Question

the point ((1, -8)) is on the graph of the function (f(x) = -\frac{8}{x}) as shown.

answer the parts below to estimate the slope of the tangent line at ((1, -8)).

(a) for each value of (x) given in the table below, find the slope of the secant line passing through ((1, -8)) and ((x, f(x))). do not round intermediate computations, and round your answers to 4 decimal places if necessary.

value of (x)	0.8	0.9	0.99	0.999	(\to 1 leftarrow)	1.001	1.01	1.1	1.2

(b) given the graph and the table above, give the apparent slope of the tangent line to the graph of (f(x)) at ((1, -8)).

(square)

Explanation:

Step1: Recall secant slope formula

The slope of the secant line through points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{f(x_2)-f(x_1)}{x_2-x_1}$. Here, $(x_1,y_1)=(1,-8)$, so $m=\frac{f(x)-(-8)}{x-1}=\frac{-\frac{8}{x}+8}{x-1}$.

Step2: Simplify the slope formula

$$\begin{align*} m&=\frac{\frac{-8 + 8x}{x}}{x-1}\\ &=\frac{8(x-1)}{x(x-1)}\\ &=\frac{8}{x} \quad (x eq1) \end{align*}$$

Step3: Calculate slopes for each x value

• For $x=0.8$: $m=\frac{8}{0.8}=10.0000$
• For $x=0.9$: $m=\frac{8}{0.9}\approx8.8889$
• For $x=0.99$: $m=\frac{8}{0.99}\approx8.0808$
• For $x=0.999$: $m=\frac{8}{0.999}\approx8.0080$
• For $x=1.001$: $m=\frac{8}{1.001}\approx7.9920$
• For $x=1.01$: $m=\frac{8}{1.01}\approx7.9208$
• For $x=1.1$: $m=\frac{8}{1.1}\approx7.2727$
• For $x=1.2$: $m=\frac{8}{1.2}\approx6.6667$

Step4: Estimate tangent slope

As $x$ approaches 1, the secant slopes approach 8.

Answer:

Part (a)

Value of $x$	Slope
0.9	8.8889
0.99	8.0808
0.999	8.0080
1.001	7.9920
1.01	7.9208
1.1	7.2727
1.2	6.6667

Part (b)

8