Let's solve the given problem step by step.

We are asked to find the expression for:

$\frac{f(2) - f(1)}{2 - 1}$

where the function $f(x) = 6 - x^2$.

Step 1: Calculate $f(2)$

Substitute $x = 2$ into the function:

$f(2) = 6 - (2^2) = 6 - 4 = 2$

Step 2: Calculate $f(1)$

Substitute $x = 1$ into the function:

$f(1) = 6 - (1^2) = 6 - 1 = 5$

Step 3: Compute the difference

Now, substitute the values into the expression:

$\frac{f(2) - f(1)}{2 - 1} = \frac{2 - 5}{1} = \frac{-3}{1} = -3$

Final Answer:

$\frac{f(2) - f(1)}{2 - 1} = -3$

Would you like more details on this or have further questions? Here are some related questions to expand your understanding:

1. What is the average rate of change of $f(x)$ over the interval [1, 3]?
2. How do we interpret the value of $\frac{f(2) - f(1)}{2 - 1}$ geometrically?
3. What are the intercepts of the graph of $f(x) = 6 - x^2$?
4. Can you find the vertex of the parabola given by $f(x) = 6 - x^2$?
5. How does the value of $f'(x)$ (the derivative) relate to the slope calculation we just performed?

Tip:

In problems like this, the expression $\frac{f(a) - f(b)}{a - b}$ represents the average rate of change of the function over the interval $[a, b]$, which geometrically is the slope of the secant line connecting the points on the curve at $x = a$ and $x = b$.