Welcome to an introduction to Taylor expansion. Taylor expansion is a method of representing a function as an infinite polynomial series. It uses the derivatives of a function at a specific point to construct a polynomial approximation. This powerful technique allows us to approximate complex functions with simpler polynomials. In this example, we see the exponential function e to the x in blue, and its Taylor approximations in red and green. The red line is the first-order approximation, which is 1 plus x. The green curve is the second-order approximation, which adds the term x squared over 2. As we include more terms, the approximation becomes more accurate.
Now, let's look at the mathematical definition of Taylor expansion. The general form of a Taylor series centered at point a is given by the formula shown. It starts with the function value at a, then adds terms involving the first, second, third, and higher derivatives at point a, multiplied by powers of x minus a and divided by factorials. When a equals zero, we get a special case called the Maclaurin series. In practice, we often use a finite number of terms, which gives us an approximation with a remainder term. In our graph, we see a function in blue, which is sine of x plus x squared. The green curve is its third-order Taylor approximation centered at x equals 2, marked by the red dot. The yellow line represents just the first-order term, which is the tangent line at that point. Notice how the approximation is very accurate near the center point but becomes less accurate as we move away.
Let's examine some common functions and their Taylor expansions. Here are the Maclaurin series for several important functions. The exponential function e to the x expands as 1 plus x plus x squared over 2 factorial, and so on. The sine function starts with x, then alternates between negative and positive terms with odd powers. The cosine function starts with 1, then alternates with even powers. The natural logarithm of 1 plus x and the geometric series 1 over 1 minus x also have well-known expansions. In our graph, we can see the exponential function in blue, along with its Taylor approximations of different orders. The red line is the first-order approximation, which is just 1 plus x. The green curve adds the second-order term x squared over 2, and the purple curve includes the third-order term. Notice how each additional term improves the approximation, especially near x equals zero, which is the center of these Maclaurin expansions.
Now let's explore some practical applications of Taylor expansions. Taylor series have numerous important uses across mathematics, science, and engineering. First, they allow us to approximate complex functions with simpler polynomials, which is particularly useful in computational mathematics and computer implementations of mathematical functions. Second, the remainder term helps us analyze and estimate the error in our approximations. Third, Taylor expansions are fundamental to many numerical methods, including solving differential equations and numerical integration. Fourth, they appear throughout physics in areas like mechanics, electromagnetism, and quantum mechanics. In our figure, we see the function square root of 1 plus x in blue, along with its first and second-order Taylor approximations. The yellow line shows the error between the actual function and its first-order approximation at x equals 1. Notice how the second-order approximation in green provides a better fit to the original function. This illustrates how Taylor series can be used to balance computational simplicity with accuracy requirements.
To summarize what we've learned about Taylor expansions: Taylor expansion is a method of representing a function as an infinite polynomial series. The coefficients of the Taylor series are determined by the derivatives of the function at the expansion point. The Maclaurin series is a special case where the expansion is centered at x equals zero. Taylor polynomials provide effective approximations for complex functions, with higher-order terms improving accuracy. These expansions have widespread applications in mathematics, physics, and engineering, from computational methods to theoretical analysis. Understanding Taylor series gives us powerful tools for approximating functions, estimating errors, and solving complex problems across various scientific disciplines.