
CDEFG
HGABI
JKGGC
LMFMF

• As a confused colourblind person. Are you saying that the letters are written on the grid ? Because I can't see a thing – user3660570 May 15 '17 at 16:43
• @user3660570 Each letter corresponds to a different color on the image. The image doesn't have any letters. – ffao May 15 '17 at 16:45
• You don't really need "nim" if you have "animals", right? (That is, "nim" gives no extra information if I understand correctly.) – Deusovi May 15 '17 at 16:55
• No, you don't. I just liked the extra free tag. – ffao May 15 '17 at 16:56
• I assume the puzzle uses standard Boggle rules? – Forklift May 15 '17 at 18:25

rtitt
imena
sartp
lgaai
fonon

Explanation:

Here's the initial grid:

ABCBB
CDEFG
HGABI
JKGGC
LMFMF

Start by counting the number of occurrences of each colour (which I'll represent using capital letters):

• A×2, B×4, C×3, D×1, E×1, F×3, G×4, H×1, I×1, J×1, K×1, L×1, M×2

and the minimum number of times we know each letter occurs in the final grid (I'll use lowercase letters for the letters in the eventual grid):

• a×3 (anagram), i×2 (termination), n×2 (termination), o×2 (nonogram), t×2 (pattern), other letters could appear once

There are a total of thirteen letters in the tags (aefgilmnoptrs), and thirteen colours on the grid, so there must be a 1-to-1 correspondence.

I started off by looking at t, o, and n. t must clearly be B or G, as no other colour allows for the adjacent ts in pattern. Looking at all the places we could possibly spell nono (from nonogram), they must either be in B and C (some way round), or else two of F/G/M.

The next letter to check was a. The requirement to spell anagram means we can't have a=C (which would force n into B with a as C, or into D which isn't a valid location for it). We also can't have a=F, as that quickly leads to a contradiction:

ABgBB    ABgBB
gDmao    gDman
HoABr or HnABr
JKoog    JKnng
Lnana    Loaoa

In other words, we've found our first letter position, a=G (immediately forcing t=B to spell pattern).

AtCtt
CDEFa
HaAtI
JKaaC
LMFMF

no now must be MF (one way round or the other). This means that p=I:

AtCtt
CDEFa
HaAtp
JKaaC
LMFMF

If M=n, then we can't spell pangram. Thus, F=n and M=o:

AtCtt
CDEna
HaAtp
JKaaC
Lonon

To be able to spell nonogram, anagram, and pangram, we need K=g:

AtCtt
CDEna
HaAtp
JgaaC
Lonon

To spell termination, we need C=i:

Atitt
iDEna
HaAtp
Jgaai
Lonon

and ADE need to spell erm in some order (for termination); to be able to fit tetris in too we need A=r, E=e, H=s:

rtitt
imena
sartp
Jgaai
Lonon

animals requires J=l, flags requires L=f:

rtitt
imena
sartp
lgaai
fonon

We can find all the words in this grid, so we know it's correct.


text:
#### Vol. 14, No. 3, 2021

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
$L^2$-boundedness of gradients of single-layer potentials and uniform rectifiability

### Laura Prat, Carmelo Puliatti and Xavier Tolsa

Vol. 14 (2021), No. 3, 717–791
##### Abstract

Let $A\left(\phantom{\rule{0.3em}{0ex}}\cdot \phantom{\rule{0.3em}{0ex}}\right)$ be an $\left(n+1\right)×\left(n+1\right)$ uniformly elliptic matrix with Hölder continuous real coefficients and let ${\mathsc{ℰ}}_{A}\left(x,y\right)$ be the fundamental solution of the PDE $divA\left(\phantom{\rule{0.3em}{0ex}}\cdot \phantom{\rule{0.3em}{0ex}}\right)\nabla u=0$ in ${ℝ}^{n+1}$. Let $\mu$ be a compactly supported $n$-AD-regular measure in ${ℝ}^{n+1}$ and consider the associated operator

${T}_{\mu }f\left(x\right)=\int {\nabla }_{x}{\mathsc{ℰ}}_{A}\left(x,y\right)\phantom{\rule{0.3em}{0ex}}f\left(y\right)\phantom{\rule{0.3em}{0ex}}d\mu \left(y\right).$

