1. 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 – x 2 / 2 – B. 1  x – C. 1  x + x 2 – D. 1  x + 2x 2 – E. 1 – x + x 2 / 2

2. Clicker Question 2 What is the degree 2 (i.e., quadratic) Taylor polynomial for f (x) =  x centered at 1? – A. 1 + x / 2 – x 2 / 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 – x 2 / 8

3. End of Semester Schedule Hand-in #5 will go out Friday, Apr 30 and be due at 4:45 pm on Tuesday, May 4. Corrected homework, clicker score, and average coming into the final exam will be available by 3:15 on Thursday, May 6. Regular office hours through next week (Tu & Th 3:15-4:45). Also Monday, May 10, 3:00-5:00. Optional review session 12:20-1:15 on Friday, May 7. Exam is Tuesday, May 11 from 9 to noon (here). You may bring TWO reference sheets.

4. How Calculators Work (4/28/10) 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 be 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. 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.