We are asked to calculate the difference quotient for the function $f(x) = \sqrt{x} + 2$ at $x = 2$, using three values for $h$. The difference quotient formula is:

$\frac{f(2 + h) - f(2)}{h}$

Step 1: Calculate $f(2)$

Since $f(x) = \sqrt{x} + 2$, we calculate $f(2)$:

$f(2) = \sqrt{2} + 2$

We'll leave $\sqrt{2}$ as an approximation later in the calculation.

Step 2: Calculate $f(2 + h)$ for different values of $h$

We need to compute this for three values: $h = 0.1$, $h = 0.01$, and $h = -0.1$. The general expression for $f(2 + h)$ is:

$f(2 + h) = \sqrt{2 + h} + 2$

Step 3: Compute the difference quotient for each $h$

Now, we calculate the difference quotient for each value of $h$ by plugging $f(2 + h)$ and $f(2)$ into the difference quotient formula:

$\frac{f(2 + h) - f(2)}{h}$

Step 4: Derivative estimate at $x = 2$

Once we have the difference quotients, we estimate the derivative at $x = 2$. It is given that the derivative is of the form $\frac{1}{n}$, so we will solve for $n$.

I'll go ahead and calculate these values now.The calculations yield the following values for the difference quotient:

• For $h = 0.1$: $\frac{f(2 + 0.1) - f(2)}{0.1} = 0.3492$
• For $h = 0.01$: $\frac{f(2 + 0.01) - f(2)}{0.01} = 0.3531$
• For $h = -0.1$: $\frac{f(2 - 0.1) - f(2)}{-0.1} = 0.3581$

Step 5: Determine $n$

The derivative of $f(x)$ at $x = 2$ seems to approach $\frac{1}{n}$. From the difference quotient values, we see that the derivative is approximately 0.35. Therefore, $n$ is likely close to 3 because $\frac{1}{n} \approx 0.35$.

Thus, $n = 3$.

Let me know if you need more details or further clarification!

Here are five related questions to explore:

1. How does the difference quotient relate to the derivative definition?
2. Why does the value of $h$ affect the accuracy of the derivative approximation?
3. What is the geometrical interpretation of the derivative at $x = 2$?
4. Can the derivative be calculated exactly using calculus techniques for $f(x) = \sqrt{x} + 2$?
5. How does the function $f(x)$ behave as $x$ increases or decreases?

Tip: For small $h$ values, the difference quotient gives a better approximation of the derivative!