Title: Limits and Their Properties
Subject/Course: Advanced Placement Calculus
Topic:
Grades:
Designers:
Limits and Their Properties
High School
Mark Pickering
Established Goals:
Standard: From the Advanced Placement AB Calculus course outline.
 How to estimate a limit using a numerical or graphical approach
 Diagram different ways that a limit can fail to exist
 Evaluate a limit using properties of limits
 Develop and use a strategy for finding limits
 Determine continuity at a pint and continuity on an open interval
 Determine one-sides limits and continuity on a closed interval
 Understand and use the Intermediate Value Theorem
 Determine infinite limits from the left and from the right
 Find and sketch the vertical asymptotes of the graph of a function
 Determine finite limits at infinity, determine the horizontal asymptotes, if any, of the graph of a function
 Determine infinite limits at infinity.
Understandings:

Students will understand that...
Functions can be represented in a variety of ways: graphical, numerical, analytical or verbal. They should understand the connections among these representations.

Technology can be utilized to help solve problems, experiment, interpret results and support conclusions.

Critical thinking will be utilized to determine the reasonableness of solutions, including sign, size,
Essential Questions:
-
 How do you use math to describe how things change?
 Is the physical universe susceptible to human understanding/mathematical description? Is it mechanical?
 Does an increasing bounded sequence always have a limit? relative accuracy and units of measurement.

Calculus is a coherent body of knowledge with limits and continuity at the foundation of this appreciable human accomplishment.
Students will know how to...
Students will be able to....
 Sketch the graph of a function
 Find the intercepts of a graph
 Test a graph for symmetry with

An intuitive understanding of the limiting process.

Calculating limits using algebra.

Estimating limits from graphs or tables of data.

Understanding asymptotes in terms of graphical behavior.

respect to an axis and the origin
 Find the points of intersection of two graphs
 Interpret mathematical models for real-life data
 Find the slope of a line passing through two points
 Write the equation of a line with a given point and slope
 Interpret slope as a ratio or as a rate in a real-life application
 Sketch the graph of a linear equation in slope-intercept form
 Write equations of lines that are parallel or perpendicular to a given line
 Use function notation to represent and evaluate a function
 Find the domain and range of a function
 Identify different types of transformations of functions
 Classify functions and recognize combinations of functions

Describing asymptotic behavior in terms of limits involving infinity.

An intuitive understanding of continuity. (The function values can be made as close as desired by taking sufficiently close values of the domain.)

Understanding continuity in terms of limits.

Geometric understanding of graphs of continuous functions (Intermediate Value Theorem).
Performance Task: o Planning calculator exercises for different concepts. o Conduct a survey of what students implement in order to solve limit problems. o
Plan a lesson for different levels of students with the different representations of symbolic, numerical, graphical, and verbal.
Other Evidence
 Quizzes on limits and continuity.
 Calculator exercises for limits and continuity concepts.
 Calculus java applets to explore concepts with a study buddy.
 Online assessments from various websites that diagnose strengths and weaknesses.
Learning Activitity: This activity is cumulative as each topic pertaining to limits and their properties involves many explorations with java applets for calculus and graphing calculator in and out of class. Thus, this activity requires the output of a review game utilizing technology that teacher will integrate into the classroom on a daily basis.
1.
G What is the goal of the task? What is it designed to assess? As a game designer with an educational company, your task is to design a review game utilizing multimedia technology such as
Voicethread, powerpoint, or GoAnimate that will most effectively review limits and continuity.
2.
R What real-world role will the student assume as he/she is performing the task? A game designer with an educational company.
3.
A Who is the audience for the task? Your target audience is the testing department for this

educational company.
4.
S What is the situation that provides the context for the task? You have a two-part challenge: (1) to develop an educational review game that integrates review of limits and continuity, and (2) to clearly communicate the process for solving each facet of their review game utilizing multimedia technology such as Voicethread, powerpoint, or GoAnimate.
5.
P What is the product or performance that is required by the task? You need to develop a written process outlining the steps in sequence for each review problem similar to an answer key.
Additionally, you need to include a graphic format to accompany the written description.
6.
S By what standards will the product or performance be judged? The review game needs to follow the criteria for good design accurately and creatively; identify the key concepts of limits and continuity; include a clear and written description of the process; and provide a review game deliverable in a multimedia format such as Voicethread, powerpoint, or GoAnimate.