The image provides an exercise about Errors of Linear Approximations, focusing on the function $f(x) = \ln(x)$ and its linearization at $x = 1$. The main goal is to calculate and analyze the error terms.

Here's a breakdown of the exercise:

Part (a)

You are asked to compute the following expressions for different values of $x$:

$\frac{R_1(1.02)}{(1.02 - 1)^2}, \quad \frac{R_1(1.002)}{(1.002 - 1)^2}, \quad \frac{R_1(1.0002)}{(1.0002 - 1)^2}$ where $R_1(x) = f(x) - L(x)$ is the error between $f(x) = \ln(x)$ and its linearization $L(x)$ at $x = 1$.

Linearization:

For $f(x) = \ln(x)$, we compute the linear approximation $L(x)$ at $x = 1$ using: $L(x) = f(1) + f'(1)(x - 1)$ Since: $f(1) = \ln(1) = 0 \quad \text{and} \quad f'(x) = \frac{1}{x} \quad \text{so} \quad f'(1) = 1,$ we get the linearization: $L(x) = 0 + 1 \cdot (x - 1) = x - 1.$ Thus, $R_1(x) = \ln(x) - (x - 1)$.

Now, we need to compute these error terms and divide them by $(x - 1)^2$ for $x = 1.02, 1.002,$ and $1.0002$.

Part (b)

The question asks to analyze Formula 1, which is: $R_1(x) = \frac{f''(c)}{2} (x - a)^2$ and determine:

• The factors that affect the magnitude of $R_1(x)$,
• The factor affecting the sign of $R_1(x)$.

Would you like me to calculate these values for part (a) and provide more detailed explanations for part (b)?