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 

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Presentation transcript:

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

Clicker Question 2 Who is buried in Grant’s tomb? A. Washington B. Lincoln C. Grant D. Dracula E. Elvis

How Calculators Work (12/10/12) 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 use Taylor series! (Actually, Taylor polynomials.)

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 a Taylor polynomial of some degree to an argument as close to the center as possible. How? It depends on the function.

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)?

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.)

Assignment for Wednesday Assignment (not hand-in) is handed out. Hand-in #4 is due Thursday at 5 pm.