![]() You can this online gif editor to make a GIF with dragging and dropping. If you have already collected images in png, jpeg and gif format and now want to create one interesting animated GIF image by merging them then can help you better. It can directly upload your resulting GIF files on popular social media sites. Those who are searching for a super quick way to develops GIFs with additional watermarks and image quality settings can prefer to move t. And you can choose different styles and animations when adding captions when editing GIF. You can adjust the duration of each frame/image, and add captions, stickers, filters to the GIF file or draw on it. You can add multiple images or paste the YouTube, Vimeo or Giphy URL to upload the video. Giphy allows you to make GIF slideshow with images as well as videos. This popular GIF search engine and interactive design tool can develop results of your choice instantly. Users can also enjoy live preview of all changes before exporting created GIF image to device, so it is a great GIF editor online to use. This photo to GIF online maker can easily process almost all media file formats like JPG, GIF and PNG so that interactive GIFs can be created. See, make gif online with Filmora online maker is easy. It may take a while according to your file size. Just position your text field to anywhere.Īfter selecting the format you want, click create to export this GIF. Besides, you can change font type, size and color. You can enter your text from inside and outside. If your uploaded file is image, you can flip and rotate it first. ![]() Drag the slider to left or right to get it. After it is uploaded, you need to trim one part you like. You can drag gif resource to the center of the page, or directly click it to upload. You can make a GIF from video or image, and Filmora meme maker supports all. To make a gif online, you need to import your media and gif material to Filmora meme maker. You just need to open it and then follow the steps below. We will let you know the steps to make a GIF online with Filmora meme maker. Some people are wondering how do I make a GIF online? Well, don't worry. Just with a few clicks, your GIF is done. You can also put text inside or outside in GIF. You can change text size, color and position to create caption when you are using this online GIF editor. It is totally FREE to use and no sign up. It helps you make image, GIF and video memes directly without watermark. Filmora meme makerįilmora meme maker is your ultimate solution. The article below includes the list of best online GIF maker tools that you can use to make GIF from images or videos with ease online. Users need not waste time uploading or downloading images because its drag and drop tools make everything much easier and laborious tasks can be completed within a fraction of seconds.Ĭheck this video tutorial to know more details:.The list of commonly used editing tools includes overlays, elements and text etc. It is possible to edit already created GIF images in Filmora and you can access inbuilt tools for this purpose.Users simply need to download their tool and start working offline. It does not demand any active internet connection.Share the GIF Video to YouTube or Vimeo directly.Export images and videos to GIF format directly, or save the gif to the mobile device for sharing.This GIF editor can process all popular media file formats with ease and it works perfectly on Windows as well as Mac devices.Below are few incredible features of Wondershare Filmora. ![]() Besides, various stickers and overlays are also available to use. If you are looking forward to working on GIF images and videos then probably Filmora can solve all your troubles. Wondershare Filmora is often recommended due to its rich feature set that allows easy customization, editing and conversion. Recommendation – Filmora: Easy GIF Maker Software Free Download for PC and Mac The following list are great makers to edit GIF from video or image. ![]() We need an easy to use and free GIF editor online to use for quickly making a GIF and share it with friends to have fun. However, there are still some GIF makers online that are not easy to use. Most of the online GIF editors are available for free and they allow users to develop customized interactive images with the advanced feature sets, which simplify the creation process of GIF images and videos. The best part is that they can be constructed with ease using a few editing tools online. GIF sharing images are gaining more and more popularity these days because of their lesser loading time and small file size.
