Question

the point (3, 1) is on the graph of the function $f(x) = -(x - 4)^3$ as shown.
answer the parts below to estimate the instantaneous rate of change of $f(x)$ at $x = 3$.
(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	$2.9, 3$	$2.99, 3$	$2.999, 3$	$3, 3.001$	$3, 3.01$	$3, 3.1$

(b) given the graph and the table above, give the apparent instantaneous rate of change of $f(x)$ at $x = 3$.
$square$

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 $f(3)=1$, so we calculate $f(x)$ for each interval endpoint first, then apply the formula.

Step1: Define average rate formula

$\text{Average rate of change} = \frac{f(b)-f(a)}{b-a}$

Step2: Calculate for $[2.9, 3]$

First find $f(2.9) = -(2.9-4)^3 = -(-1.1)^3 = 1.331$
$\frac{f(3)-f(2.9)}{3-2.9} = \frac{1 - 1.331}{0.1} = -3.31$

Step3: Calculate for $[2.99, 3]$

First find $f(2.99) = -(2.99-4)^3 = -(-1.01)^3 = 1.030301$
$\frac{f(3)-f(2.99)}{3-2.99} = \frac{1 - 1.030301}{0.01} = -3.0301$

Step4: Calculate for $[2.999, 3]$

First find $f(2.999) = -(2.999-4)^3 = -(-1.001)^3 = 1.003003001$
$\frac{f(3)-f(2.999)}{3-2.999} = \frac{1 - 1.003003001}{0.001} = -3.0030$

Step5: Calculate for $[3, 3.001]$

First find $f(3.001) = -(3.001-4)^3 = -(-0.999)^3 = 0.997002999$
$\frac{f(3.001)-f(3)}{3.001-3} = \frac{0.997002999 - 1}{0.001} = -2.9970$

Step6: Calculate for $[3, 3.01]$

First find $f(3.01) = -(3.01-4)^3 = -(-0.99)^3 = 0.970299$
$\frac{f(3.01)-f(3)}{3.01-3} = \frac{0.970299 - 1}{0.01} = -2.9701$

Step7: Calculate for $[3, 3.1]$

First find $f(3.1) = -(3.1-4)^3 = -(-0.9)^3 = 0.729$
$\frac{f(3.1)-f(3)}{3.1-3} = \frac{0.729 - 1}{0.1} = -2.71$

Step8: Estimate instantaneous rate

As the interval width approaches 0, the average rates approach -3.

Answer:

Part (a)

Interval	Average rate of change
$[2.99, 3]$	$-3.0301$
$[2.999, 3]$	$-3.0030$
$[3, 3.001]$	$-2.9970$
$[3, 3.01]$	$-2.9701$
$[3, 3.1]$	$-2.7100$

Part (b)

$-3$