Taylor approximation is a fundamental technique in mathematics that allows us to approximate complex functions using simple polynomial functions. Instead of working with complicated functions like sine, cosine, or exponential functions, we can use polynomials that are much easier to calculate and manipulate. The general Taylor series formula shows how we can express any smooth function as an infinite sum of polynomial terms, each involving derivatives of the function at a specific point.
Now let's examine the components of the Taylor series formula in detail. The general form shows that any function can be expressed as an infinite sum, where each term has a specific structure. The zeroth term is simply the function value at point a. The first term involves the first derivative multiplied by x minus a. The second term uses the second derivative divided by two factorial, multiplied by x minus a squared. Each subsequent term follows this pattern with higher derivatives, larger factorials, and higher powers. The factorial terms in the denominator ensure the series converges properly, while the derivatives capture the local behavior of the function at point a.
The first-order Taylor approximation is the simplest and most intuitive case. It represents the function using only the constant term and the linear term, which geometrically corresponds to the tangent line at the expansion point. The formula T1 of x equals f of a plus f prime of a times x minus a. This is exactly the point-slope form of a line passing through the point a, f of a with slope f prime of a. While this linear approximation is quite simple, it provides a good approximation for the function's behavior in a small neighborhood around point a. The accuracy depends on how much the function curves away from the tangent line.
Higher-order Taylor approximations provide increasingly accurate representations of the original function. The second-order approximation adds a quadratic term involving the second derivative, which captures the curvature of the function at point a. The third-order approximation includes a cubic term with the third derivative, providing even better accuracy. As we can see in the graph, each additional term makes the polynomial approximation follow the original function more closely over a wider range. The second-order polynomial captures the basic curvature, while the third-order polynomial provides an even better fit. This progressive improvement demonstrates how Taylor series can approximate complex functions with arbitrary precision by including more terms.
Let's work through a complete example using f of x equals e to the x expanded around x equals zero. First, we calculate the derivatives at x equals zero. Since the derivative of e to the x is itself, all derivatives equal one at x equals zero: f of zero, f prime of zero, f double prime of zero, and f triple prime of zero all equal one. Next, we substitute these values into the Taylor formula to build our approximating polynomials. The first-order approximation is T1 of x equals one plus x. The second-order is T2 of x equals one plus x plus x squared over two. The third-order is T3 of x equals one plus x plus x squared over two plus x cubed over six. As we can see in the graph, each successive approximation follows the exponential function more closely over a wider range, demonstrating the power of Taylor series.