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In the example below, the photo on the left is taken using the maximum f/2.8 aperture for a shallow depth of field – a shallow focus – while the photo on the right minimum is f/11 and provides a large depth of field. Simply put, you might want to use a large aperture in dark environments to maximize brightness or for an attractive shallow focus in portraits. Conversely, a small aperture is ideal for landscape photography to ensure the entire scene is in sharp focus. These examples are not rules though – photography is a creative practice, after all, and you can take some great photos by using aperture and f-stops in less common ways, such as shallow focus landscapes. Do you want a more personal and professional email address, but without the fuss of managing another email account? We have a solution to that! With $5 Forwards, you can keep using your current email provider, but with your own Email is the easiest and least expensive way to get your own personalized email address. 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There’s no hassle of managing multiple inboxes in different places! Your emails will instantly appear in the account you’re already managing, like Appear more professionalįor blogs, portfolios or resumes, your domain email address will stand out from the rest while still being super easy to manage. When you become a subscriber you'll be the first to know about exclusive special offers, new product and more. Our awesome customer support team makes it easy for you to set up forward-only email. ![]() We’ll painlessly set you up with a email that directs all your mail to another specified address. There are all kinds of ways this could be useful! Plus, a real customer support person will always answer your calls if you ever have any questions! The possibilities are endless!Īt just five dollars each, you can create multiple emails under your domain that forward to any email address that’s currently in use. You can use email forwarding to direct emails to teams or departments that operate under individual email addresses. For example, messages sent to can go to the members of your sales team’s current email accounts. If you wear different hats at your company, then you can forward different email addresses to the same account while keeping communication more professional, like and If you do some freelance work on the side, you could create another address that pipes into your main account. Or, if you simply decide you want a new username, you can create one without having to send one of those “Hey everyone I’ve ever emailed, my email address has changed!” emails. ![]() But if they make a mistake and destroy the wrong objects, it will be them who die.Īfter each round, each team will change their roles: the hunters will become props, and the props, hunters. Midnight Ghost Hunt s prop hunting can be hard but haunting is a hoot. The red team has 30 seconds to disguise and camouflage itself as an element of the scenario, while the blue team has to find the hidden players and kill them. this fall and winterCDC: Monkeypox in the US unlikely to be eliminated in the. bind ' ' 'gmspawn (paste the prop name here)' so, a completed example would look like. Install About this game arrowforward The essence of the game is in the early rounds, one player becomes the hunter and the other props, the purpose of players who play for the props have to. right click on it, copy it to keyboard the prop name and then open up console. open the spawn menu and pick a prop you desire to spawn. With this GMod extension installed on our Windows PC, we can take part in a kind of hide and seek game played by 2 teams. yo, gonna teach you how to spawn props from pressing a key on your keyboard or a button on your mouse. This game mode is similar to what we can find in other open world games like Minecraft, and its Hide n' Seek mod, that has become tremendously popular thanks to videos by different youtubers, including PewDiePie or Markiplier, among others. Includes: phmotelblackev3 phno1office phratrunsewers AND Ratrun Material. NOTE: This map pack requires Counter Strike: Source for textures and models. With this add-on you'll have access to a new gaming mode within GMod in which you'll play hide and seek inside the title developed by Garry Newman.Īnother GMod mod together with TTT and DarkRP. Thread starter Similar threads Forum Replies Date FREE MOD Prop Hunt Multiplayer Free 1.025b Mega Mod APK GodMode: GAMES DELETED ON PLAYSTORE: 48: Apr 30, 2018: FREE MOD Hide Prop Seek Online 1.2. Trifles play theme, Graco tranzitions 3 in 1 harness booster manual, Sketchware mod apk latest version download Lyrics to songs from grease, Randolph new. Prop Hunt Classic Map Pack Created by eelikay For the people who want the classic prop hunt maps but not the unofficial Prop Hunt gamemode. Prop Hunt is an extension for the popular sandbox game Garry's Mod launched back in 2004. ![]() All of the numbers in cells of a particular color must only contain one of each digit from 1 to 9. ![]() Color-Sudoku: Assigns a color to each individual cell.They overlap the board in the shape of a percentage sign. Percent-Sudoku: A mixture of X-Sudoku and Hyper-Sudoku in that the additional constraints are a single diagonal and two extra 3x3 regions.Hyper-Sudoku: Adds 4 extra 3x3 grids that overlap at the intersection of the default 