We show that if ${T}_{\mu }$ is bounded in ${L}^{2}\left(\mu \right)$, then $\mu$ is uniformly $n$-rectifiable. This extends the solution of the codimension-$1$ David–Semmes problem for the Riesz transform to the gradient of the single-layer potential. Together with a previous result of Conde-Alonso, Mourgoglou and Tolsa, this shows that, given $E\subset {ℝ}^{n+1}$ with finite Hausdorff measure ${\mathsc{ℋ}}^{n}$, if ${T}_{{\mathsc{ℋ}}^{n}{|}_{E}}$ is bounded in ${L}^{2}\left({\mathsc{ℋ}}^{n}{|}_{E}\right)$, then $E$ is $n$-rectifiable. Further, as an application we show that if the elliptic measure associated to the above PDE is absolutely continuous with respect to surface measure, then it must be rectifiable, analogously to what happens with harmonic measure.

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/apde

We have not been able to recognize your IP address 3.238.204.31 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recommendation form. Or, visit our subscription page for instructions on purchasing a subscription.

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Or, you may purchase this single article for USD 40.00:

##### Keywords
David–Semmes problem, uniform rectifiability, gradient of the single-layer potential, elliptic measure, rectifiability
##### Mathematical Subject Classification 2010
Primary: 28A75, 31B15, 35J15, 42B37
##### Milestones
Received: 26 November 2018
Revised: 5 September 2019
Accepted: 21 November 2019
Published: 18 May 2021
##### Authors
 Laura Prat Departament de Matemàtiques Universitat Autònoma de Barcelona and Centre de Recerca Matemàtica Bellaterra (Barcelona) Catalonia Carmelo Puliatti Departament de Matemàtiques and BGSMath Universitat Autònoma de Barcelona Bellaterra (Barcelona) Catalonia Xavier Tolsa ICREA Bellaterra (Barcelona) Catalonia Departament de Matemàtiques and BGSMath Universitat Autònoma de Barcelona Bellaterra (Barcelona) Catalonia


text:
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

A few months ago I attended the 2019 rstudio::conf, including the shiny train-the-trainer workshop. It was a two day workshop and it inspired me in many ways. The first day of the workshop focused on the very basics of teaching (R or anything else), and for me it put the spotlight on things I never considered before.

One of the important takeways from the workshop was how to approach educating others: preparing for a course, things you can do during the lessons, and how to self-learn and improve my own teaching methods afterwards.

This led me to create the teachR’s cheatsheet. It outlines the basics of teaching and I chose to give it the flavour of R (in the examples and illustrations within the cheatsheet).

I have contributed it to RStudio’s cheat sheet repo, so you can download it directly from: https://github.com/rstudio/cheatsheets/raw/master/teachR.pdf.

In the cheat sheet you will find three segments:

1. Preparing a new course / workshop / lesson.
2. Things you can do during the lesson itself.
3. Things you should do when the course is completed in order to improve your own teaching methods.

I previously blogged about some of the things learned at the train-the-trainer, and not everything made it to the cheat sheet, so if you’re interested you can read more here.

Here’s an example for some of the things you can find in the cheat sheet.

Designing a new course

The cheat sheet covers the various steps of designing a course, i.e.:

1. Persona analysis of your learners.
2. Defining the course’s goals using Bloom’s taxonomy.
3. Using conceptual maps to grasp what the the course should look like and what related terms/materials should appear.
4. Writing the final exam, the slides, check-ins and faded examples.

Here are some examples relating to 1-2:

Persona analysis

Take a while to understand and characterize your learners: are the novice? advanced? false experts?

What are the learner’s goals from the course? what prior knowledge you can assume (and what not), and do they have any special needs.

If end up with too many personas anticipate trouble – it’s hard to accomodate a diverse crowd, what are you going to miss out on?

Define goals using Bloom’s taxonomy

Bloom’s taxonomy illustrates the levels of learning new concepts or topics.

The Vanderbilt University Center for Teaching has a nice illustration for it.

You can visit the Vanderbilt website for a more thorough explanation about the taxonomy, but suffice it to say that “remember” is the most basic form of acquired knowledge, and the highest levels (at the top of the pyramid) are evaluate and create (being able to evaluate someone else’s work, or create your own noval work).

If we translate that to R, “remember” might translate to: “learners will be able to state the main packages in tidyverse and their purpose” versus “create” which in that context would translate to: “learners will be able to contribute to a tidyverse packages or create their own tidy package.” You can see that the first is something you can teach an R beginner but the latter is much more complex and can be mastered by an advanced useR.

