Glossary
Lagrange Error Bound
A formula used to determine the maximum possible error when a function is approximated by a Taylor polynomial, providing an upper bound for the remainder.
Example:
After approximating e^0.5 with a Taylor polynomial, you'd use the Lagrange Error Bound to guarantee that your approximation is accurate to within a certain number of decimal places.
Maclaurin Polynomial
A special type of Taylor polynomial that is centered specifically at x=0, providing a polynomial approximation of a function around the origin.
Example:
To approximate the value of cos(0.1), you might use a third-degree Maclaurin polynomial for cos(x), which simplifies the calculations since it's centered at zero.
Remainder of a Taylor Polynomial
The difference between the actual value of a function and the value given by its Taylor polynomial approximation, representing the error in the approximation.
Example:
When approximating sin(x) with a third-degree Taylor polynomial, the remainder of a Taylor polynomial quantifies how much the polynomial's output deviates from the true sin(x) value.
Taylor Polynomial
A polynomial approximation of a function at a specific point 'a', constructed using the function's derivatives evaluated at that center point.
Example:
If you need to estimate sqrt(4.1) without a calculator, you could use a second-degree Taylor polynomial for sqrt(x) centered at x=4 to get a very close approximation.