Question

the point $(-5, 8)$ is on the graph of the function $f(x) = -dfrac{8}{4 + x}$ as shown.

answer the parts below to estimate the slope of the tangent line at $(-5, 8)$.

(a) for each value of $x$ given in the table below, find the slope of the secant line passing through $(-5, 8)$ and $(x, f(x))$. do not round intermediate computations, and round your answers to 4 decimal places if necessary.

value of $x$	$-5.2$	$-5.1$	$-5.01$	$-5.001$	$\to -5 leftarrow$	$-4.999$	$-4.99$	$-4.9$	$-4.8$

(b) given the graph and the table above, give the apparent slope of the tangent line to the graph of $f(x)$ at $(-5, 8)$.
$square$

Explanation:

Step1: Recall secant slope formula

The slope of the secant line between $(-5, 8)$ and $(x, f(x))$ is $\frac{f(x)-8}{x-(-5)}=\frac{f(x)-8}{x+5}$, where $f(x)=-\frac{8}{4+x}$.

Step2: Calculate for $x=-5.2$

First find $f(-5.2)=-\frac{8}{4+(-5.2)}=-\frac{8}{-1.2}=\frac{20}{3}\approx6.6667$.
Slope: $\frac{\frac{20}{3}-8}{-5.2+5}=\frac{-\frac{4}{3}}{-0.2}=\frac{20}{3}\approx6.6667$

Step3: Calculate for $x=-5.1$

$f(-5.1)=-\frac{8}{4+(-5.1)}=-\frac{8}{-1.1}=\frac{80}{11}\approx7.2727$.
Slope: $\frac{\frac{80}{11}-8}{-5.1+5}=\frac{-\frac{8}{11}}{-0.1}=\frac{80}{11}\approx7.2727$

Step4: Calculate for $x=-5.01$

$f(-5.01)=-\frac{8}{4+(-5.01)}=-\frac{8}{-1.01}=\frac{800}{101}\approx7.9208$.
Slope: $\frac{\frac{800}{101}-8}{-5.01+5}=\frac{-\frac{8}{101}}{-0.01}=\frac{800}{101}\approx7.9208$

Step5: Calculate for $x=-5.001$

$f(-5.001)=-\frac{8}{4+(-5.001)}=-\frac{8}{-1.001}=\frac{8000}{1001}\approx7.9920$.
Slope: $\frac{\frac{8000}{1001}-8}{-5.001+5}=\frac{-\frac{8}{1001}}{-0.001}=\frac{8000}{1001}\approx7.9920$

Step6: Calculate for $x=-4.999$

$f(-4.999)=-\frac{8}{4+(-4.999)}=-\frac{8}{-0.999}=\frac{8000}{999}\approx8.0080$.
Slope: $\frac{\frac{8000}{999}-8}{-4.999+5}=\frac{\frac{8}{999}}{0.001}=\frac{8000}{999}\approx8.0080$

Step7: Calculate for $x=-4.99$

$f(-4.99)=-\frac{8}{4+(-4.99)}=-\frac{8}{-0.99}=\frac{800}{99}\approx8.0808$.
Slope: $\frac{\frac{800}{99}-8}{-4.99+5}=\frac{\frac{8}{99}}{0.01}=\frac{800}{99}\approx8.0808$

Step8: Calculate for $x=-4.9$

$f(-4.9)=-\frac{8}{4+(-4.9)}=-\frac{8}{-0.9}=\frac{80}{9}\approx8.8889$.
Slope: $\frac{\frac{80}{9}-8}{-4.9+5}=\frac{\frac{8}{9}}{0.1}=\frac{80}{9}\approx8.8889$

Step9: Calculate for $x=-4.8$

$f(-4.8)=-\frac{8}{4+(-4.8)}=-\frac{8}{-0.8}=10$.
Slope: $\frac{10-8}{-4.8+5}=\frac{2}{0.2}=10$

Step10: Estimate tangent slope

Observe secant slopes approach 8 as $x\to-5$.

Answer:

Part (a) Slopes (rounded to 4 decimals):

Value of $x$	-5.2	-5.1	-5.01	-5.001	-4.999	-4.99	-4.9	-4.8

Part (b)

8