During the course

Some tips I learned at the train-the-trainer workshop, for when you are during the lesson itself.

Sticky notes

At the start of the lesson, give each learner three sticky notes (green, red, and blue). The learners put them on their computers according to their progress:

• Green = I’m doing fine / finished teh exercise.
• Red = Something is wrong, I need help.
• Blue = I need a break

If you see a lot of greens – try to up the pace. If you see a lot of reds, maybe take it easier.

Check-ins

Try to set a few check-ins every hour, to evaluate the progress and make sure that the learners are “with you”. You can even use some kind of online surveying tool to turn this into a “game”.

After the course

Make sure you debrief properly, and learn from your experience. Use surveys to collect feedback. Also measure the time each chapter really takes you, so you can better estimate the time for each type of lesson.

Conclusion

Teaching can be challenging, but it is also rewarding and fun.

It is important to come well prepared, and this cheat sheet can help you checklist what you need to do: https://github.com/rstudio/cheatsheets/raw/master/teachR.pdf

Teaching is an iterative process in which you can keep improving each time, if you measure and learn from your mistakes.


text:
[This article was first published on R on The broken bridge between biologists and statisticians, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

# A measure of joint variability

In statistics, dependence or association is any statistical relationship, whether causal or not, between two random variables or bivariate data. It is often measured by the Pearson correlation coefficient:

$\rho _{X,Y} =\textrm{corr} (X,Y) = \frac {\textrm{cov}(X,Y) }{ \sigma_X \sigma_Y } = \frac{ \sum_{1 = 1}^n [(X – \mu_X)(Y – \mu_Y)] }{ \sigma_X \sigma_Y }$

Other measures of correlation can be thought of, such as the Spearman $$\rho$$ rank correlation coefficient or Kendall $$\tau$$ rank correlation coefficient.

# Assumptions for the Pearson Correlation Coefficient

The Pearson correlation coefficients makes a few assumptions, which should be carefully checked.

1. Interval-level measurement. Both variables should be measured on a quantitative scale.
2. Random sampling. Each subject in the sample should contribute one value on X, and one value on Y. The values for both variables should represent a random sample drawn from the population of interest.
3. Linearity. The relationship between X and Y should be linear.
4. Bivarlate normal distribution. This means that (i) values of X should form a normal distribution at each value of Y and (ii) values of Y should form a normal distribution at each value of X.

# Hypothesis testing

It is possible to test whether $$r = 0$$ against the alternative $r 0$. The test is based on the idea that the amount:

$T = \frac{r \sqrt{n – 2}}{\sqrt{1 – r^2}}$ is distributed as a Student’s t variable.

Let’s take the two variables ‘cyl’ and ‘mpg’ from the ‘mtcars’ data frame. The correlation is:

r <- cor(mtcars$cyl, mtcars$gear)
r
## [1] -0.4926866

The T statistic is:

T <- r * sqrt(32 - 2) / sqrt(1 - r^2)
T
## [1] -3.101051

The p-value for the null is:

2 * pt(T, 30)
## [1] 0.004173297

which is clearly highly significant. The null can be rejected.

As for hypothesis testing, it should be considered that the individuals where couple of measurements were taken should be independent. If they are not, the t test is invalid. I am dealing with this aspect somewhere else in my blog.

# Correlation in R

We have already seen that we can use the usual function ‘cor(matrix, method=)’. In order to obtain the significance, we can use the ‘rcorr()’ function in the Hmisc package

