Each of the following expressions represents the derivative of some function
y = f(x) for some value, x = c. Identify the function y = f(x) and the value for x = c. lim h->0 ((7-2(2+h)-3)/(h))

We are given the expression for the derivative:

\[
\lim_{h \to 0} \frac{7 - 2(2 + h) - 3}{h}
\]

### Step 1: Simplify the expression inside the limit

First, distribute the \( 2 \) inside the parentheses:

\[
7 - 2(2 + h) - 3 = 7 - (4 + 2h) - 3 = 7 - 4 - 2h - 3 = 0 - 2h = -2h
\]

So, the expression becomes:

\[
\lim_{h \to 0} \frac{-2h}{h}
\]

### Step 2: Simplify the fraction

\[
\frac{-2h}{h} = -2
\]

Thus, the expression simplifies to:

\[
\lim_{h \to 0} (-2) = -2
\]

### Step 3: Interpretation as a derivative

The given expression represents the derivative of some function \( f(x) \) at a point \( x = c \). Based on the limit form, this is the difference quotient:

\[
f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h}
\]

By comparing this to the given expression, we can identify:

- \( f(c + h) = 7 - 2(2 + h) \)
- \( f(c) = 7 - 2(2) = 3 \)

Thus, the function is:

\[
f(x) = 7 - 2x
\]

The point where we are evaluating the derivative is \( c = 2 \), since the expression inside the limit involves \( 2 + h \).

### Conclusion

- The function is \( f(x) = 7 - 2x \)
- The value of \( c \) is \( 2 \)

The derivative \( f'(x) = -2 \), and indeed the limit confirms that the derivative at \( x = 2 \) is \( -2 \).

---

Would you like further details on any part of the explanation?

Here are 5 related questions:
1. How do you differentiate other linear functions like \( f(x) = 3x + 4 \)?
2. Can the derivative of \( f(x) = 7 - 2x \) change if the function was nonlinear?
3. What does the value of the derivative tell us about the graph of \( f(x) \)?
4. How would you approach the derivative of a quadratic function?
5. How do limits relate to the concept of continuity?

**Tip:** The derivative of a constant linear function \( f(x) = ax + b \) is always equal to the coefficient of \( x \), which is \( a \).

Each of the following expressions represents the derivative of some function y = f(x) for some value, x = c. Identify the function y = f(x) and the value for x = c. lim h->0 ((7-2(2+h)-3)/(h))

Learn how to identify the function f(x) and the value of c in the expression lim_{h -> 0} ((7 - 2(2 + h) - 3) / h). This problem explores the limit definition of derivatives, suitable for students in Grades 11-12 studying calculus.

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