Find the value of f(x+h) - f(x) /H, if f(x)=x2+7$$  $$$$  $$

To find the value of $\frac{f(x+h) - f(x)}{h}$ for the function $f(x) = x^2 + 7$, we need to carefully evaluate each step. Here's the detailed process:

1. **Evaluate $f(x + h)$:**

   Given $f(x) = x^2 + 7$, substitute $x + h$ into the function:
   \[
   f(x + h) = (x + h)^2 + 7
   \]

2. **Expand $f(x + h)$:**

   \[
   f(x + h) = (x + h)^2 + 7 = x^2 + 2xh + h^2 + 7
   \]

3. **Find $f(x + h) - f(x)$:**

   Now, compute the difference:
   \[
   f(x + h) - f(x) = (x^2 + 2xh + h^2 + 7) - (x^2 + 7)
   \]
   Simplify the expression:
   \[
   f(x + h) - f(x) = x^2 + 2xh + h^2 + 7 - x^2 - 7 = 2xh + h^2
   \]

4. **Divide by $h$:**

   \[
   \frac{f(x + h) - f(x)}{h} = \frac{2xh + h^2}{h}
   \]
   Simplify the fraction:
   \[
   \frac{f(x + h) - f(x)}{h} = 2x + h
   \]

So, the value of $\frac{f(x + h) - f(x)}{h}$ is $2x + h$.

If you need further clarification or have any questions, feel free to ask!

Here are some related questions to explore:

1. What is the derivative of $f(x) = x^2 + 7$ using the limit definition of a derivative?
2. How does the expression $\frac{f(x + h) - f(x)}{h}$ help in understanding the concept of a secant line?
3. What happens to $\frac{f(x + h) - f(x)}{h}$ as $h$ approaches 0?
4. Can you generalize the process of finding $\frac{f(x + h) - f(x)}{h}$ for any polynomial function?
5. How would you interpret the result $2x + h$ in the context of the rate of change?
6. What are the implications of this result for the function's behavior near $x$?
7. Can you find $\frac{f(x + h) - f(x)}{h}$ for the function $f(x) = \sin(x)$?
8. How does the concept of a limit apply to finding derivatives?

**Tip:** Always double-check your algebraic manipulations, especially when expanding expressions and simplifying fractions. This attention to detail ensures accuracy in your calculations.

Discover how to calculate the difference quotient $\frac{f(x+h) - f(x)}{h}$ for the function f(x) = x^2 + 7. This problem involves expanding the function, simplifying algebraic expressions, and understanding the concept of limits. Suitable for advanced high school students.

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