# Correlations with significance levels
library(Hmisc)
corr2 <- rcorr(as.matrix(mtcars), type="pearson")
print(corr2$r, digits = 2) ## mpg cyl disp hp drat wt qsec vs am gear carb ## mpg 1.00 -0.85 -0.85 -0.78 0.681 -0.87 0.419 0.66 0.600 0.48 -0.551 ## cyl -0.85 1.00 0.90 0.83 -0.700 0.78 -0.591 -0.81 -0.523 -0.49 0.527 ## disp -0.85 0.90 1.00 0.79 -0.710 0.89 -0.434 -0.71 -0.591 -0.56 0.395 ## hp -0.78 0.83 0.79 1.00 -0.449 0.66 -0.708 -0.72 -0.243 -0.13 0.750 ## drat 0.68 -0.70 -0.71 -0.45 1.000 -0.71 0.091 0.44 0.713 0.70 -0.091 ## wt -0.87 0.78 0.89 0.66 -0.712 1.00 -0.175 -0.55 -0.692 -0.58 0.428 ## qsec 0.42 -0.59 -0.43 -0.71 0.091 -0.17 1.000 0.74 -0.230 -0.21 -0.656 ## vs 0.66 -0.81 -0.71 -0.72 0.440 -0.55 0.745 1.00 0.168 0.21 -0.570 ## am 0.60 -0.52 -0.59 -0.24 0.713 -0.69 -0.230 0.17 1.000 0.79 0.058 ## gear 0.48 -0.49 -0.56 -0.13 0.700 -0.58 -0.213 0.21 0.794 1.00 0.274 ## carb -0.55 0.53 0.39 0.75 -0.091 0.43 -0.656 -0.57 0.058 0.27 1.000 print(corr2$P, digits = 2)
##          mpg     cyl    disp      hp    drat      wt    qsec      vs
## mpg       NA 6.1e-10 9.4e-10 1.8e-07 1.8e-05 1.3e-10 1.7e-02 3.4e-05
## cyl  6.1e-10      NA 1.8e-12 3.5e-09 8.2e-06 1.2e-07 3.7e-04 1.8e-08
## disp 9.4e-10 1.8e-12      NA 7.1e-08 5.3e-06 1.2e-11 1.3e-02 5.2e-06
## hp   1.8e-07 3.5e-09 7.1e-08      NA 1.0e-02 4.1e-05 5.8e-06 2.9e-06
## drat 1.8e-05 8.2e-06 5.3e-06 1.0e-02      NA 4.8e-06 6.2e-01 1.2e-02
## wt   1.3e-10 1.2e-07 1.2e-11 4.1e-05 4.8e-06      NA 3.4e-01 9.8e-04
## qsec 1.7e-02 3.7e-04 1.3e-02 5.8e-06 6.2e-01 3.4e-01      NA 1.0e-06
## vs   3.4e-05 1.8e-08 5.2e-06 2.9e-06 1.2e-02 9.8e-04 1.0e-06      NA
## am   2.9e-04 2.2e-03 3.7e-04 1.8e-01 4.7e-06 1.1e-05 2.1e-01 3.6e-01
## gear 5.4e-03 4.2e-03 9.6e-04 4.9e-01 8.4e-06 4.6e-04 2.4e-01 2.6e-01
## carb 1.1e-03 1.9e-03 2.5e-02 7.8e-07 6.2e-01 1.5e-02 4.5e-05 6.7e-04
##           am    gear    carb
## mpg  2.9e-04 5.4e-03 1.1e-03
## cyl  2.2e-03 4.2e-03 1.9e-03
## disp 3.7e-04 9.6e-04 2.5e-02
## hp   1.8e-01 4.9e-01 7.8e-07
## drat 4.7e-06 8.4e-06 6.2e-01
## wt   1.1e-05 4.6e-04 1.5e-02
## qsec 2.1e-01 2.4e-01 4.5e-05
## vs   3.6e-01 2.6e-01 6.7e-04
## am        NA 5.8e-08 7.5e-01
## gear 5.8e-08      NA 1.3e-01
## carb 7.5e-01 1.3e-01      NA

We could also use these functions with two matrices, to obtain the correlations of each column in one matrix with each column in the other

# Correlation matrix from mtcars
x <- mtcars[1:3]
y <- mtcars[4:6]
cor(x, y)
##              hp       drat         wt
## mpg  -0.7761684  0.6811719 -0.8676594
## cyl   0.8324475 -0.6999381  0.7824958
## disp  0.7909486 -0.7102139  0.8879799

# Relationship to slope in linear regression

The correlation coefficient and slope in linear regression bear some similarities, as both describe how Y changes when X is changed. However, in correlation, we have two random variables, while in regression we have Y random, X fixed and Y is regarded as a function of X (not the other way round).