3x3 grids, requiring you to make sure that those extra regions only contain one of each digit from 1 to 9.X-Sudoku: Adds an additional constraint to the board such that the diagonals from top-left-to-bottom-right and top-right-to-bottom-left must contain every digit from 1 to 9. Play Offline with Web Sudoku Deluxe Download for Windows and Mac.Each region must still contain only one of each digit from 1 to 9. Squiggly Sudoku: Jumbles up the default 3x3 regions into misshapen squigglies, forcing you to think differently.If you try to place a picture where it already exists on that row, column, or small square, all occurrences will flash. Each picture may appear only once in each row, column, and 3x3 box. Each puzzle has a unique solution and can be solved with pure logic, no guessing required. Color Sudoku Puzzle Fill all of empty squares in the grid with a picture. Color Sudoku Puzzle online is optimized for use on PC, Android and iOS devices, including tablets and mobile phones. Any cells that are reduced to one possible candidate are solved. Unknown squares are filled with 'candidates' - possible solutions. Click on Take Step to step through the solution. You can also pick examples from the list above. Fiendish Sudoku posts five new puzzles every day. Use the 'Import a Sudoku' button or type in a Sudoku puzzle in the small board. Every row, every column, and every 3 x 3 box should contain one of each digit. Follow along and lets see if you can follow. Place digits from 1 to 9 in each empty cell. Remember that “Extra Regions” setting from before? This is what it does: Sudoku Solver - Explained Squiggly Sudoku Hard SudokuI now will explain how I solve this squiggly sudoku puzzle. Please contact us a if you need a copy of this program conveyed to you.Now here is where Andoku really sets itself apart from every other Sudoku game I’ve played. You should have received a copy of the GNU General Public License GNU General Public License for more details. Color regions Show errors Allow notes Show and prune possibles. Please kindly note that the permissions needed for this game is. ![]() You should be able to solve any puzzle here without writing anything down, and they all have only one solution. Standard Sudoku, Sudoku-x, Hyper Sudoku, Percent Sudoku, Color Sudoku and Squiggly Sudoku. Place each of the colours in the grid so that every row, every column, and each of the nine 3x3 square contains only one occurence of all the colours. MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. Color Sudoku Puzzle Welcome to our colour Sudoku. This program is distributed in the hope that it will be useful,īut WITHOUT ANY WARRANTY without even the implied warranty of of a 9x9 Sudoku grid containing areas surrounded by the same background color. The Free Software Foundation, either version 3 of the License, or Printable hard sudoku 4 to 6 puzzles per page, in pdf or html format. It under the terms of the GNU General Public License as published by This program is free software: you can redistribute it and/or modify Hope you like it please rate us if you do!įree Sudoku - a sudoku puzzle game for Android based of the original Andoku, by Markus Wiederkehr, and the Sudoku Explainer by Nicolas JuilleratĬopyright (C) 2006 - 2007 Nicolas JuilleratĬopyright (C) 2009 - 2011 Markus WiederkehrĬopyright (C) 2011 - 2017 Cool Android Appz The application can be downloaded for free from the Android Market. This game is completely free, and will eventually be supported by unobtrusive adds. Something all many advanced users requested in previous versions of this awesome game. now you can enter pencil marks with just one digit. The Free Sudoku extension includes 15000 new puzzles, App to SD, a new Sudoku Strategy Tutorial, which will help you learn how to solve from the easiest puzzles for beginners to the Fiendish puzzles for advanced users.įinally, the Interface has also been enhanced, solving a popular demand. Chose between Classic, Monster, Kids and Squiggly, and easy, medium. The most popular Sudoku game on the Market has been polished to perfection.įree Sudoku is an extension of the great Andoku, offering the same cool features and game variations: In a classic Sudoku puzzle, the task is to fill the numbers in a 9 x 9 grid laced. ![]() Your YouTube to MP3 converter must have high-quality options. Some of the best YouTube converters on this page support many formats like ACC, OGG, FLAC, AIFF, and more! Audio Quality But if you can also get more variants to choose from, then that’s better! If you get MP3 and WAV as the standard output formats, that’s good. They also work faster than online tools, since they don’t process data on a server. They can process multiple YouTube videos at once, so batch conversions and downloads are possible. You’ll save storage space on your computer, too.