Calc._Lesson_1.10.pdf - LESSON 1.10 Continuity(Part 2 UNIT 1 NOTES WARM-UP PROPERTIES OF CONTINUITY Discuss the continuity of the following function = 2

# Calc._Lesson_1.10.pdf - LESSON 1.10 Continuity(Part 2 UNIT...

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LESSON 1.10 Continuity (Part 2). EXERCISES. Find the constant ? , or the constants ? and ? , such that the function is continuous on the entire real line. 1. ?(?) = { ? 3 , ? ≤ 2 ?? 2 , ? > 2 2. ?(?) = { 2, ? ≤ 1 ?? + ?, − 1 < ? < 3 −2, ? ≥ 3 Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of ? guaranteed by the theorem. 3. ?(?) = ? 2 + ? − 1 , [0,5] , ?(?) = 11 4. ?(𝑡) = 𝑥 2 +𝑥 𝑥−1 , [0, 3] , ?(?) = 4 HOMEWORK. Discuss the continuity of the composite function ℎ(?) = ?(?(?)) . 1. ?(?) = ? 2 , ?(?) = ? − 1 2. ?(?) = 1 √𝑥 , ?(?) = ? − 1 3. ?(?) = sin ? , ?(?) = ? 2 Find the constant ? , or the constants ? and ? , such that the function is continuous on the entire real line. 4. ?(?) = { 𝑥 2 −𝑎 2 𝑥−𝑎 , ? ≠ ? 8, ? = ? LESSON OBJECTIVE. * In this lesson you will be able to justify conclusions about continuity at a point using the properties of continuity * You will also be able to determine intervals over which a function is continuous. ESSENTIAL QUESTION(S). * Write the definition of continuity with your own words. NOTES. PROPERTIES OF CONTINUITY. The following types of functions are continuous at every point in their domain. 1. Polynomial functions: ?(?) = ? 𝑛 ?