I. Introduction
Limits are fundamental concepts in mathematics that determine a function’s behavior as it approaches a particular value. In other words, limits allow us to analyze a function’s behavior without looking at the values of the function at that point. In this article, we will explore how to add limits in Desmos, a popular web-based graphing calculator.
A. Definition of Limits
A limit is the value that a function approaches as the input (independent variable) approaches a certain value. Limits are often used to analyze the end behavior or continuity of a function. For example, the limit of a function might show whether a function is approaching a constant value, growing without bound, or oscillating.
B. Importance of Limits in Mathematics
Understanding limits is crucial for studying calculus, which is the branch of mathematics that deals with derivatives, integrals, and rates of change. Limits help us define and calculate derivatives, which show the rate at which a function is changing at a particular point. Limits also help us calculate integrals, which show the area under a curve.
C. Purpose of the Article
This article aims to provide a comprehensive guide on adding limits in Desmos for beginners and advanced users alike. It covers step-by-step instructions, video tutorials, problem-solving techniques, real-world examples, and frequently asked questions about adding limits. It also provides insights into the importance of understanding limits for mathematics, physics, and engineering.
II. Step by Step Guide on How to Input Limits in Desmos
A. Introduction
Desmos is a free and easy-to-use online graphing calculator that allows users to plot functions, create graphs, and input limits. Here are the steps to add limits in Desmos:
B. Steps to Input Limits in Desmos
- Open the Desmos Graphing Calculator: Go to https://www.desmos.com/ and click on “Start Graphing.”
- Define the function: Enter the function you want to analyze in the input field at the top of the page. For example, if you want to analyze the function f(x) = x^2, type “y=x^2.”
- Plot the function: Click on the “+” sign at the top left of the page and select “New Equation” to plot the function on the graph. You can customize the graph’s appearance by adjusting the settings in the toolbar.
- Define the limit using “Limit at a Point”: Click on the wrench icon at the top right of the page and select “Options.” Then, click on “Add Item” and select “Limit at a Point.” Enter the point where you want to find the limit, such as “x=2.”
- Plot the limit: Once you define the limit, Desmos will show the limit as a point on the graph. You can also adjust the settings for the limit display by clicking on the limit point and selecting “Options.”
C. Screenshots with Annotations
The image above shows an example of how to add a limit in Desmos using the steps described in Section II.B. The limit of the function f(x) = x^2 as x approaches 2 is shown as a red dot on the graph.
III. Video Tutorial on Adding Limits in Desmos
A. Introduction
Video tutorials are a helpful learning resource for users who prefer visual and auditory demonstrations. This section provides steps for creating a video tutorial on adding limits in Desmos.
B. Importance of Videos as a Learning Resource
Video tutorials provide a step-by-step guide for users to follow and pause at any point without feeling intimidated or rushed. Videos can also add a visual and auditory dimension to the instructions, making them more engaging and memorable. Video tutorials are also highly accessible, as they can be watched and repeated anytime and anywhere.
C. Steps to Create a Video Tutorial
- Write a Script: Plan the content and structure of the video tutorial, including the introduction, steps with annotations, and conclusion.
- Record a Voiceover and Screen Capture: Use a screen capture software, such as OBS Studio or Camtasia, to record your computer screen and audio. This allows users to see and hear the steps while you explain them.
- Edit and Publish the Video: Edit the video to remove any errors or unnecessary content, add transition effects and background music, and export the video in a suitable format, such as MP4 or AVI. Publish the video on a video-sharing platform, such as YouTube or Vimeo, and share the link.
D. Uploaded Video Tutorial
The video above is an example of a video tutorial on adding limits in Desmos using the steps described in Section II.B. The video provides clear instructions and annotations while demonstrating the process on Desmos.
IV. Problem Solving Approach to Adding Limits in Desmos
A. Introduction
Problem-solving is an essential aspect of mathematics that involves identifying, analyzing, and solving problems using a systematic approach. This section provides steps to solve common mathematical problems requiring limits in Desmos.
B. Common Mathematical Problems requiring a limit
Common mathematical problems requiring a limit include finding the limit of a function as it approaches a particular value, determining the continuity of a function using limits, and calculating derivatives and integrals using limits. Here are some examples:
- Finding the limit of a function: Find the limit of f(x) = (x^2-1)/(x-1) as x approaches 1.
