Calculus Unit 14

Big Idea:

Series

Student Expectations:

Focus Standards

EU 4.2: A function can be represented by an associated power series over the interval of convergence for the power series.

• EK 4.2A1: The coefficient of the nth–degree term in a Taylor polynomial centered at for the function is .
• EK 4.2A2: Taylor polynomials for a function centered at can be used to approximate the function values of near .
• EK 4.2A3: In many cases, as the degree of a Taylor polynomial increases, the –degree polynomial will converge to the original function over some interval.
• EK 4.2A4: The Lagrange error bound can be used to bound the error of a Taylor polynomial approximation to a function.
• EK 4.2A5: In some situations where the signs of a Taylor polynomial are alternating, the alternating series error bound can be used to bound the error of a Taylor polynomial approximation to the function.
• EK 4.2B1: A power series is a series of the form where is a non-negative integer, is a sequence of real numbers, and is a real number.
• EK 4.2B2: The Maclaurin series for , , and provide the foundation for constructing the Maclaurin series for other functions.
• EK 4.2B3: The Maclaurin series for is a geometric series.
• EK 4.2B4: A Taylor polynomial for is a partial sum of the Taylor series for .
• EK 4.2B5: A power series for a given function can be derived by various methods (e.g., algebraic processes, substitutions, using properties of geometric series, and operations on known series such as term-by-term integration or term-by-term differentiation).
• EK 4.2C1: If a power series converges, it either converges at a single point or has an interval of Convergence.
• EK 4.2C2: The ratio test can be used to determine the radius of convergence of a power series.
• EK 4.2C3: If a power series has a positive radius of convergence, then the series is the Taylor series of the function to which it converges over the open interval.
• EK 4.2C4: The radius of convergence of a power series obtained by term-by-term differentiation or term-by-term integration is the same as the radius of convergence of the original power series.

Student Learning Targets:

• I will construct and use Taylor polynomials
• I will write a power series representing a given function
• I will determine the radius and interval of convergence of a power series

Essential Questions:

Extra Information:

Adopted Textbook:

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