Problem 142 - Thomas Calculus in SI Units

Question

The linearization of \(\log _{3} x\) a. Find the linearization of \(f(x)=\log _{3} x\) at \(x=3 .\) Then round its coefficients to two decimal places. b. Graph the linearization and function together in the windows \(0 \leq x \leq 8\) and \(2 \leq x \leq 4\)

Step-by-Step Solution

Answer

Linearization at \(x=3\) is \(L(x) \approx 1 + 0.30(x - 3)\). Graph both in specified windows.

Step 1: Understand Linearization Concept

Linearization is the process of approximating a function using a tangent line at a point. For a given function \(f(x)\) at point \(x = a\), the linearization, \(L(x)\), is given by the formula \(L(x) = f(a) + f'(a)(x - a)\).

Step 2: Determine Function and Derivative

Given \(f(x) = \log_{3} x\), we first express this using natural logarithms as \(f(x) = \frac{\ln x}{\ln 3}\). The derivative is \(f'(x) = \frac{1}{x \ln 3}\).

Step 3: Evaluate at the Given Point

Calculate \(f(3)\) and \(f'(3)\). \[ f(3) = \frac{\ln 3}{\ln 3} = 1 \] \[ f'(3) = \frac{1}{3 \ln 3} \]

Step 4: Construct the Linearization Formula

Substitute \(f(3)\), \(f'(3)\), and \(a = 3\) into the linearization formula:\[ L(x) = 1 + \frac{1}{3 \ln 3}(x - 3) \]

Step 5: Round the Coefficients

Calculate \(\frac{1}{3 \ln 3}\) and round to two decimal places. First, compute \(3 \ln 3 \approx 3 \times 1.0986 \approx 3.2958\), hence the coefficient is approximately \(\frac{1}{3.2958} \approx 0.303\). So, the linearization is:\[ L(x) \approx 1 + 0.30(x - 3) \]

Step 6: Graph the Functions Together

Graph \(f(x) = \log_{3} x\) and \(L(x) = 1 + 0.30(x - 3)\) on the specified intervals \(0 \leq x \leq 8\) and \(2 \leq x \leq 4\). Use graphing software or a calculator to visualize the function and its linear approximation.

Key Concepts

Derivative

The derivative of a function measures the rate at which the function's value changes as the input changes. In simpler terms, it tells us how steep the graph of the function is at any given point. When we think about the derivative, it's like finding the slope of a tangent line that just barely "touches" the graph at a specific point.
For the function \(f(x) = \log_{3} x\), we can re-write this in terms of natural logarithms: \(f(x) = \frac{\ln x}{\ln 3}\). This transformation is crucial as it converts the logarithmic base to natural logarithms, making differentiation simpler.

• To find the derivative \(f'(x)\), we apply the rules of differentiation for \(\ln x\), resulting in: \(f'(x) = \frac{1}{x \ln 3}\).
• This derivative tells us the slope of the tangent line to the curve \(f(x)\) for any value of \(x\).

Understanding how to differentiate logarithmic functions is essential for situations requiring precise calculations of change rates.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions, and they have a base which dictates their growth and rate of change. The logarithmic function \(f(x) = \log_{3} x\) is based on base 3.
Logarithmic functions have several important properties:

• They grow very slowly compared to polynomial and exponential functions.
• The function \(\log_{3} x\) tells us how many times we need to multiply 3 to get \(x\).

The conversion of logarithms to a natural base using the property \(\log_{b} x = \frac{\ln x}{\ln b}\) allows for ease in calculus operations.
By rewriting \(\log_{3} x\) as \(\frac{\ln x}{\ln 3}\), we simplify it enough to find its derivative using standard techniques.

Tangent Line Approximation

The tangent line approximation, or linearization, is a technique used to approximate a given function with a simpler linear function, particularly near a point of interest. In mathematical terms, linearization is represented by the equation:

• \(L(x) = f(a) + f'(a)(x - a)\)

Where \(f(a)\) is the value of the function at point \(a\) and \(f'(a)\) is the slope of the tangent line at \(a\).
For the function \(f(x) = \log_{3} x\) at \(x=3\):

• The value \(f(3) = 1\) because \(\log_{3} 3 = 1\).
• The slope \(f'(3) = \frac{1}{3 \ln 3}\).

Hence, the linear approximation, \(L(x)\), becomes \(1 + 0.30(x - 3)\) after rounding the derivative's coefficient. This line closely approximates \(\log_{3} x\) near \(x = 3\), demonstrating how tangent lines can simplify complex functions in a local interval.