1. determine the slope of the secant line passing through the curve $y = \\frac{1}{x}$ at $x = 1$ and $x =…

Explanation:

Step1: Find the y - values

When $x = 1$, $y_1=\frac{1}{1}=1$. When $x = 3$, $y_2=\frac{1}{3}$.

Step2: Use the slope formula

The slope $m$ of a secant line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Here, $x_1 = 1,y_1 = 1,x_2 = 3,y_2=\frac{1}{3}$. So $m=\frac{\frac{1}{3}-1}{3 - 1}$.

Step3: Simplify the expression

First, simplify the numerator: $\frac{1}{3}-1=\frac{1 - 3}{3}=-\frac{2}{3}$. Then, $m=\frac{-\frac{2}{3}}{2}=-\frac{1}{3}$.

Answer:

$-\frac{1}{3}$