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 instantaneous rate of change of $f(x)$ at $x = -5$.
(a) find the average rate of change of $f(x)$ over each given interval in the table below. do not round intermediate computations, and round your answers to 4 decimal places if necessary.

interval	$-5.1, -5$	$-5.01, -5$	$-5.001, -5$	$-5, -4.999$	$-5, -4.99$	$-5, -4.9$

(b) given the graph and the table above, give the apparent instantaneous rate of change of $f(x)$ at $x = -5$.
$square$

Explanation:

The average rate of change of a function $f(x)$ over an interval $[a,b]$ is given by $\frac{f(b)-f(a)}{b-a}$, and we know $f(-5)=8$.

Step1: Rate for $[-5.1,-5]$

Calculate $f(-5.1)$ first:
$f(-5.1)=-\frac{8}{4+(-5.1)}=-\frac{8}{-1.1}=\frac{80}{11}\approx7.2727$
Average rate: $\frac{f(-5)-f(-5.1)}{-5-(-5.1)}=\frac{8-\frac{80}{11}}{0.1}=\frac{\frac{88-80}{11}}{0.1}=\frac{\frac{8}{11}}{0.1}=\frac{80}{11}\approx7.2727$

Step2: Rate for $[-5.01,-5]$

Calculate $f(-5.01)$:
$f(-5.01)=-\frac{8}{4+(-5.01)}=-\frac{8}{-1.01}=\frac{800}{101}\approx7.9208$
Average rate: $\frac{f(-5)-f(-5.01)}{-5-(-5.01)}=\frac{8-\frac{800}{101}}{0.01}=\frac{\frac{808-800}{101}}{0.01}=\frac{\frac{8}{101}}{0.01}=\frac{800}{101}\approx7.9208$

Step3: Rate for $[-5.001,-5]$

Calculate $f(-5.001)$:
$f(-5.001)=-\frac{8}{4+(-5.001)}=-\frac{8}{-1.001}=\frac{8000}{1001}\approx7.9920$
Average rate: $\frac{f(-5)-f(-5.001)}{-5-(-5.001)}=\frac{8-\frac{8000}{1001}}{0.001}=\frac{\frac{8008-8000}{1001}}{0.001}=\frac{\frac{8}{1001}}{0.001}=\frac{8000}{1001}\approx7.9920$

Step4: Rate for $[-5,-4.999]$

Calculate $f(-4.999)$:
$f(-4.999)=-\frac{8}{4+(-4.999)}=-\frac{8}{-0.999}=\frac{8000}{999}\approx8.0080$
Average rate: $\frac{f(-4.999)-f(-5)}{-4.999-(-5)}=\frac{\frac{8000}{999}-8}{0.001}=\frac{\frac{8000-7992}{999}}{0.001}=\frac{\frac{8}{999}}{0.001}=\frac{8000}{999}\approx8.0080$

Step5: Rate for $[-5,-4.99]$

Calculate $f(-4.99)$:
$f(-4.99)=-\frac{8}{4+(-4.99)}=-\frac{8}{-0.99}=\frac{800}{99}\approx8.0808$
Average rate: $\frac{f(-4.99)-f(-5)}{-4.99-(-5)}=\frac{\frac{800}{99}-8}{0.01}=\frac{\frac{800-792}{99}}{0.01}=\frac{\frac{8}{99}}{0.01}=\frac{800}{99}\approx8.0808$

Step6: Rate for $[-5,-4.9]$

Calculate $f(-4.9)$:
$f(-4.9)=-\frac{8}{4+(-4.9)}=-\frac{8}{-0.9}=\frac{80}{9}\approx8.8889$
Average rate: $\frac{f(-4.9)-f(-5)}{-4.9-(-5)}=\frac{\frac{80}{9}-8}{0.1}=\frac{\frac{80-72}{9}}{0.1}=\frac{\frac{8}{9}}{0.1}=\frac{80}{9}\approx8.8889$

Step7: Estimate instantaneous rate

Observe the average rates approach 8 as intervals narrow around $x=-5$.

Answer:

Part (a)

Interval	Average rate of change
$[-5.01, -5]$	$7.9208$
$[-5.001, -5]$	$7.9920$
$[-5, -4.999]$	$8.0080$
$[-5, -4.99]$	$8.0808$
$[-5, -4.9]$	$8.8889$

Part (b)

$8$