Question

score on last try: 0 of 1 pts. see details for more. > next question get a similar question you can retry this question below consider the graph of f given below. evaluate the following limit. $lim_{delta x
ightarrow0}\frac{f(7 + delta x)-f(7)}{delta x}=$

Explanation:

Step1: Recall the definition of the derivative

The given limit $\lim_{\Delta x
ightarrow0}\frac{f(7 + \Delta x)-f(7)}{\Delta x}$ is the definition of the derivative of the function $y = f(x)$ at $x = 7$, i.e., $f^{\prime}(7)$. Geometrically, the derivative of a function at a point is the slope of the tangent line to the graph of the function at that point.

Step2: Identify two - points on the tangent - like line at $x = 7$

To find the slope of the tangent line at $x = 7$, we consider the part of the graph around $x = 7$. The graph of the function is a straight - line segment from $x=6$ to $x = 8$. The two points on this line segment are $(6,2)$ and $(8,- 4)$.

Step3: Calculate the slope

The slope $m$ of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by the formula $m=\frac{y_2 - y_1}{x_2 - x_1}$. Here, $x_1 = 6,y_1 = 2,x_2=8,y_2=-4$. So, $m=\frac{-4 - 2}{8 - 6}=\frac{-6}{2}=-3$.

Answer:

$-3$