To solve the given problem, we need to understand the expressions and what they represent. Here's a breakdown of the expressions:

1. Expression Interpretation:
   • $f(x) - 2x - 1$: This expression represents a function $f(x)$ minus $2x + 1$. We need the explicit form of $f(x)$ to further analyze it.
   • $\frac{f(x+h) - f(x)}{h}$: This expression is the difference quotient of the function $f(x)$. It is used to calculate the derivative of $f(x)$ as $h$ approaches 0, which is the definition of the derivative.

Since you are working with these expressions, let’s assume we want to find the derivative of $f(x)$ and analyze it in the context of the given function.

Step-by-Step Solution:

1. Assume the function $f(x)$: Let's assume $f(x) = 2x + 1$ to match the context of the expressions provided.

2. Calculate $f(x+h)$: $f(x+h) = 2(x+h) + 1 = 2x + 2h + 1$

3. Compute the Difference Quotient: Using the difference quotient formula $\frac{f(x+h) - f(x)}{h}$, we substitute $f(x+h)$ and $f(x)$:

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

   Simplify the expression: $= \frac{2x + 2h + 1 - 2x - 1}{h} = \frac{2h}{h} = 2$

4. Interpretation: The derivative $f'(x)$ of the function $f(x) = 2x + 1$ is 2, which confirms that the slope of the function is constant.

Summary:

• The expression $f(x) - 2x - 1$ simplifies to 0 if $f(x) = 2x + 1$.
• The difference quotient $\frac{f(x+h) - f(x)}{h}$ simplifies to 2, which is the derivative of $f(x) = 2x + 1$.

Would you like more details, or do you have any questions?

Here are 5 relative questions for further exploration:

1. What if the function $f(x)$ was not linear? How would the derivative change?
2. How does the difference quotient relate to the concept of limits in calculus?
3. Can we find the second derivative of a function from its first derivative?
4. How would the analysis change if $f(x) = 3x^2 + 2x + 1$?
5. What are some applications of derivatives in real-world problems?

Tip: The difference quotient is a fundamental concept for understanding derivatives, which measure the rate at which a function changes with respect to a variable.