ĭesktop apps, on the other hand, are better for long-term or heavy-duty us e. They are more convenient since you don’t need to download and install the program on your desktop to use it. Web-based converters like Vidmate Online are good for one-time or casual users. Although all solutions passed my speed tests, the fastest YouTube to MP3 converter is Any Video Converter Ultimate. With a stable internet connection, a good program should be able to convert any track within 30 seconds to a minute. Programs like Any Video Converter Ultimate and Wondershare Unicoverter also offer an array of extra functions which may come in handy. ![]() With this, you can just search for your videos straight from the interface.īatch processing is another feature that will allow you to convert and download multiple videos at once. If it has an in-program browser to directly extract audio from YouTube, then that’s a keeper. As soon as you paste the link, the program immediately tracks the video and provides multiple quality options ready for downloading. One is automatic video analysis, which you’ll see in software like 4K YouTube to MP3. There are certain specs that make a program of this kind stand out. These factors will help you narrow down your many options to one. But with the best YouTube audio converter, you can make it even easier, safer, and quicker! Converting a YouTube video to an MP3 track is easy. ![]() Set User Output 25 to AES A25 and continue to patch 1 to 1 through User Output 30. This is what feeds the last 8 channels to the Waves system. I usually leave these open for Talkback or MC mics anyway, they can be patched from local sources on the mixer or in the example above I am using channels 31 and 32 on a D元2. Input 31 and Input 32 can be set to any preamp and used for channels that do not include Waves Inserts. Set Input 25 to Card 25 and patch 1 to 1 through to Input 30. Next we can skip down to User In Bank 25-32. These will be our returns from the Main L/R insert. ![]() Set Input 1 to Card 31 and Input 2 to Card 32. Here we set the Aux Output to be Inserts. In this example I am going to use AES A1-32 to feed all 32 channels to the mixer.Ĭard Out 1-8: AES A1-8 (this send the signal coming off the preamps to Waves)Ĭard Out 25-32: User Out 25-32 Patch Points: AUX This is where we set the sources that will feed the Expansion Card and ultimately our channels. You could use User In 1-2 and use the others as inserts somewhere else but that would make for routing that is more complex than it is probably worth.Īux In: User 1-2 (Aux in remap will not let you route anything to Aux 7+8, but in the configuration tab for the Aux it is possible to assign any source.) Expansion Card Output The Aux In Remap only allows Selection of Card or User sources starting at 1 so we will use User In 1-6. Input Blocks get routed like the picture above. scn file from the downloads section of the website if you would like. ![]() See video of process or scroll down to read the article:Īlso, you can download the. You could also change their source for instance to Local Inputs to handle playback or something but if you do not change the source then you need to mute them before using the Aux In Remap.*** ***IMPORTANT NOTE- Before going forward mute the Aux In Channels 1+2 so you do not create a feedback loop. This would only change the Card Out and User Out Routing. You could substitute Local channels or sources from AESA and AESB in addition or instead of. In this example I am using a Midas D元2 for all 32 channels on AES A. There are many different ways to route sources in the X32 and this is just one example, but I have tried to come up with a system that will be the easiest to keep straight. ![]() Much like part 1, we are going to use the new User Input routing patch system introduced in Firmware Version 4 as well as the Aux in Remap to achieve 30 channels of Waves processing as well as Bus Processing– in this example on the Left Right Main Bus. This setup will still cause the overall latency to be doubled, which on a native system could very well be too much for your program, but the routing would be the same for a Soundgrid Server System from Waves which is very low latency and more reliable than a native system. You can also use the last two input channels- they will not have waves inserts but would still sum and be included in the mix before the Main L/R Waves Insert. Following the instructions below you can setup an X32 with Waves processing on 30 channels as well as inserts including Main Left/Right bus. Synopsis: This Article shows how to setup Waves MultiRack or SuperRack on an X32 to process channels and buses at the same time. *** 10% off Waves Plugins with the following code: *** ![]() ![]() Toon Boom is premium software that is excellent for pro-league animators and is one of Adobe Animates biggest competitors. The common opinion amongst reviewers is that Toon Boom offers many more features than Adobe Animate. Reviewers say compared to Adobe Animate.Superb animation software with dozens of features that would attract freelance animators and professionals. Toom Boom is rated excellent on G2 with 4.7 stars.Ĭan run with Windows 7 & 10, macOS High Sierra, Catalina, Mojave.