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Before the age of calculators, studying functions such as sin x, cos x , ex , and ln x was quite time consuming. The graphs of these functions are important when studying their characteristics. James Gregory, a Scottish mathematician in the 17th century, made an important discovery about these functions. Using calculus, he wrote a series of terms to approximate very closely the graph of the curve. His main focus was with the function ln x ; he was able to calculate any positive value of x using a polynomial series. Brook Taylor, an English mathematician, generalized the Maclaurin series, devised by Colin Maclaurin. However, Gregory had actually known about them long before Taylor came into the picture. Taylor invented the method for expanding functions in terms of polynomials about an arbitrary point known as Taylor Series, which he published in 1715. Computing values of polynomials is much easier and less time consuming than evaluating a function like sin x. In this paper, I will look at the background needed before one can truly understand polynomials, the definition of Taylor polynomials, and how to use Taylor polynomials to approximate the functions I mentioned above.