evaluate the difference quotient  f(x)=2x   f(x+h)-f)x)/h

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.

evaluate the difference quotient  f(x)=2x   f(x+h)-f(x)/h

Explore how to evaluate the difference quotient for the linear function f(x) = 2x. This problem demonstrates key calculus concepts such as the difference quotient and rate of change. Perfect for students in Grades 10-12 looking to strengthen their understanding of fundamental calculus principles.

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