Lesson 16: 1-16 Intermediate Value Theorem
Duration of Days: 1
Lesson Objective
The student will be able to:
Explain the requirements for the IVT to apply (continuity on a closed interval).
Verify the existence of a specific function value y = k between f(a) and f(b).
Apply the IVT to prove that a function has at least one "zero" or "root" on a given interval.
Communicate a formal mathematical justification that follows the "If-Then" structure required for AP FRQs.
If you are 5 feet tall at age 10 and 6 feet tall at age 20, did you ever have to be exactly 5.5 feet tall? Why?
Why is continuity the essential condition for this theorem? What happens if there is a jump or a hole?
Does the IVT tell us how many times a function hits a certain value, or just that it hits it at least once?
Intermediate Value Theorem (IVT)
Existence Theorem
Closed Interval
Intermediate Value
Root / Zero / x-intercept
Hypothesis vs. Conclusion
F-IF.B.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
F-IF.C.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
F-IF.C.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
Description: The IVT states that if a function is continuous on [a, b], it must take on every y-value between f(a) and f(b). Its most common application is the Location of Roots Theorem: if f(a) is negative and f(b) is positive, the function must cross the x-axis (the zero) at least once.
Purpose: This lesson trains students to think like mathematicians. On the AP Exam, they are frequently asked, "Must there be a time t where the velocity is 50 mph?"
Students must learn to check the "hypotheses" (Is it continuous?) before they can claim the "conclusion."
DOK Levels
DOK Level 2 (Skill/Concept): Using a table of values to determine if a function must have a root on a specific interval.
DOK Level 3 (Strategic Thinking): Writing a formal justification for the existence of a value, citing the continuity of the function as the necessary condition.
For Struggling Learners (Scaffolding):The "Teleportation" Analogy: Explain that a continuous function cannot "teleport" over a y-value. To get from y=1 to y=5, you have to pass through 2, 3, and 4.
For Advanced Learners (Extension):The Fixed Point Theorem: Challenge them to use the IVT to prove that if f is continuous on [0, 1] and its range is also [0, 1], there must be a point where f(x) = x.
AP College Board Assessments