To evaluate the difference quotient $\frac{f(x+h) - f(x)}{h}$ for the function $f(x) = 2x$, follow these steps:

1. Substitute $f(x)$ and $f(x+h)$:

   • $f(x) = 2x$
   • $f(x+h) = 2(x+h)$

2. Calculate $f(x+h) - f(x)$: $f(x+h) - f(x) = 2(x+h) - 2x$ Simplify the expression: $f(x+h) - f(x) = 2x + 2h - 2x = 2h$

3. Divide by $h$ to get the difference quotient: $\frac{f(x+h) - f(x)}{h} = \frac{2h}{h}$ Simplify: $\frac{f(x+h) - f(x)}{h} = 2$

So, the difference quotient $\frac{f(x+h) - f(x)}{h}$ for the function $f(x) = 2x$ is $2$.

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

Here are 5 related questions to explore further:

1. What is the difference quotient of $f(x) = 3x^2$?
2. How does the difference quotient relate to the derivative?
3. What does the difference quotient represent geometrically?
4. Can you evaluate the difference quotient for $f(x) = \sin(x)$?
5. How would you compute the difference quotient for a function $f(x)$ that is a polynomial of degree 3?

Tip: The difference quotient is a fundamental concept in calculus, representing the average rate of change of a function over a small interval. It is a stepping stone towards understanding derivatives.