Let's solve each problem step by step.
1) Find the rate of change between points A(3,9) and B(2, 4)
To find the rate of change between two points, we use the formula for the slope of the secant line:
\[ \text{Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
Here, \( A(3, 9) \) corresponds to \( x = 3 \) and \( B(2, 4) \) corresponds to \( x = 2 \):
- \( f(3) = 3^2 = 9 \)
- \( f(2) = 2^2 = 4 \)
Now substituting into the formula:
\[ \text{Rate of Change} = \frac{f(2) - f(3)}{2 - 3} = \frac{4 - 9}{2 - 3} = \frac{-5}{-1} = 5 \]
So, the rate of change between points A and B is 5.
2) Find \( f'(x) \) using first principles
To find the derivative using first principles, we use the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \]
For \( f(x) = x^2 \):
\[ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - x^2}{h} \]
Expanding \( (x + h)^2 \):
\[ = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} = \lim_{h \to 0} \frac{2xh + h^2}{h} = \lim_{h \to 0} (2x + h) \]
As \( h \to 0 \):
\[ f'(x) = 2x \]
So, \( f'(x) = 2x \).
3) Estimate the rate of change at (2, 4) using a graphical method
Using the graphical method, we can draw the tangent line at the point \( B(2, 4) \) on the curve \( y = f(x) = x^2 \). The slope of this tangent line can be estimated around that point.
From the derivative we calculated:
\[ f'(2) = 2 \cdot 2 = 4 \]
Thus, the estimated rate of change (the slope of the tangent line) at the point \( B(2, 4) \) is approximately 4.
4) Find \( \lim_{x \to 4} \frac{x^2 + x - 12}{x + 4} \)
To solve this limit, we can first try substituting \( x = 4 \):
\[ \lim_{x \to 4} \frac{x^2 + x - 12}{x + 4} = \frac{4^2 + 4 - 12}{4 + 4} = \frac{16 + 4 - 12}{8} = \frac{8}{8} = 1 \]
So, \( \lim_{x \to 4} \frac{x^2 + x - 12}{x + 4} = 1 \).
5) Compare findings from questions #2 and #3
From question #2, we found that \( f'(2) = 4 \) using first principles. From question #3, we estimated the slope at (2, 4) using a graphical method and found it to be approximately 4 as well.
The derivative \( f'(2) \) calculated using first principles gives the exact rate of change, which coincides with the graphical estimate of the tangent's slope, confirming consistency.
Conclusion: Yes, the findings from questions #2 and #3 are the same, indicating that both methods are valid for determining the rate of change of the function at the given point. The reason they are the same is that they both represent the same underlying mathematical concept of instantaneous rate of change or the slope of the tangent line.