) Calculate f ′ (x) for the function f(x) = 3x 2 −7, using the limit definition. Show all of your work. Simply writing down what f ′ (x) is will get you no points.

To calculate the derivative \( f'(x) \) of the function \( f(x) = 3x^2 - 7 \) using the limit definition of a derivative, we follow these steps.

### Limit Definition of Derivative:
The derivative of a function \( f(x) \) is given by:

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

### Step-by-Step Solution:

1. **Write down the function**: 
   \[
   f(x) = 3x^2 - 7
   \]

2. **Substitute \( f(x+h) \) into the limit definition**:
   We first need to find \( f(x+h) \). 

   \[
   f(x + h) = 3(x+h)^2 - 7
   \]

   Expand \( (x+h)^2 \):

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

   Now substitute this into the expression for \( f(x + h) \):

   \[
   f(x + h) = 3(x^2 + 2xh + h^2) - 7 = 3x^2 + 6xh + 3h^2 - 7
   \]

3. **Set up the difference quotient**:

   Now calculate the difference \( f(x+h) - f(x) \):

   \[
   f(x+h) - f(x) = \left( 3x^2 + 6xh + 3h^2 - 7 \right) - \left( 3x^2 - 7 \right)
   \]

   Simplifying this:

   \[
   f(x+h) - f(x) = 3x^2 + 6xh + 3h^2 - 7 - 3x^2 + 7 = 6xh + 3h^2
   \]

4. **Divide by \( h \) to form the difference quotient**:

   \[
   \frac{f(x+h) - f(x)}{h} = \frac{6xh + 3h^2}{h}
   \]

   Simplify by dividing each term by \( h \):

   \[
   \frac{f(x+h) - f(x)}{h} = 6x + 3h
   \]

5. **Take the limit as \( h \to 0 \)**:

   \[
   f'(x) = \lim_{h \to 0} (6x + 3h)
   \]

   As \( h \to 0 \), the term \( 3h \) vanishes, leaving:

   \[
   f'(x) = 6x
   \]

### Final Answer:
The derivative of \( f(x) = 3x^2 - 7 \) is:

\[
f'(x) = 6x
\]

Would you like any further clarification, or details on any step?

### Expanding Questions:
1. How would the process change if the function was \( f(x) = 3x^2 - 7x + 2 \)?
2. Can you apply the limit definition of a derivative to a trigonometric function, such as \( f(x) = \sin(x) \)?
3. What is the geometric interpretation of the derivative you just calculated?
4. How would the second derivative of \( f(x) = 3x^2 - 7 \) be calculated using the same limit definition?
5. What does the derivative \( f'(x) = 6x \) tell you about the slope of the tangent line to the curve \( y = 3x^2 - 7 \)?

### Tip:
Always expand \( (x+h)^n \) carefully when using the limit definition to avoid algebraic mistakes.

Calculate f ′ (x) for the function f(x) = 3x^2 −7, using the limit definition. Show all of your work. Simply writing down what f ′ (x) is will get you no points.

Learn how to calculate the derivative of the function f(x) = 3x^2 - 7 using the limit definition. This detailed, step-by-step solution covers key calculus concepts and the limit definition of the derivative. Suitable for high school students in Grades 11-12.

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