Question

Use the definition to find the Taylor series (centered at \(c\) ) for the function. $$ f(x)=\tan x, \quad c=0 \text { (first three nonzero terms) } $$

Short answer

The first three nonzero terms of the Taylor series of \(f(x) = tan(x)\) centered at \(c = 0\) are \(x + \frac{1}{3}x^3\).

Step-by-step solution

Compute the Derivatives of \(f(x)\)

Compute the zeroth, first, second, and third derivatives of \(f(x) = tan(x)\). They are given by:\n\n- \(f(x) = tan(x)\)\n\n- \(f'(x) = \frac{d}{dx}tan(x) = sec^2(x)\)\n\n- \(f''(x) = \frac{d^2}{dx^2}tan(x) = \frac{d}{dx}[sec^2(x)] = 2sec^2(x)tan(x)\)\n\n- \(f'''(x) = \frac{d^3}{dx^3}tan(x) = \frac{d}{dx}[2sec^2(x)tan(x)] = 2sec^2(x)tan^2(x) + 4sec^4(x)\)

Evaluate the Derivatives at \(x = 0\)

Evaluate each of the calculated derivatives at \(x = 0\). Note that \(sec(0) = 1\) and \(tan(0) = 0\). Hence,\n\n- \(f(0) = tan(0) = 0\)\n\n- \(f'(0) = sec^2(0) = 1\)\n\n- \(f''(0) = 2sec^2(0)tan(0) = 0\)\n\n- \(f'''(0) = 2sec^2(0)tan^2(0) + 4sec^4(0) = 4\)

Apply the Taylor Series Formula

Apply the results from the previous step to the Taylor series formula to find the first three nonzero terms. The \(n\)th term of the Taylor series is [\(\frac{f^{(n)}(0)}{n!}(x-0)^n\)]. \n\nSince \(f(0) = 0\), the first nonzero term will arise from the first derivative. Thus, the first three nonzero terms of the Taylor Series will be: \n\n- \(\frac{f^{(1)}(0)}{1!}(x-0)^1 = x\)\n\n- \(\frac{f^{(3)}(0)}{3!}(x-0)^3 = \frac{1}{3}x^3\)\n\nTherefore, the expansion of \(tan(x)\) centered at \(x = 0\) is \(x + \frac{1}{3}x^3\) (excluding higher order terms).

Key concepts

Derivative Calculations

Understanding derivatives is fundamental to calculus, and it is crucial when we explore the behavior of functions. The derivative of a function at a point measures the rate at which the function's value is changing at that point. It's like the mathematical equivalent of speed for a changing quantity.

For example, in our exercise, we calculated the derivatives of the tangent function, \( \tan(x) \). Using basic differentiation rules, we found that the first derivative \( f'(x) \) is \( \text{sec}^2(x) \). Subsequently, we derived the second derivative \( f''(x) \) and third derivative \( f'''(x) \) using the chain rule and product rule. These derivative calculations are necessary for constructing the Taylor series, as they give us the coefficients that will be used in the series expansion.

Taylor Series Expansion

The Taylor series expansion is a powerful tool that allows us to approximate functions using polynomials. This approximation becomes exact if we include infinitely many terms. When we center the series at a point \( c \), we're essentially zooming in on the function's behavior near that point and expressing it as an infinite sum of its derivatives evaluated at \( c \).

In simpler terms, think of Taylor series as a way to mimic the function's shape with a polynomial that has the same slope, curvature, and so on, at the center point \( c = 0 \) in our example. We compute the first few derivatives of the function, plug in the center point into these derivatives to find the coefficients, and then multiply them by powers of \( (x - c) \). The exercise leads us to the first three nonzero terms of the \( \tan(x) \) Taylor series at the origin, which are essential to understanding the function's local behavior.

Tangent Function Properties

The tangent function, \( \tan(x) \), has unique properties that influence its series expansion. It is periodic, with a period of \( \text{π} \), and it has asymptotes where the cosine function equals zero. Unlike polynomials, its graph has no fixed upper or lower bounds.

When discussing its Taylor series, it is worth acknowledging that \( \tan(x) \) is an odd function, meaning that it is symmetric with respect to the origin. This symmetry is reflected in the series as well: only odd powers of \( x \) appear in its expansion. This property simplifies the series because it implies that all the even-order derivatives evaluated at \( x = 0 \) are zero. Our exercise highlights this property by showing that the zeroth and second derivatives at the origin equal zero, resulting only in terms with odd powers of \( x \) in the Taylor series expansion.