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Chapter 3: Problem 8

Find the linearization of \(f(x)\) at \(x=a\). \(f(x)=\tan x, a=0\)

Short Answer

Expert verified
The linearization of \(f(x) = \tan x\) at \(a = 0\) is \(L(x) = x\).

Step by step solution

01

- Understand the Formula for Linearization

The linearization of a function at a point is given by the formula: \[ L(x) = f(a) + f'(a) (x - a) \] where \( a \) is the point around which the function is being linearized.
02

- Evaluate the Function at \( x = a \)

We need to find \( f(a) \) for \( f(x) = \tan x \) at \( a = 0 \). So, \( f(0) = \tan(0) = 0 \).
03

- Compute the Derivative of the Function

The derivative of \( f(x) = \tan x \) is \( f'(x) = \frac{d}{dx} (\tan x) = \frac{1}{\tan^2 x + 1} = \frac{1}{\frac{1}{\tan^2 x} + 1} \).
04

- Evaluate the Derivative at \( x = a \)

We now need \( f'(a) \) for \( f(x) = \tan x \) at \( a = 0 \). So, \( f'(0) = \frac{1}{\tan^2 (0) + 1} = \frac{1}{0 + 1} = 1 \).
05

- Construct the Linearization

We have \( f(a) = 0 \) and \( f'(a) = 1 \). Using the linearization formula: \[ L(x) = 0 + 1 \times (x - 0) \]. Thus, the linearization is \( L(x) = x \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tangent Function
Understanding the tangent function, or \(\tan x\), is crucial in calculus. The tangent function relates the angle in a right triangle to the ratio of the opposite side to the adjacent side. In calculus, it鈥檚 especially important because it behaves in predictable patterns which we can manipulate using derivatives to find linear approximations or linearizations.
Derivative Calculation
Calculating a derivative is like finding how a function changes at any point. For \(\tan x\), its derivative is \(\frac{d}{dx} (\tan x) = \sec^2 x\). This comes from applying the rules of differentiation to the tangent function. Remember, derivatives help us understand the slope of the tangent line to the curve of the function at a specific point.
Calculus for Life Sciences
Calculus is widely used in life sciences to model growth, understand rates of change, and predict behavior in biological systems. Linearization, for example, helps simplify complex nonlinear models, making them easier to analyze and interpret. It's not just about solving equations but applying these techniques to understand and solve real-world problems.

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