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End of Semester Schedule (4/23/14)

End of Semester Schedule (4/23/14). Hand-in #5 will go out Friday, April 25 and be due at on Monday, April 28 at 4 pm. Corrected homework, clicker score, and average coming into the final exam will be available outside my office door sometime on Tues or Wed (I’ll email you when they are ready).

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End of Semester Schedule (4/23/14)

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  1. End of Semester Schedule (4/23/14) • Hand-in #5 will go out Friday, April 25 and be due at on Monday, April 28 at 4 pm. • Corrected homework, clicker score, and average coming into the final exam will be available outside my office door sometime on Tues or Wed (I’ll email you when they are ready). • Exam is Wednesday, May 7 from 9:00 to noon (here).You may bring TWO reference sheets. It is comprehensive.

  2. Clicker Question 1 • What is the degree 2 (i.e., quadratic) Taylor polynomial for f (x) = 1 / (x + 1) centered at 0? • A. 1 + x – x2 / 2 • B. 1  x • C. 1  x + x2 • D. 1  x + 2x 2 • E. 1 – x + x2 / 2

  3. Clicker Question 2 • What is the degree 2 (i.e., quadratic) Taylor polynomial for f (x) = x centered at 1? • A. 1 + x / 2 – x2 / 8 • B. 1 + (x – 1) / 2 – (x – 1)2 / 4 • C. 1 + (x – 1) / 2 – (x – 1)2 / 8 • D. 1 – (x – 1) / 2 + (x – 1)2 / 8 • E. 1 + x / 2 – x2 / 8

  4. How Calculators Work • Hand calculators can add, subtract, multiply, divide, and raise to whole powers easily. • They can easily store constants like the values of e, , ln(10), and so on. • In order to compute the values of power functions to non-whole exponents and to compute any of the transcendental functions, calculators useTaylor Series! (Actually, Taylor polynomials.)

  5. Working Near the Center • Taylor series are exact, but Taylor polynomials are approximations, and they are most accurate near the center of the interval of convergence. • Hence calculators do what they can by applying the Taylor polynomial to an argument as close to the center as possible. • How? It depends on the function.

  6. An Example: The Sin Function • Sin is periodic, with period 2, and calculators know that. • The standard Taylor series for the sin is centered at 0, so we alter the argument to be as near 0 as possible. • What can a calculator do to get the best estimate of sin(34)?

  7. Another Example: ln(x) • The standard Taylor series for ln(x) is centered at 1 and only converges on (0, 2], so to apply this technique the argument must be within that interval (and as close to 1, the center, as possible). • What can a calculator do to get the best estimate of ln(834)? (Calculators can be taught the rules of logs, of course.)

  8. Assignment for Friday • Assignment (not hand-in) is handed out.

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