Given the function
f ( x )

Given the function $f(x) = 2x + 1$:

(a) Calculate the average rate of change of the function from 1 to $x$:

The average rate of change of a function between two points $x_1$ and $x_2$ is calculated using the formula:

$\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

In this case, we want the average rate of change from $x_1 = 1$ to $x_2 = x$, so:

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

Substitute the values of $f(x)$ and $f(1)$:

$f(x) = 2x + 1, \quad f(1) = 2(1) + 1 = 3$

Now calculate:

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

For $x \neq 1$, the $(x - 1)$ terms cancel out, and we are left with:

$\text{Average Rate of Change} = 2$

So, the average rate of change of the function from 1 to $x$ is constant and equals $2$, for $x \neq 1$.

(b) Find the limit of the average rate of change of the function from $x$ to 1:

We want to find:

$\lim_{x \to 1} \frac{f(x) - f(1)}{x - 1}$

We already know the expression for the average rate of change:

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

Since the average rate of change is always $2$ for all $x \neq 1$, the limit as $x \to 1$ is simply:

$\lim_{x \to 1} 2 = 2$

Final Answers:

• (a) The average rate of change of the function from $1$ to $x$ is $2$.
• (b) The limit of the average rate of change of the function from $x$ to $1$ is $2$.

Would you like further clarification on any part of this solution?

Here are five related questions:

1. What is the geometric interpretation of the average rate of change for linear functions?
2. How would the average rate of change differ for non-linear functions?
3. Can you find the derivative of the function $f(x) = 2x + 1$ and explain its significance?
4. How does the concept of average rate of change relate to the slope of a line?
5. How would you calculate the instantaneous rate of change at a point for a more complex function?

Tip: The average rate of change of a linear function is always equal to its constant slope!