Problem 37 - Calculus: One and Several Variables

Question

Find the Lagrange form of the remainder \(R_{n}(x)\). $$f(x)=\tan x ; \quad n=2$$

Step-by-Step Solution

Answer

The Lagrange form of the remainder for \(f(x) = \tan x\), where \(n = 2\), is \(R_2(x) = \frac{2 \sec^2(c)(\tan^2(c) + 1)}{6}(x - a)^3\), where \(c\) is some number between \(x\) and \(a\).

Step 1: Understand the Lagrange form of the remainder

The Lagrange form of the remainder is given by \(R_n(x) = \frac{f^{(n+1)}(c)}{(n + 1)!}(x - a)^{n + 1}\), where \(c\) is some number between \(x\) and \(a\). Here, \(f^{(n + 1)}\) represents the \((n + 1)\)th derivative of \(f\).

Step 2: Differentiate the function

Determine the derivative of the function \(f(x) = \tan x\) for the first three times. The first derivative of \(f(x)\) is \(f'(x) = \sec^2 x\). The second derivative is \(f''(x) = 2\sec x \tan x\). The third derivative is \(f'''(x) = 2\sec^2(x)(\tan^2(x) + 1)\). The third derivative is used because \(n = 2\) in this case.

Step 3: Substitute in Lagrange remainder

One cannot compute the exact value of the remainder because the absolute value of \(x - a)\) must be less than \(R\), but one can form the general expression using the Lagrange remainder formula. Substituting all these values in the formula will give \(R_2(x) = \frac{f^{3}(c)}{3!}(x - a)^3 = \frac{2 \sec^2(c)(\tan^2(c) + 1)}{6}(x - a)^3\).

Key Concepts

Lagrange Remainder

The Lagrange Remainder is an essential part of understanding Taylor series approximations. It helps us estimate the error or "remainder" when approximating a function with a Taylor polynomial.
The formula for the Lagrange Remainder is \( R_n(x) = \frac{f^{(n+1)}(c)}{(n + 1)!}(x - a)^{n + 1} \), where \( n \) is the degree of the polynomial, \( a \) is the point of expansion, and \( c \) is an unknown value between \( x \) and \( a \).
Key points to remember:

• \( f^{(n+1)} \) is the \( n + 1 \) derivative of the function.
• \( x - a \) is the distance from the point of expansion.
• It helps us understand how accurate our polynomial approximation is.

This remainder term can inform us when the Taylor polynomial accurately represents the function and when it might miss out on important details. It becomes particularly significant for functions with complex behavior.

Calculus Derivatives

In calculus, derivatives measure how a function changes as its input changes. For our specific problem, derivatives play a crucial role in forming the polynomial approximation. Let's explore derivatives of the tangent function:
For \( f(x) = \tan x \):

• The first derivative, \( f'(x) = \sec^2 x \), tells us the slope of the tangent line at any given point.
• With the second derivative, \( f''(x) = 2\sec x \tan x \), we start understanding the concavity of the function.
• The third derivative, \( f'''(x) = 2\sec^2(x)(\tan^2(x) + 1) \), becomes vital as it is used directly in the remainder formula for the Taylor series.

Using these derivatives allows us to approximate functions and comprehend their local behavior more finely. Recognizing the crucial derivatives necessary for each Taylor polynomial term leads to deeper insights into the function's behavior at points of interest.

Tangent Function

The tangent function, \( \tan x \), is a fundamental trigonometric function with unique characteristics. Understanding its properties is vital for accurate approximations using Taylor series.

• The function is periodic, with the period \( \pi \), and it exhibits vertical asymptotes due to the function's undefined nature at odd multiples of \( \frac{\pi}{2} \).
• It's essential to always note where the function is continuous and where it shows discontinuities, as this may affect our polynomial approximations and the reliability of the Lagrange Remainder.

When expanding \( \tan x \) into a Taylor series, choosing an appropriate center \( a \) is crucial. Typically, centers such as \( x = 0 \) are favorable as they simplify calculations and guarantee the function's derivatives are well-defined. This clear understanding of the tangent function assists us in predicting its behavior and effectively utilizing derivatives to express the function's details.