Problem 18 - Essential Calculus

Question

In Exercises \(7-18\), find the Maclaurin polynomial of degree \(n\) for the function. $$ f(x)=\tan x, \quad n=3 $$

Step-by-Step Solution

Answer

The Maclaurin polynomial of degree 3 for the function \( f(x) = \tan x \) is \( x + \frac{x^3}{3} \).

Step 1: Compute the function value and derivative at \( x = 0 \)

Compute the function value at \( x = 0 \): \( f(0) = \tan 0 = 0 \). Calculate the first derivative of \( f(x) = \tan x \): \( f'(x) = \sec^2 x \), and then evaluate at \( x = 0 \): \( f'(0) = \sec^2 0 = 1 \).

Step 2: Compute the second derivative

The second derivative of \( f(x) \) is given by the derivative of \( f'(x) \). Thus, \( f''(x) = 2\tan x \cdot \sec^2 x \) and when evaluated at \( x = 0 \), we get \( f''(0) = 0 \).

Step 3: Compute the third derivative

The third derivative of \( f(x) \) is then \( f'''(x) = 2\sec^2 x + 4\tan^2 x \sec^2 x \). Evaluate this at \( x = 0 \) to get \( f'''(0) = 2 \).

Step 4: Construct the Maclaurin Polynomial

Combine these results to form the Maclaurin polynomial of degree 3: \( P_3(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} = 0 + 1x + 0x^2 + \frac{2x^3}{6} = x + \frac{x^3}{3} \).

Key Concepts

Calculus

Calculus is a branch of mathematics that studies continuous change. It provides a framework to model systems that change and helps in calculating various important quantities such as areas, volumes, and critical points. In the context of Maclaurin polynomials, calculus allows us to approximate functions using a series expansion.
A Maclaurin polynomial is a type of Taylor polynomial centered at zero. It is specifically useful when evaluating a function or its derivatives at zero. The procedure involves taking successive derivatives of a function, evaluating them at zero, and then constructing a polynomial based on these values.
In this exercise, we are constructing a Maclaurin polynomial for the function \( f(x) = \tan x \) up to degree 3. Calculus provides the tools to find the necessary derivatives, which are crucial as they determine the coefficients of the polynomial terms.

Derivative

In calculus, a derivative represents the rate at which a function is changing at any given point. It is a fundamental concept for understanding the behavior of functions. Calculating derivatives allows us to derive formulas that describe how functions change and predict values.
To construct a Maclaurin polynomial, it is essential to calculate derivatives systematically. For example, for \( f(x) = \tan x \), we use the fact that the first derivative is \( f'(x) = \sec^2 x \), which simplifies to 1 when \( x = 0 \). Second and third derivatives are computed by taking higher-order derivatives and evaluating them at zero to find \( f''(0) \) and \( f'''(0) \), respectively.
Each derivative provides the next term in the Taylor series expansion. Understanding how to calculate derivatives and their roles in constructing the polynomial is crucial for solving such problems efficiently.

Taylor Series Expansion

The Taylor Series Expansion is a way of representing a function as an infinite sum of its derivatives at a particular point. The Maclaurin series is a special case of the Taylor Series, where the expansion is taken around zero. This type of series allows us to approximate complex functions using simpler polynomial forms.

• Start with the function and evaluate it at the expansion point.
• Take the first derivative, evaluate, and use it to find the linear term.
• Continue with higher derivatives for more terms.

In this exercise, the Maclaurin polynomial for \( f(x) = \tan x \) is formulated as \( P_3(x) = x + \frac{x^3}{3} \). This polynomial provides an approximation of \( \tan x \) near \( x = 0 \) up to the third degree.
The expansion provides a simplified way to handle functions using only a few terms, which is especially helpful in computational applications or when handling functions that are otherwise complicated to compute directly. By understanding how to form a Maclaurin series, you gain insight into the behavior of functions and the power of polynomial approximations.