Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.
ln 1.05

Verified step by step guidance

1. Identify the function you want to approximate. In this case, it's the natural logarithm function, \( f(x) = \ln(x) \).
2. Choose a value of \( a \) close to 1.05 where the function is easy to compute. A good choice is \( a = 1 \) because \( \ln(1) = 0 \).
3. Find the derivative of the function, \( f'(x) = \frac{1}{x} \). This will be used to find the slope of the tangent line at \( x = a \).
4. Calculate the slope of the tangent line at \( x = a \) using the derivative: \( f'(a) = \frac{1}{1} = 1 \).
5. Use the linear approximation formula \( L(x) = f(a) + f'(a)(x - a) \) to estimate \( \ln(1.05) \). Substitute \( a = 1 \), \( f(a) = 0 \), and \( f'(a) = 1 \) into the formula to get \( L(x) = 0 + 1 \cdot (x - 1) = x - 1 \). Evaluate \( L(1.05) = 1.05 - 1 \).

Key Concepts

Linear Approximation

Linear approximation is a method used to estimate the value of a function near a given point using the tangent line at that point. It is based on the idea that a function can be closely approximated by a linear function when the input is near a specific value. The formula for linear approximation is f(x) ≈ f(a) + f'(a)(x - a), where 'a' is the point of tangency.

Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is a fundamental concept in calculus that provides the slope of the tangent line to the function at that point. For the natural logarithm function, the derivative is given by f'(x) = 1/x, which is essential for calculating the linear approximation.

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base 'e', where 'e' is approximately 2.71828. It is a key function in calculus and is used to solve problems involving exponential growth and decay. Understanding the properties of the natural logarithm, such as its behavior near 1, is crucial for effectively using linear approximations to estimate values like ln(1.05).