Let's solve the given problem step by step.

The function provided is:

$f(x) = \frac{1}{x+2}$

Part (a): Average rate of change on the interval $[1, 4]$

The formula for the average rate of change of a function $f(x)$ over an interval $[a, b]$ is given by:

$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

For the interval $[1, 4]$:

• $a = 1$
• $b = 4$

Now, we compute the values of $f(1)$ and $f(4)$:

$f(1) = \frac{1}{1+2} = \frac{1}{3}$ $f(4) = \frac{1}{4+2} = \frac{1}{6}$

Now, using the formula for the average rate of change:

$\text{Average Rate of Change} = \frac{f(4) - f(1)}{4 - 1} = \frac{\frac{1}{6} - \frac{1}{3}}{3}$

Simplifying the difference:

$\frac{1}{6} - \frac{1}{3} = \frac{1}{6} - \frac{2}{6} = -\frac{1}{6}$

So, the average rate of change is:

$\text{Average Rate of Change} = \frac{-\frac{1}{6}}{3} = -\frac{1}{18}$

Part (b): Instantaneous rate of change at $x = 3$

The instantaneous rate of change of a function at a particular point is given by the derivative $f'(x)$. Let's find the derivative of $f(x) = \frac{1}{x+2}$ using the chain rule.

1. We can rewrite $f(x)$ as $f(x) = (x + 2)^{-1}$.
2. Differentiate using the power rule:

$f'(x) = -1 \cdot (x+2)^{-2} \cdot 1 = -\frac{1}{(x+2)^2}$

Now, we substitute $x = 3$ into the derivative to find the instantaneous rate of change at $x = 3$:

$f'(3) = -\frac{1}{(3+2)^2} = -\frac{1}{5^2} = -\frac{1}{25}$

Final Answers:

• (a) The average rate of change on the interval $[1, 4]$ is $-\frac{1}{18}$.
• (b) The instantaneous rate of change at $x = 3$ is $-\frac{1}{25}$.

Would you like further details or clarifications on these steps?

Here are five related questions you might want to explore:

1. How does the average rate of change differ from the instantaneous rate of change conceptually?
2. Can you find the instantaneous rate of change at $x = 2$ for the same function?
3. What is the behavior of $f'(x)$ as $x$ approaches infinity?
4. How does the average rate of change vary with different intervals?
5. Can you sketch the graph of $f(x) = \frac{1}{x+2}$ and its derivative?

Tip: For smooth functions, the derivative (instantaneous rate of change) at a point is the slope of the tangent line to the curve at that point.