Teaching Taylor’s Theorem Mark Howell Gonzaga College High School Washington, DC

What Does Taylor’s Theorem Say? If g is a function with n + 1 continuous derivatives on [a,b], then there is a number c in (a, b) such that…

Or…

Ifis the degree Taylor Polynomial for g at x = a, then where is the “remainder”…

LaGrange Form of the Remainder derived from the integral form c is found in (a,b) gives rise to the LaGrange error bound IS tested in AP Calculus BC

NOT tested in AP Calculus

Historic Notes James Gregory first stated a form of the result in 1671 Brook Taylor stated it in 1712 (without the remainder) LaGrange and Cauchy gave their remainder forms about a century later

Seek Comfort in the Familiar Taylor’s Theorem, like the Intermediate Value, Extreme Value, and Mean Value Theorem is an existence theorem.

Find the c, just like the MVT See the first page of the handout. We’ll do this in a moment.

Use the LaGrange Error Bound See released FR questions. Approximate the square root of e to within Determine how many terms are needed. We need with 0 < c < 0.5. Verify it works with the calculator. calculator

Foreshadow with Activities See the Topic Focus materials on Approximation at AP Central. Here’s the gist of it…

Why?

Let n = 0 in Taylor’s Theorem. Alas… or for some c in (a,b). Hmmm… Leverage MVT Connection

MVT Connection

Watch the movies… Taylor 1 st through 10 th Degrees

Error movie