Chapter 2, Problem 1RE

Approximating the Slope of a Graph In Exercises 1-4, approximate the slope of the graph at the point (x, y). (Each square on the grid is 1 unit by 1 unit.)

To determine

To calculate: The slope of the provided graph at the given point $\left(x,y\right)$.

The provided graph is shown below,

Answer to Problem 1RE

Solution:

The slope of the given curve at point $\left(x,y\right)$ is $-2$.

Explanation of Solution

Given Information:

The provided graph is shown below,

Each grid in the curve represents 1 unit by 1 unit as dimensions

Formula used:

Consider the equation of the given curve as $y=f\left(x\right)$, the slope of the curve is given by $f^{\prime }\left(x\right)$, that is given by,

$f^{\prime }\left(x\right)=\frac{\Delta y}{\Delta x}$

Here $\Delta y$ is the change in the y direction and $\Delta x$ is the change in the x direction.

Calculation:

The change in y is 4 units decreasing for 2 unit increase in x direction.

Hence, using formula for the slope of a function,

$f^{\prime }\left(x\right)=\frac{\Delta y}{\Delta x}$

Here $\Delta y$ is the change in the y direction and $\Delta x$ is the change in the x direction.

The slope of the given graph is,

$\begin{array}{l}{f}^{\prime }\left(x\right)=\frac{-4}{2}\\ f^{\prime }\left(x\right)=-2\end{array}$

Now, as there is a decrease of 2 units in y as per increase in a unit of x

Therefore, the slope at point $\left(x,y\right)$ is $-2$.