Taylor Series Exploration

A Grade 12 Calculus worksheet exploring Taylor series, focusing on definition, expansion, and common applications.

Grade 12 Math CalculusTaylor Series

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Includes

Fill in the BlanksMultiple Choice2 Short AnswerTrue / False

Standards

CCSS.MATH.CONTENT.HS.C.A.2

Topics

CalculusTaylor SeriesSeries ExpansionGrade 12 Math

7 sections · Free to use · Printable

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Taylor Series Exploration

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Read each question carefully and provide detailed answers. Show all your work for full credit. Remember the Taylor series expansion formula: f(x) = Σ [fⁿ(a) / n!](x - a)ⁿ from n=0 to ∞.

1. A Taylor series is a representation of a function as an infinite sum of calculated from the values of the function's derivatives at a single point.

2. When the Taylor series is centered at a = 0, it is called a series.

3. The radius of convergence of a Taylor series determines the interval for which the series to the function.

1. Which of the following functions is represented by the Taylor series expansion Σ (xⁿ / n!) centered at x = 0?

sin(x)

cos(x)

eˣ

ln(1+x)

2. What is the value of 'a' when expanding a Maclaurin series?

Any real number

Undefined

1. Find the first three non-zero terms of the Taylor series for f(x) = sin(x) centered at a = π/2.

2. Determine the Taylor series for f(x) = eˣ centered at a = 1.

1. The Taylor series for a polynomial function is always the polynomial itself.

True

False

2. A Maclaurin series is a special case of a Taylor series where the series is centered at any point 'a'.

True

False

1. Use a Taylor series to approximate the value of √e to three decimal places. (Hint: Use the Taylor series for eˣ centered at a = 0 and evaluate at x = 0.5)