a) Determine the general equation, in simplified form, for the slope of the secant line on the graph of f(x) = x2 between the point with x-coordinate a = 3, and the point (a + h, f(a + h)). Do not use a specific value of h. (4 marks)

b) Use the result of part a) to estimate the slope of the tangent line at the point with x-coordinate 3. (1 mark)

### Part (a) - Secant Line Slope:

We are given the function \( f(x) = x^2 \), and we need to find the general equation for the slope of the secant line between the point with x-coordinate \( a = 3 \) and the point \( (a + h, f(a + h)) \).

The slope of a secant line between two points on a graph is given by:

\[
m_{\text{secant}} = \frac{f(a+h) - f(a)}{(a+h) - a}
\]

Here, \( f(x) = x^2 \), and the points are \( (a, f(a)) = (3, f(3)) \) and \( (a+h, f(a+h)) = (3+h, f(3+h)) \).

1. **Step 1**: Calculate \( f(a) = f(3) \).
   \[
   f(3) = 3^2 = 9
   \]

2. **Step 2**: Calculate \( f(a+h) = f(3+h) \).
   \[
   f(3+h) = (3+h)^2 = 9 + 6h + h^2
   \]

3. **Step 3**: Apply the secant line formula.
   \[
   m_{\text{secant}} = \frac{f(3+h) - f(3)}{(3+h) - 3}
   \]
   Substituting the values:
   \[
   m_{\text{secant}} = \frac{(9 + 6h + h^2) - 9}{h}
   \]

4. **Step 4**: Simplify the expression.
   \[
   m_{\text{secant}} = \frac{9 + 6h + h^2 - 9}{h} = \frac{6h + h^2}{h}
   \]

5. **Step 5**: Factor out \( h \) and simplify further.
   \[
   m_{\text{secant}} = \frac{h(6 + h)}{h} = 6 + h
   \]

Thus, the general equation for the slope of the secant line is:
\[
m_{\text{secant}} = 6 + h
\]

---

### Part (b) - Estimate the Slope of the Tangent Line:

To estimate the slope of the tangent line at the point where \( x = 3 \), we can take the limit of the secant slope as \( h \to 0 \). The slope of the tangent line is:

\[
m_{\text{tangent}} = \lim_{h \to 0} (6 + h)
\]

Taking the limit:
\[
m_{\text{tangent}} = 6 + 0 = 6
\]

Thus, the slope of the tangent line at \( x = 3 \) is:
\[
m_{\text{tangent}} = 6
\]

---

Do you have any questions or want more details on any of the steps? 

Here are five related questions for further practice:
1. How do you find the slope of a secant line for a cubic function?
2. How does the difference between secant and tangent line slopes evolve as \( h \) decreases?
3. What is the derivative of \( f(x) = x^2 \) at any point \( x = a \)?
4. What is the geometric interpretation of the tangent line on a curve?
5. How do limits apply in the definition of the derivative?

**Tip**: The slope of the tangent line at any point on a function represents the instantaneous rate of change at that point.

Determine the general equation, in simplified form, for the slope of the secant line on the graph of f(x) = x^2 between the point with x-coordinate a = 3, and the point (a + h, f(a + h)). Do not use a specific value of h. Use the result to estimate the slope of the tangent line at the point with x-coordinate 3.

Explore how to determine the general equation for the slope of the secant line on the graph of f(x) = x^2 between two points, followed by estimating the tangent slope at x = 3 using limits. A comprehensive guide for high school calculus students.

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