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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.

1. If the position of a particle is given by the following function f(x) = 1/(x + 2), (a) find the average rate of change on the interval [1, 4], (b) find the instantaneous rate of change at x = 3 using the techniques presented in class.

Explore the solution for finding the average rate of change of the function f(x) = 1/(x+2) over the interval [1, 4], and determine the instantaneous rate of change at x = 3 using derivatives. This problem includes step-by-step calculations and explanations, ideal for Grades 11-12.

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