īasic - $25/month Advanced - $63/month Premium - $115/month. Toon Boom can import dozens of different file formats like BMP, JPEG, PSD, TGA, SGI, TIFF, TVG and PAL. The extremely instinctive interface where you can navigate using various effects like zoom in & out, pan and reset your views.Īdvanced Cut out+Bitmap and Vector Drawing+Bitmap and Vector Drawing+Access to Harmony Server+Access to Harmony Server Part 2 Top 10 Adobe Animate Alternatives to Improve Animation Experience No. Check out our list and find out which alternative suits your level of animation skills. This is exactly where free Alternatives to Adobe Animate enters the scene. It also lacks sufficient facilities for practicing 3D animations. This becomes an obstacle to those who are just starting out and are not that confident in getting their hands on a professional league interface like Animate. When it comes to choosing the best premium animation software, Adobe Animate CC is the obvious choice, but what makes beginners hesitant in using Adobe Animate is the fact that you need to subscribe to Adobe Creative Cloud. What Lay Behind Adobe Animate Alternative? These high-quality fonts are available in HTML5 canvas documents. Now you can use a vast library of Adobe web fonts with Adobe Animate. Though it resembles the same functionality with Adobe Illustrator, Adobe Animate has renewed the tool with the ability to use tilt and pressure effects when drawing with a stylus or pen. Using this feature, Adobe animate users can now control their animation camera movements giving your animation videos a more realistic feel.Įxporting your animations to your desktop is a convenience every software should provide.Īdobe Animate enables you to synchronize audio to your animations completely easily which is yet another feature that has been recently added. Adding to Adobe's line of exclusivity is its all-new Adobe Virtual Camera. ![]() Once the Lighten stacking is done, you can review the image up to 1:1 size. The 'Start Processing' button is 4th in from the upper left (see pic below). I'd start with the Lighten option first and review the results. The Lighten mode retains the brightest pixel from the image stack when creating the star trails. In the evolution of the application, 'Lighten' came first. Choose the Blending Tab and then select the 'Blending Mode' options - the two you're likely most interested in are 'Lighten' and 'Gap Filling'. On the right side, you should see a preference panel with a 'Blending' tab. With the images loaded, you're ready to blend. ![]() If you properly numbered your files in sequence, they will load in proper order with the first image at the top. They can be dragged and dropped into the app. If a user wants to delete a Mac-partition to free up some hard drive space, then using the help of Partition Manager Mac they can delete the pre-existing volume. Free Download Mac OS X Partition Manager to Create a new partition on Macintosh computer without any data loss problems. Partition software, free download - Recovery Partition Creator, Mac Free Disk Partition Recovery, IUWEshare Mac Disk Partition Recovery Wizard, and many more programs. Fortunately it's available for both platforms and so the following discussion applies to both Mac and PC users. As a Mac devotee (10 years on Apple's Web Marketing team will do that), I had one choice: Star Stax. They're both free but donations are encouraged. The two well regarded stacking software apps are StarStax (Mac or Windows) and Startrails.exe (Windows only). img-001.jpg, img-002.jpg etc.) and you're ready to stack. Now export those images out as JPEGs or TIFFs into a folder using an ordered number sequence (i.e. Once you have that image where you want it, copy the image adjustment settings and then apply them to the rest of the images in the sequence. The key at this point is having the ability to apply adjustments easily across all images. If you're using RAW source files or you wish to aesthetically adjust your entire sequence of images, you will need to first import the images into Adobe Lightroom or an equivalent image editor. Once you've made your exposures, you're ready to post process. In this tutorial I will show you my workflow to do star trails! First, check out my friend- Emeric Le Bars' web site on time lapse. Star Trails - App to generate easy and quick impressive Star Trails pictures and videos. Free download Star Trails Star Trails for Mac OS X.Instead of taking one long exposure one can take many shorter exposures. With modern digital cameras is it relatively easy to get decent startrail-images.This hardcore RPG features dozens of talents and spells, distinct races and character classes as well as a challenging isometric turn-based combat system. Star Trail is a faithful remake of Realms of Arkania’s classic sequel. Yet not all is as simple as it seems: As the party approaches an abandoned dwarven mine that may hold the answers to their questions, someone else makes it clear that not everyone in