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Calculus II (MAT 146) Dr. Day Wednesday December 4, 2013

Calculus II (MAT 146) Dr. Day Wednesday December 4, 2013. Transforming Series into Functions: Using Derivatives and Integrals Taylor Series and Maclaurin Series Assignments and Announcements. Geometric Power Series. If we let c n = 1 for all n , we get a familiar series:.

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Calculus II (MAT 146) Dr. Day Wednesday December 4, 2013

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  1. Calculus II (MAT 146)Dr. Day Wednesday December 4, 2013 • Transforming Series into Functions: Using Derivatives and Integrals • Taylor Series and Maclaurin Series • Assignments and Announcements MAT 146

  2. Geometric Power Series • If we let cn = 1 for all n, we get a familiar series: • This geometric series has common ratio x and we know the series converges for |x| < 1. • We also know the sum of this series: MAT 146

  3. Geometric Power Series MAT 146

  4. Geometric Power Series MAT 146

  5. Beyond Geometric Series Connections:Taylor Series How can we describe the cnso a power series can represent OTHER functions? ANY functions? Now we go way back to the ideas that motivated this chapter’s investigations and connections: Polynomial Approximators! MAT 146

  6. MAT 146

  7. MAT 146

  8. Taylor Series Demo #1 Taylor Series Demo #2 Taylor Series Demo #3 Taylor Series Demo #4 MAT 146

  9. Taylor Series Look at characteristics of the function in question and connect those to the cn. Example: f(x) = ex, centered around x = 0. MAT 146

  10. Taylor Series Example: f(x) = ex, centered around x = 0. And…how far from x = 0 can we stray and still find this re-expression useful? MAT 146

  11. General Form: Coefficients cn MAT 146

  12. Examples: Determining the cn f(x) = cos(x), centered around x = 0. MAT 146

  13. Examples: Determining the cn f(x) = sin(x), centered around x = 0. MAT 146

  14. Examples: Determining the cn f(x) = ln(1-x), centered around x = 0. MAT 146

  15. Assignments Tasks • This Week: WA 11.9 (today: 12/4) and WA 11.10 (Fri 12/6) • By the Day of Semester Exam: Four WA Review Assignments Also • Help Session Sun 12/8: STV 126, 6:30 pm – 8 pm • Semester Extra-Credit Video: Cover Page Required MAT 146

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