Taylor and Maclaurin Series Quiz

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Questions and Answers

Who introduced the Taylor series in 1715?

Colin Maclaurin
Isaac Newton
Leonhard Euler
Brook Taylor (correct)

What is the nth Taylor polynomial of a function?

The derivative of the function at the nth term
A polynomial of degree n formed by the first n + 1 terms of the Taylor series (correct)
The sum of the Taylor series up to the nth term
A polynomial of degree n+1 formed by the first n terms of the Taylor series

What do Taylor polynomials serve as for a function?

Derivatives of the function
Exact representations of the function
Infinite series expansions of the function
Approximations of the function (correct)

When is a Taylor series said to be convergent?

When its sum is the limit of the function (D) Signup and view all the answers

What does Taylor's theorem provide estimates on?

The error introduced by the use of Taylor polynomial approximations (C) Signup and view all the answers

Match the mathematician with the type of Taylor series named after them:

Brook Taylor = Taylor series Colin Maclaurin = Maclaurin series Signup and view all the answers

Match the following with their role in Taylor series:

Taylor polynomial = Approximation of a function Taylor's theorem = Estimates the error in approximations Convergent Taylor series = Has a sum as the limit of the series Partial sum of Taylor series = Polynomial of degree n Signup and view all the answers

Match the function with its property near the point of equality with the Taylor series:

Common functions = Equal to the sum of its Taylor series near a point Taylor series = Infinite sum of terms expressed in terms of derivatives at a single point Signup and view all the answers

Match the following with their usage in Taylor series:

Taylor series = Infinite sum of terms expressed in terms of function's derivatives at a single point Taylor polynomials = Approximations of a function Signup and view all the answers

Match the mathematician with their contribution to the Taylor series:

Brook Taylor = Introduced Taylor series in 1715 Colin Maclaurin = Made extensive use of Maclaurin series in the mid-18th century Signup and view all the answers

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Study Notes

History of Taylor Series

Brook Taylor introduced the Taylor series in 1715.

Taylor Polynomials

The nth Taylor polynomial of a function is an approximation of the function at a point.
Taylor polynomials serve as an approximation of a function around a point.

Convergence of Taylor Series

A Taylor series is said to be convergent if it converges to the original function at a point.

Taylor's Theorem

Taylor's theorem provides estimates on the accuracy of the Taylor polynomial approximations.

Mathematicians and Their Contributions

Brook Taylor: introduced the Taylor series in 1715.
Colin Maclaurin: named after the Maclaurin series, a special case of the Taylor series.

Key Concepts in Taylor Series

Center point: the point around which the Taylor series is expanded.
Lagrange remainder: used to estimate the error in approximating a function with its Taylor polynomial.