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Teacher Tools > Lesson Plans > Lesson Plans By Subject > Math Lesson Plans

Continuity theorems

Continuity theorems preview
Continuity theorems preview
Continuity theorems video preview 2
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Look Inside · 8 pages

Continuity theorems title slide with objectives for composite-function limits and the intermediate value theorem
Opening slide stating the lesson objectives
Composite function theorem lesson slide with a worked limit showing cos(x − π/2) approaches 1 as x approaches π/2
Composite function theorem and first worked example
Second worked example applies the composite function theorem to show e^(sin x) approaches 1 as x approaches 0
Second example of evaluating a composite limit
Intermediate value theorem lesson slide states the theorem and graphs three points where f(c) equals M
Theorem statement and graph of intermediate values
Root-existence example introduces p(x) = 2x³ − 5x² − 10x + 5 and asks for proof of a root in [−1, 2]
Polynomial root example: setting up the proof
Polynomial root proof uses p(−1) = 8 and p(2) = −19 with continuity to establish a zero between −1 and 2
Polynomial root example: endpoint values and proof
Polynomial root example adds a calculator graph showing a zero near x = 0.425 after the existence proof
Polynomial root example: calculator graph
Final supplied lesson slide proves x − cos x has a zero using opposite signs at the endpoints 0 and π/2
Worked proof that x − cos x has at least one zero
Attributes
Subject

Math, Calculus

Grades

Grade 12

Types

Teacher Tools, Lesson Plans

File

Microsoft PowerPoint

Editable
No
Author
Maths Support Centre
Rating
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About This Product

PowerPoint presentation, 9 slides stating the theorem for limits of composite functions and providing an example of the intermediate value theorem.

What's Included

PowerPoint presentation, 9 slides

Resource Tags

calculus intermediate value theorem continuity theorems

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