Problem 5 - Thomas Calculus

Question

Find the linearization \(L(x)\) of \(f(x)\) at \(x=a.\) \(f(x)=\tan x, \quad a=\pi\)

Step-by-Step Solution

Answer

The linearization is \(L(x) = x - \pi\).

Step 1: Understand the Linearization Formula

The linearization of a function at a point provides an approximation using the tangent line at that point. It is given by the formula:\[ L(x) = f(a) + f'(a)(x-a) \] where \(f(a)\) is the function value at \(x=a\) and \(f'(a)\) is the derivative of the function evaluated at \(x=a\).

Step 2: Calculate the Function Value at \(x = a\)

We need to find \(f(a)\) for \(a = \pi\). The function is \(f(x) = \tan(x)\), so:\[ f(\pi) = \tan(\pi) = 0 \].

Step 3: Compute the Derivative of the Function

The derivative of \(f(x) = \tan(x)\) is \(f'(x) = \sec^2(x)\).

Step 4: Evaluate the Derivative at \(x = a\)

Substitute \(x = \pi\) into the derivative:\[ f'(\pi) = \sec^2(\pi) = 1 \] because \(\sec(\pi) = \frac{1}{\cos(\pi)} = -1\). Thus, \( \sec^2(\pi) = (-1)^2 = 1 \).

Step 5: Construct the Linearization \(L(x)\)

Using the linearization formula:\[ L(x) = f(\pi) + f'(\pi)(x - \pi) \]\[ L(x) = 0 + 1(x - \pi) \]Thus, the linear approximation \(L(x)\) is:\[ L(x) = x - \pi \].

Key Concepts

Tangent Line Approximation

Linearization of a function through tangent line approximation is a fundamental concept to approximate a function around a specific point. Imagine you have a curve representing a function, and you want to understand its behavior around a particular point without dealing with the complexities of the curve itself.
This is where the tangent line comes into play. It's a straight line that just touches the curve at that point, reflecting the slope and direction of the function at that exact spot.
By using the formula:

• \( L(x) = f(a) + f'(a)(x-a) \)

we can calculate the linear approximation. This equation tells us how the function behaves close to the point \( a \), leveraging the value of the function and its derivative at \( a \).
For instance, for \( f(x) = \tan(x) \) at \( x = \pi \), the tangent line approximation gives us a simple linear expression \( L(x) = x - \pi \).
By replacing \( x \) with values near \( \pi \), you can estimate \( \tan(x) \) without complicated calculations.

Derivative Evaluation

Understanding derivative evaluation is crucial in building the linearization of a function. The derivative represents the rate of change or the slope of the function at a particular point.
To compute the linear approximation, we need to evaluate the derivative \( f'(x) \) at \( x = a \).
Here’s a step-by-step method:

• Identify the original function, say \( f(x) = \tan(x) \).
• Calculate the derivative, which for \( \tan(x) \) is \( f'(x) = \sec^2(x) \).
• Substitute \( x = a \) into this derivative to find \( f'(a) \), e.g., \( f'(\pi) = \sec^2(\pi) = 1 \).

At \( x = \pi \), the derivative of our function provides the slope of the tangent line. This slope is critical for constructing an accurate tangent line approximation.
Through derivative evaluation, such as with \( \tan \), it becomes evident how changes around a point are modeled accurately, simplifying complex analyses.

Trigonometric Functions

Trigonometric functions, like \( \tan(x) \), play a significant role in a wide range of mathematical applications. These functions relate the angles of a triangle to the lengths of its sides, but they also serve well in various calculus concepts.
Key aspects:

• \( \tan(x) \), the tangent function, represents the ratio of the opposite side to the adjacent side in a right triangle.
• Its properties and periodic behavior make it useful in wave patterns and oscillations.
• The derivative, \( \sec^2(x) \), reflects the rate at which the tangent function changes, instrumental in understanding the function’s behavior near specific points.

When we evaluated \( \tan(\pi) \) and found it equals zero, it helped in constructing the linearization. Knowing these fundamental properties aids not only in solving math problems but also in applying these concepts to real-world scenarios involving cycles and periodic trends.