Without neglecting their different meaning, it may be useful to show the algebraic relationship between the correlation coefficient and the slope in regression. Let’s simulate a dataset with two variables, coming from a multivariate normal distribution, with means respectively equal to 10 and 2, and variance-covariance matrix of:

library(MASS)
cov <- matrix(c(2.20, 0.48, 0.48, 0.20), 2, 2)
cov
##      [,1] [,2]
## [1,] 2.20 0.48
## [2,] 0.48 0.20

We use the ‘mvrnomr()’ function to generate the dataset.

set.seed(1234)
dataset <- data.frame( mvrnorm(n=10, mu = c(10, 2), Sigma = cov) )
names(dataset) <- c("X", "Y")
dataset
##            X        Y
## 1  11.756647 2.547203
## 2   9.522180 2.199740
## 3   8.341254 1.862362
## 4  13.480005 2.772031
## 5   9.428296 1.573435
## 6   9.242788 1.861756
## 7  10.817449 2.343918
## 8  10.749047 2.451999
## 9  10.780400 2.436263
## 10 11.480301 1.590436

The correlation coefficient and slope are as follows:

r <- with(dataset, cor(X, Y))
b1 <- coef( lm(Y ~ X, data=dataset) )[2]
r
## [1] 0.6372927
b1
##         X
## 0.1785312

The equation for the slope is:

$b_1 = \frac{ \sum_{i = 1}^n \left[ ( X-\mu_X )( Y-\mu_Y )\right] }{ \sigma^2_X }$

From there, we see that:

$r = b_1 \frac{\sigma_X}{ \sigma_Y }$

and:

$b_1 = r \frac{\sigma_Y}{\sigma_X}$

Indeed:

sigmaX <- with(dataset, sd(X) )
sigmaY <- with(dataset, sd(Y) )
b1 * sigmaX / sigmaY
##         X
## 0.6372927
r * sigmaY / sigmaX
## [1] 0.1785312

It is also easy to see that the correlation coefficient is the slope of regression of standardised Y against standardised X:

Yst <- with(dataset, scale(Y, scale=T) )
summary( lm(Yst ~ I(scale(X, scale = T) ), data = dataset) )
##
## Call:
## lm(formula = Yst ~ I(scale(X, scale = T)), data = dataset)
##
## Residuals:
##       Min        1Q    Median        3Q       Max
## -0.082006 -0.067143 -0.036850  0.009214  0.237923
##
## Coefficients:
##                          Estimate Std. Error t value Pr(>|t|)
## (Intercept)            -5.633e-18  3.478e-02   0.000   1.0000
## I(scale(X, scale = T))  1.785e-01  7.633e-02   2.339   0.0475 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.11 on 8 degrees of freedom
## Multiple R-squared:  0.4061, Adjusted R-squared:  0.3319
## F-statistic: 5.471 on 1 and 8 DF,  p-value: 0.04748

# Intra-class correlation (ICC)

It describes how strongly units in the same group resemble each other. While it is viewed as a type of correlation, unlike most other correlation measures it operates on data structured as groups, rather than data structured as paired observations. The intra-class correlation coefficient is:

$IC = {\displaystyle {\frac {\sigma _{\alpha }^{2}}{\sigma _{\alpha }^{2}+\sigma _{\varepsilon }^{2}}}.}$

where $$\sigma _{\alpha }^{2}$$ is the variance between groups and $$\sigma _{\varepsilon }^{2}$$ is the variance within a group (better, the variance of one observation within a group). The sum of those two variances is the total variance of observations. In words, the intra-class correlation coefficient measures the joint variability of subjects in the same group (that relates on how groups are different from one another), with respect to the total variability of observations. If subjects in one group are very similar to one another (small $$\sigma_{\varepsilon}$$) but groups are very different (high $$\sigma_{\alpha}$$), the ICC is very high.

The existence of grouping of residuals is very important in ANOVA, as it means that independence is violated, which calls for the use of mixed model.

But … this is a totally different story …


text:
Sum of powers of unity root

Let $G_n(1)=\{q\in Z_n|(q,n)=1\}$ $S\subset G_n (1)$ (strict subset). Prove that:
$
where $\omega_n=e^{i\frac{2\pi}n}.$


text:
# I'm so behind in physics, Circular Rotation help.

1. Oct 2, 2011

### GRice40

1. The problem statement, all variables and given/known data
The "Giant Swing" at a county fair consists of a vertical central shaft with a number of horizontal arms attached at its upper end. Each arm supports a seat suspended from a 5.00m long cable, the upper end of which is fastened to the arm at a point 3.00 m from the central shaft. Find the time of one rotation if the angle produced from the cable connecting to the arm is 30 degrees from vertical.

2. Relevant equations
Circumference= distance = 2 (pi) R
Acceleration = V^2/R
F = m(V^2/R)

3. The attempt at a solution
I honestly have no clue how to do this one. I keep working myself in circles, which doesn't seem to be getting anything accomplished. Any help on where I should start?

