The difference quotient of a function f(x) at x=a is given by
(2+h)2−22h

Find f(x) and a.

The given expression for the difference quotient is:

\[
\frac{(2+h)^2 - 2^2}{h}
\]

This matches the general form of the difference quotient:

\[
\frac{f(a+h) - f(a)}{h}
\]

To identify \( f(x) \) and \( a \), we can compare terms.

1. The term \( (2+h)^2 \) corresponds to \( f(a+h) \), which suggests that \( f(x) = x^2 \) (since \( (a+h)^2 \) implies a quadratic function).
2. The value of \( a \) must be 2, as indicated by the expression \( 2^2 \) in the difference quotient.

Thus, we have:

- \( f(x) = x^2 \)
- \( a = 2 \)

Would you like further explanation on how this conclusion was reached or more details about the difference quotient?

Here are 5 questions to deepen your understanding:
1. How would the difference quotient change for a different function like \( f(x) = x^3 \)?
2. What is the geometric meaning of the difference quotient?
3. How does the difference quotient relate to the derivative?
4. Can you compute the difference quotient for \( f(x) = x^2 \) at \( a = 3 \)?
5. What is the next step after finding the difference quotient in calculus?

**Tip:** The difference quotient is the foundation for calculating the derivative of a function!

The difference quotient of a function f(x) at x=a is given by (2+h)^2−2^2/h. Find f(x) and a.

Explore how to determine the function f(x) and the value of a from a given difference quotient expression: (2+h)^2−2^2/h. This problem illustrates key calculus concepts, including the difference quotient and its relation to the derivative, using quadratic functions. Suitable for Grades 11-12.

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