Introduction to Taylor and Maclaurin Series Approximations
In calculus, a Taylor series is an infinite sum of terms expressed in terms of a function's derivatives at a single point. If the series is centered at zero, it is called a Maclaurin series. These series matter because they let us approximate non-polynomial functions (like sine, cosine, and exponential functions) using simple polynomial terms, which are easy to compute, differentiate, and integrate.
An approximation using the first few terms of a Taylor series is called a Taylor polynomial. Its accuracy depends on two factors: the degree of the polynomial (the number of terms included) and the distance between the evaluation point x and the center point a. As the degree increases, the approximation generally converges toward the true value of the function within its interval of convergence.
This calculator computes Taylor and Maclaurin polynomial approximations for four functions: e^x, sin(x), cos(x), and ln(1+x). Enter the center a, the polynomial degree n (1–8), and the target evaluation point x, and it shows the term-by-term derivative expansion, the polynomial's approximate value, and the absolute and relative error against the true function value.