Taylor Polynomial
Definition
A Taylor polynomial is a polynomial approximation of a function centered around a specific point. It is used to estimate the value of the function at nearby points.
Analogy
Think of a Taylor polynomial as an artist's sketch of a landscape. The sketch captures the main features and contours of the landscape, but it may not be an exact replica. Similarly, a Taylor polynomial captures the essential behavior of a function near a point, but it may not perfectly match the actual function.
Related terms
Derivative: The derivative measures how fast a function is changing at each point. It plays a crucial role in determining the coefficients of a Taylor polynomial.
Maclaurin Series: A Maclaurin series is a special case of a Taylor series where the center point is 0 (zero). It provides an approximation for functions around this specific point.
Remainder Term: The remainder term represents the difference between the actual value of the function and its approximation using a Taylor polynomial. It helps quantify how accurate our estimation is.
"Taylor Polynomial" appears in:
Study guides (2)
Practice Questions (8)
Which term is essential for the first-degree Taylor polynomial of a function f(x) at x = a?
What is the maximum error of the 3rd degree Taylor polynomial of sin(x) centered at the point a = 1, with error evaluated at the point x = 1?
What is the maximum error of the 1st degree Taylor polynomial of sqrt(x^2 + 4) centered at the point a = 0, with error evaluated at the point x = 1?
Which term is included in the second-degree Taylor polynomial for the function f(x) at x = a?
What is the general formula for the nth-degree Taylor polynomial for a function f(x) centered at x = a?
What is the relationship between the degree of a Taylor polynomial and its accuracy in approximating a function?
What happens to the accuracy of a Taylor polynomial approximation as the degree of the polynomial increases?
Which concept is fundamental in finding Taylor polynomial approximations of functions?
Resources
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.