2. Oct 2, 2011

### issacnewton

is there any figure given with this.......i have trouble imagining this....

3. Oct 2, 2011

### Staff: Mentor

It's always best to start with a diagram. Draw a cross section showing the vertical pole, a cross beam, and one cable (at 30 degrees from vertical) with a mass on the end. Identify the center of rotation of the mass and the radius of the circle it will follow.

After that you'll be drawing an FBD and working out the components of the accelerations acting.

4. Oct 2, 2011

### GRice40

There was a figure with the problem, but I can't save it to upload here.

I've been working through it, and it seems that if I could find the velocity, I'd be able to solve it.

Any tips on how to go about finding velocity?

5. Oct 2, 2011

### GRice40

Here's what I have so far:

Radius: 3 + 5sin(30) = 5.5 m

V = d/t = 34.56/t

Circumference = 2(pi)(5.5)= 34.56

Acceleration = V^2/R = (34.56/t)^2/5.5

That's about as far as I can get without another variable that I know =/

6. Oct 2, 2011

### issacnewton

without figure I can't help..... difficult to see the situation

7. Oct 2, 2011

### Staff: Mentor

Good so far! Now, ask yourself why the cables are hanging at an angle of 30° to the vertical rather than flying out horizontally.

8. Oct 2, 2011

### GRice40

Hope this works, we'll see!

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9. Oct 2, 2011

### GRice40

Hmm, gravity pulling them down would be my guess?

10. Oct 3, 2011

### issacnewton

that helps....... set up free body diagram for the girl........there is tension in the wire

11. Oct 3, 2011

### GRice40

I assume that because she is staying in place in the y direction, that the y direction is in equilibrium.

So here's what I have:

Tcos(30) = m * (9.8)

(Another possibility that is coming to mind is: Force(out) + W = T)

(Here's the other possibility running through my mind: Force(out) = Tsin(30) for the x direction)

12. Oct 3, 2011

### PeterO

It would be useful to know the other formula for centripetal acceleration

Ac = V2/R or 4∏2R/T2

Where T is the period.

13. Oct 3, 2011

### issacnewton

which component of tension is acting as a centripetal force ?

14. Oct 3, 2011

### GRice40

The centripetal force should be Tsin(30), I think?

15. Oct 3, 2011

### PeterO

That's true. Did you read post #12

16. Oct 3, 2011

### issacnewton

yes........ we also know that the centripetal force is

$$\frac{mv^2}{r}$$

and you already said $$T\cos(30)=mg$$ so can you manipulate these

17. Oct 3, 2011

### GRice40

Ok here's what I got:

F(out) = m*a1 = Tsin(30)

W = M*a2 = Tcos(30)

So if you divide the two you get: (m*a1/m*a2) = Tsin(30)/Tcos(30)
which reduces to
(a1/a2) = sin(30)/cos(30) or (a1/9.8) = .577

.577(9.8) = a1 = 5.66 m/s^2

So 5.66 = 4(pi)^2R/t^2,
5.66 = 217.13/t^2
t^2 = 217.13/5.66
t = sqrt(217.13/5.66) = 6.19 seconds

That's what the answer is supposed to be. Thank you so much guys, I've been working on that problem for HOURS!

18. Oct 3, 2011

### issacnewton

don't forget to put figures when they are provided......without the figures its difficult for people here to understand........always post the problem AS it appears in the text.....
don't psycho-analyse the problem and write what YOU think the problem statement is...

you will likely get more help when you present the problem and your attempted work in organised manner.....

good luck $$\smile$$

19. Oct 3, 2011

### PeterO

You have the right answer, but I do not like the bit I highlited red. It looks like you think there is an outward force - the centrifugal force. There is no such force. The net force on the people on the device is INWARDS - the centripetal Force.

20. Oct 3, 2011

### vela

Staff Emeritus
From what you've written, it also looks like you don't understand how to apply ƩF=ma. What are a1 and a2 supposed to represent? The horizontal and vertical components of acceleration?

Back in post #11, you said correctly that the acceleration in the vertical direction was 0 because the girl wasn't moving vertically, so the equations should be of the form:
\begin{align*}
\sum F_x &= ma_x = m\left(\frac{v^2}{r}\right) \\
\sum F_y &= ma_y = m(0)
\end{align*}
where ax=v2/r is the centripetal acceleration and ay=0.


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