the country favors a cooperation between the two races. To unite the races of the elves and the dwarves in the struggle against a common enemy, they must recover an artifact known as the Salamander Stone. An elf ambassador summons the party to the city of Kvirasim. Now the orcs have regrouped and attacked other, more vulnerable areas – Lowangen is occupied and half the Svellt Valley lies in ashes. The united orc tribes attempted to invade the region of Thorwal, but a group of valiant adventurers thwarted their plans during their previous quest. ![]() allowing numeric and banded amounts), and sending them a follow-up online survey to validate answers given in telephone interviewsĪ consideration of the cost of an organisation’s most disruptive cyber security breach or attack beyond the immediate direct costs (i.e. ![]() The inclusion of micro and small businesses, and low-income charities, which ensures that the respective findings are not disproportionately skewed towards larger organisationsĪ data collection approach predominantly conducted by telephone, which aims to also include businesses and charities with less of an online presence (compared to online-only surveys)Ī comprehensive attempt to obtain accurate frequency and cost data from respondents, giving respondents flexibility in how they can answer (e.g. The use of random probability sampling and interviewing to avoid selection bias The 2023 survey shares the same strengths as previous surveys in the series: Using partially representative sampling or online-only data collection methods.īy contrast, the Cyber Security Breaches Survey series is intended to be statistically representative of UK businesses of all sizes and all relevant sectors, and of UK registered charities in all income bands. While there have been other surveys about cyber security in organisations in recent years, these have often been less applicable to the typical UK business or charity for several methodological reasons, including:įocusing on larger organisations employing cyber security or IT professionals, at the expense of small organisations (with under 50 staff) that typically do not employ a professional in this role – these small organisations make up the overwhelming majority of the business and charity populationsĬovering several countries alongside the UK, which leads to a small sample size of UK organisations ![]() 1.2 Strengths and limitations of the survey overall Nevertheless, the core approach – data collected from organisations via a random-probability survey, predominantly conducted by telephone – is unchanged, and therefore the survey is still considered comparable across years. This year’s survey sees more substantial changes to aspects of the survey methodology than in previous years (see Section 1.4). These exclusions are consistent with previous years. In addition, businesses with no IT capacity or online presence were deemed ineligible. Sole traders and public-sector organisations were outside the scope of the survey. We carried out 44 in-depth interviews between December 2022 and January 2023, to gain further qualitative insights from some of the organisations that answered the survey. The data for businesses and charities have been weighted to be statistically representative of these two populations. We undertook a random probability telephone and online survey of 2,263 UK businesses, 1,174 UK registered charities and 554 education institutions from 27 September 2022 to 18 January 2023. It also considers the different cyber attacks and cyber crimes these organisations face, as well as how these organisations are impacted and respond.įor this latest release, the quantitative survey was carried out in winter 2022/23 and the qualitative element in early 2023.Ġ20 7215 1000 Chapter 1: Overview 1.1 Summary of methodologyĪs in previous years, there were two strands to the Cyber Security Breaches Survey: The study explores the policies, processes and approach to cyber security, for businesses, charities and educational institutions. It is primarily used to inform government policy on cyber security, making the UK cyberspace a secure place to do business. The Cyber Security Breaches Survey is a research study for UK cyber resilience, aligning with the National Cyber Strategy. Since 2023, the study has also been co-funded and co-developed by the Home Office. ![]() In February 2023, the parts of UK government responsible for cyber security policy moved to the new department, DSIT. This research was previously commissioned by the former Department for Digital, Culture, Media and Sport (DCMS). There is another Education Institutions Findings Annex, available on the same GOV.UK page, that covers the findings for schools, colleges and universities. The annex supplements a main Statistical Release published by the Department for Science, Innovation and Technology (DSIT), covering the this year’s results for businesses and charities. It covers the quantitative survey (fieldwork carried out in winter 2022 and