Given the function  
f
(
x
)
=
2
x
+
1
 
(a) Calculate the average rate of change of the function from  
1
  to  
x
 
(b) Find the limit of the average rate of change of the function from  
x
  to  
1
 .

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!

Given the function f(x) = 2x + 1 (a) Calculate the average rate of change of the function from 1 to x. (b) Find the limit of the average rate of change of the function from x to 1.

This tutorial demonstrates how to calculate the average rate of change of the function f(x) = 2x + 1 from x = 1 to a variable x, and find the limit of this average rate as x approaches 1. Explore key algebraic concepts and learn about limits and rates of change. Suitable for students in Grades 10-12.

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