Question

the point (4, 9) is on the graph of the function $f(x) = 4sqrt{x} + 1$ as shown.
answer the parts below to estimate the instantaneous rate of change of $f(x)$ at $x = 4$.
(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	3.9, 4	3.99, 4	3.999, 4	4, 4.001	4, 4.01	4, 4.1

(b) given the graph and the table above, give the apparent instantaneous rate of change of $f(x)$ at $x = 4$.
$square$

Explanation:

The average rate of change of a function $f(x)$ over an interval $[a,b]$ is given by $\frac{f(b)-f(a)}{b-a}$. We know $f(4)=9$, so we calculate this for each interval.

Step1: Calculate for [3.9,4]

First find $f(3.9)$:
$f(3.9)=4\sqrt{3.9}+1 \approx 4\times1.97484+1 \approx 7.89936+1=8.89936$
Average rate of change:
$\frac{f(4)-f(3.9)}{4-3.9}=\frac{9-8.89936}{0.1}=\frac{0.10064}{0.1}=1.0064$

Step2: Calculate for [3.99,4]

First find $f(3.99)$:
$f(3.99)=4\sqrt{3.99}+1 \approx 4\times1.997498+1 \approx 7.98999+1=8.98999$
Average rate of change:
$\frac{f(4)-f(3.99)}{4-3.99}=\frac{9-8.98999}{0.01}=\frac{0.01001}{0.01}=1.0010$

Step3: Calculate for [3.999,4]

First find $f(3.999)$:
$f(3.999)=4\sqrt{3.999}+1 \approx 4\times1.99975+1 \approx 7.9990+1=8.9990$
Average rate of change:
$\frac{f(4)-f(3.999)}{4-3.999}=\frac{9-8.9990}{0.001}=\frac{0.001}{0.001}=1.0000$

Step4: Calculate for [4,4.001]

First find $f(4.001)$:
$f(4.001)=4\sqrt{4.001}+1 \approx 4\times2.00025+1 \approx 8.001+1=9.001$
Average rate of change:
$\frac{f(4.001)-f(4)}{4.001-4}=\frac{9.001-9}{0.001}=\frac{0.001}{0.001}=1.0000$

Step5: Calculate for [4,4.01]

First find $f(4.01)$:
$f(4.01)=4\sqrt{4.01}+1 \approx 4\times2.002498+1 \approx 8.00999+1=9.00999$
Average rate of change:
$\frac{f(4.01)-f(4)}{4.01-4}=\frac{9.00999-9}{0.01}=\frac{0.00999}{0.01}=0.9990$

Step6: Calculate for [4,4.1]

First find $f(4.1)$:
$f(4.1)=4\sqrt{4.1}+1 \approx 4\times2.024846+1 \approx 8.09938+1=9.09938$
Average rate of change:
$\frac{f(4.1)-f(4)}{4.1-4}=\frac{9.09938-9}{0.1}=\frac{0.09938}{0.1}=0.9938$

Step7: Estimate instantaneous rate

As intervals approach $x=4$, average rates approach 1.

Answer:

(a)

Interval	Average rate of change
$[3.99, 4]$	1.0010
$[3.999, 4]$	1.0000
$[4, 4.001]$	1.0000
$[4, 4.01]$	0.9990
$[4, 4.1]$	0.9938

(b) 1