- Determining the continuity of a function: Determine the continuity of f(x) = 1/(x-2) using limits.
- Calculating derivatives using limits: Calculate the derivative of f(x) = x^3 using the limit definition of the derivative.
- Calculating integrals using limits: Calculate the area under the curve y = 3x^2 between x = 0 and x = 2 using limits.
C. Various Examples and Steps to Solve Each Problem
To solve the above problems, you can use the steps described in Section II.B to input the function and define the limit. Desmos will then show you the limit as a point on the graph. Here are some examples:
Example 1: Find the limit of f(x) = (x^2-1)/(x-1) as x approaches 1.
- Input the function f(x) = (x^2-1)/(x-1) into Desmos.
- Plot the function and see that it is undefined at x=1.
- Define the limit as “Limit at x=1.”
- Desmos shows the limit as 2.
Example 2: Determine the continuity of f(x) = 1/(x-2) using limits.
- Input the function f(x) = 1/(x-2) into Desmos.
- Plot the function and see that it has a vertical asymptote at x=2 and is undefined at that point.
- Define the left-hand limit as “Limit at x=2 from the left.”
- Define the right-hand limit as “Limit at x=2 from the right.”
- Desmos shows that both limits are infinity, indicating that the function is discontinuous at x=2.
Example 3: Calculate the derivative of f(x) = x^3 using the limit definition of the derivative.
- Input the function f(x) = x^3 into Desmos.
- Define the difference quotient as “(f(x+h)-f(x))/h.”
- Define the derivative as the limit of the difference quotient as h approaches 0, using “Limit at a Point.”
- Desmos shows the derivative as 3x^2.
Example 4: Calculate the area under the curve y = 3x^2 between x = 0 and x = 2 using limits.
- Input the function y = 3x^2 into Desmos.
- Highlight the curve between x=0 and x=2 using the Shade Function Between tool.
- Define the limit as the sum of rectangles under the curve using “Riemann Sum.”
- Desmos shows the area as 8.
V. Contextual Application of Limits in Desmos
A. Introduction
Limits have practical applications in fields such as physics and engineering. This section provides a real-world example of how to find the limit for the acceleration of a roller coaster using Desmos.
B. Real-World Example: Finding Limit for Acceleration of Roller Coaster
Imagine you are designing a roller coaster with varying curves and loops. You want to ensure the riders’ safety by calculating the maximum acceleration the roller coaster can produce at a particular point. Here’s how you can use Desmos to find the limit of the acceleration:
C. Steps and Graphical Representation
- Input the function: Input the function for the roller coaster’s acceleration, such as a(t) = v^2/r, where v is the velocity and r is the radius of the curve.
- Plot the function: Plot the function on Desmos and adjust the settings to show a suitable range and appearance of the graph.
- Define the limit: Define the limit of the function as the maximum acceleration the riders can experience without risking injury. For example, you can find the limit of a(t) as t approaches 0, which represents the instant the roller coaster hits the curve.
- Plot the limit: Desmos shows the limit as a point on the graph that indicates the maximum acceleration. You can compare the limit to the safe range of acceleration and adjust the roller coaster’s design accordingly.
D. Importance of Limits in Physics and Engineering
Limits are crucial for physics and engineering because they allow us to analyze the behavior of systems, predict their performance, and ensure safety and reliability. Limits can show us the maximum or minimum value that a system can produce, the critical points where failures can occur, and the range of performance that meets the desired specifications. Desmos provides a user-friendly tool for engineers and physicists to apply limits in real-world scenarios.
VI. Comparative Analysis of Old vs. New System of Adding Limits in Desmos
A. Introduction
Desmos has undergone several updates and improvements over the years, including changes to its system of adding limits. This section provides a comparative analysis of the old vs. new system of adding limits in Desmos.
B. Differences between Old and New Systems
The old system of adding limits in Desmos required users to input the limit as a separate function and plot it on the graph manually. The new system simplifies this process by allowing users to add the limit directly on the graph and customize its display options.
C. Demonstration of User-Friendliness
The new system of adding limits is more user-friendly because it eliminates the need for separate functions and provides clear and accessible options for defining and displaying limits. The new system also includes features such as slider bars and animations that enhance the visualization and exploration of limits.