early 2023) and qualitative element (carried out in early 2023), and copies of the main survey instruments (in the appendices) to aid with interpretation of the findings. This Technical Annex provides the technical details of the Cyber Security Breaches Survey 2023. ![]() Theorem of calculus tells us that this is going toīe equal to lowercase f of x. Taking the derivative of the right hand side. Of the left hand side, it's the same thing as This with respect to x, which is equal to Is the same thing as taking the derivative of So if I were to take theĭerivative of capital F with respect to x, which Of our capital F, so the derivative- let me make Us that if we were to take the derivative Not an abbreviation- theorem of calculus tells ![]() Write this down because this is a big deal. Now the cool part, theįundamental theorem of calculus. That's between a and b, it'll tell you theĪrea under lowercase f of t between a and x. Over here is going to be another function of x. Going to be a function of x- and let me make Here, we can say is the definite integralįrom a to x of f of t dt. Now before we come up with the conclusionīetween two endpoints. The area under the curve between two endpoints? Well, we just use ourĭefinite integral. Some new function to capture the area under New function that's the area under the curveīetween a and some point that's in our interval. We're including that endpoint, let me make them bold My horizontal axis t so we can save x for later. So it also includes a and b in the interval. So, while we can create a mathematical model and imagine cutting it into infinitely many pieces just so we can minimize our approximation errors, that doesn't mean that something in the real world can be cut up into infinitely many slices. And, if current scientific hypotheses are correct, we will never get close to measuring time that small. But, the dispute amongst scientists is not over the measurement, but over whether there even exists a time period shorter than 1 Planck time.Īt present, we don't have the technology to measure time anywhere near as brief as a Planck time (though we have other reasons for knowing that is the limit of measurement). So, you could only ever measure whole number increments of a Planck time. What is fairly well accepted by scientists is that it is impossible by any means to measure time shorter than 1 Planck time, whether or not there is such a thing as part of a Planck time. This unit is the Planck time, which is equal to about 5.39×10⁻⁴⁴ sec. While this is not a well-confirmed idea, and many scientists disagree, there are some scientists who think that there is a smallest unit of time, something like the time equivalent of an atom - you just can't get any briefer a period of time. These are mathematical models of reality, not reality itself. When the curve is at y = 3, for example, the area under the curve is expanding at a rate of 3. Ultimately we're trying to prove something that is, in a way, quite obvious: that the rate of change in the area under a curve is given by the value of the curve. The reason is that we're trying to get at the rate of change in F(x), and the rate at which F(x) is changing doesn't depend on how much area there is to the left of x. ![]() The other thing that many people find confusing is that we get the same result no matter what point we choose for a, the starting point of the region we're studying. We use t in place of x for the curve, and sometimes this is called a dummy variable. ![]() We set it up initially as f(t) because we're using x for a different purpose (area under the curve) and it would create a logical contradiction for x to be both the independent variable that produces the curve and the independent variable that produces the area under the curve. It can operate on t or x or any other variable or constant within its domain. The key to understanding this is to realize that f, by itself, is a function. Some people have a hard time with this duality of f(t) and f(x). We're trying to show that if we have a function F(x) that provides the area under the curve f(t), the derivative of F(x) is f(x). If you get that much firmly in mind, the rest should be easier, but there are a couple of other points of confusion. We're given a function f(t) and asked to think about another function, not displayed, that is the area under the curve, not the value displayed on the curve. The graph doesn't show F(x) at all in fact, it doesn't have an x-axis. As x moves to the right, this area increases, even if f(t) is decreasing. Instead, F(x) is the area under this graph between point a and point x. So the first thing I would offer in trying to understand this better is to get a clear picture that this graph does not depict F(x). In this case, we're studying a function F(x) but looking at a graph of a different function, f(t). Part of the problem is that in almost all our other work we're looking at a graph of the function we're studying. |
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