Use the limit process to find the slope of the graph of the following function at the point $( 5 , - 27 )$.
$$g ( x ) = - 7 - 4 x$$
A) $- 11$
B) $- 27$
C) $- 3$
D) $- 4$
E) 5

Question

Quizplus · Accepted Answer

The slope of a function at a point is given by the derivative of the function at that point. Alternatively, we can use the limit process to find the slope.

The slope of the tangent line to the graph of $g(x)$ at $x = 5$ is given by the limit:

$$\lim_{h \rightarrow 0} \frac{g(5+h) - g(5)}{h}$$

We have:
\begin{align*}
g(5+h) &= -7 -4(5+h)\
&= -27-4h
\end{align*}

And:
\begin{align*}
g(5) &= -7-4(5)\
&= -27
\end{align*}

Substituting these into the limit expression, we get:

$$\lim_{h \rightarrow 0} \frac{(-27-4h) - (-27)}{h} = \lim_{h \rightarrow 0} (-4) = -4$$

Therefore, the slope of the graph of $g(x)$ at the point $(5,-27)$ is $-4$. Thus, the best choice for this question is D) $-4$.

Use the Limit Process to Find the Slope of the Graph