Calculus-17- Limits/Continuity: Continuity (Definition), Graphical Interpretation,
Application (IVT)
Prerequisites:
Students must know how to plot a graph and analyze it in order to understand the concept of continuity.
Objectives1. Students will learn to identify continuous functions and will find how it is applicable to the
Intermediate Value Theorem.
2. Students will understand the Intermediate Value Theorem and its application relative to graphs.
3. Students will later apply the concept of continuity to the Mean Value Theorem.
Book Pages to Read. Book Hw. 5 Problems
Unit 2.4, Continuity, Pages 117-125, Problems 3, 10, 11, 25, 31
Mastermathmentor worksheet
Unit 11, Continuity and Differentiation Problems 1-11
Alternate Worksheet
Alternate Continuity Worksheet
Videos
Continuity and One-Sided Limits
Continuity Part 2
Intermediate Value Theorem
Roshan Continuity
Powerpoints
Continuity PPTh
Calculus Applet
Continuity Applet
Continuity Calculus Applet
Continuity using limits
General Webpage that tutors the material
IVT
Real life application of Continuity (Enrichment) / Graphing calculator material
Continuity in Real Life
(A graphing calculator may be used to find areas where a graph is discontinuous).
Smartboard Download Lesson Link
Smartboard Continuity
Main Lesson
1. First 5 minutes- Warm up quiz- Use the Continuity in Real Life above for a list of problems that
may be used; preferably, use the first three examples where the functions are all continuous in
order to allow students to grasp the concept of continuity.
2. Next 20 min- Teach the MMM Classwork (Unit 11), and allow students to work on the problems.
3. Next 10 min- Use the Alternate Continuity Worksheet to see if students really understand
continuity.
4. Next 10 min- Explain the IVT with the Intermediate Value Theorem video. It is very clear,
concise, and goes slowly.
5. Last 9 minutes- Use number 35 and 36 (pg 127) from the textbook as a wrap-up quiz to finish
explaining the IVT.
Enrichment Lesson
Continuity in Real Life has fantastic examples in the end for practical, at home, real life applications to
continuity.
Ex. Sketch the graph of the population of the earth as a function of time. Is this a continuous function.
Why is it reasonable to represent this graph as a continuous function?
Extras
VOCABULARY :
Continuity: If a function is defined at all points that are connected, the line is continuous.
Certain conditions need to be met:
-The limit of f(x) as x approaches c must exist
- f(c) must exist
- the limit of f(x) as x approaches c must equal f(c).
Intermediate Value Theorem: If a function f is continuous on [a,b], and L is any number between f(a) and
f(b), then there is at least one number x in (a,b) such that f(x)=L
FORMULAS/RULES/PICTURES: Images, Equations
Even root functions are discontinuous where f(x)<0
Odd root functions are continuous everywhere.
The denominator of a function must not equal zero.
Extra Videos
Detailed Explanation of Continuity Part 1
Detailed Explanation of Continuity Part 2