Math Problem Statement

Find the linear approximation of f(x) = ln(x) at x = 1 and use it to estimate ln(1.46).

Solution

The problem asks us to find the linear approximation of f(x)=ln(x)f(x) = \ln(x) at x=1x = 1 and then use it to estimate ln(1.46)\ln(1.46).

Step 1: Linear Approximation Formula

The linear approximation of a function f(x)f(x) around a point x=ax = a is given by:

L(x)=f(a)+f(a)(xa)L(x) = f(a) + f'(a)(x - a)

In this case, f(x)=ln(x)f(x) = \ln(x) and a=1a = 1.

Step 2: Calculate f(1)f(1)

Since f(x)=ln(x)f(x) = \ln(x), we have:

f(1)=ln(1)=0f(1) = \ln(1) = 0

Step 3: Calculate f(x)f'(x)

The derivative of f(x)=ln(x)f(x) = \ln(x) is:

f(x)=1xf'(x) = \frac{1}{x}

Thus, f(1)=11=1f'(1) = \frac{1}{1} = 1.

Step 4: Form the Linear Approximation L(x)L(x)

Using the formula L(x)=f(a)+f(a)(xa)L(x) = f(a) + f'(a)(x - a), we substitute a=1a = 1, f(1)=0f(1) = 0, and f(1)=1f'(1) = 1:

L(x)=0+1(x1)=x1L(x) = 0 + 1(x - 1) = x - 1

Thus, the linear approximation is:

L(x)=x1L(x) = x - 1

Step 5: Estimate ln(1.46)\ln(1.46)

To estimate ln(1.46)\ln(1.46), we substitute x=1.46x = 1.46 into the linear approximation L(x)L(x):

L(1.46)=1.461=0.46L(1.46) = 1.46 - 1 = 0.46

Thus, the estimated value of ln(1.46)\ln(1.46) is approximately 0.460.46.

Final Answers:

  1. The linear approximation L(x)=x1L(x) = x - 1.
  2. The estimated value of ln(1.46)0.46\ln(1.46) \approx 0.46.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

  1. What is the error between the actual value of ln(1.46)\ln(1.46) and the linear approximation?
  2. How does linear approximation change if we expand around a different point, say x=2x = 2?
  3. Why does linear approximation work well for values of xx close to 1?
  4. How does the linear approximation compare with higher-order approximations (e.g., Taylor series)?
  5. What is the second derivative of ln(x)\ln(x) and how does it impact the curvature near x=1x = 1?

Tip: Linear approximation is only accurate for values close to the point of approximation. For better accuracy, consider using more terms in the Taylor series.

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Math Problem Analysis

Mathematical Concepts

Calculus
Linear Approximation
Derivatives
Natural Logarithm

Formulas

L(x) = f(a) + f'(a)(x - a)
f(x) = ln(x)
f'(x) = 1/x

Theorems

Linear Approximation Formula

Suitable Grade Level

Grades 11-12 (High School Calculus)