Part 1: Understanding the Function and the Secant Lines

Given the function $f(x) = -4x^2$, the slope of a secant line between two points $x = a$ and $x = b$ on the curve is given by:

$\text{Slope of the secant line} = \frac{f(b) - f(a)}{b - a}$

Part 2: Calculating Slopes of Secant Lines

Let's calculate the slopes of the secant lines for the intervals provided:

Interval $[1, 2]$

• $a = 1$, $b = 2$
• $f(1) = -4(1)^2 = -4$
• $f(2) = -4(2)^2 = -16$
• Slope $= \frac{-16 - (-4)}{2 - 1} = \frac{-16 + 4}{1} = \frac{-12}{1} = -12$

Interval $[1.5, 2]$

• $a = 1.5$, $b = 2$
• $f(1.5) = -4(1.5)^2 = -4(2.25) = -9$
• $f(2) = -16$
• Slope $= \frac{-16 - (-9)}{2 - 1.5} = \frac{-16 + 9}{0.5} = \frac{-7}{0.5} = -14$

Part 3: Completing the Table

The table below shows the slopes of the secant lines for different intervals approaching $x = 2$:

Interval	Slope of Secant Line
$[1, 2]$	$-12$
$[1.5, 2]$	$-14$
$[1.9, 2]$	$-15.6$
$[1.99, 2]$	$-15.96$
$[1.999, 2]$	$-15.996$

Part 4: Conjecture About the Slope of the Tangent Line at $x = 2$

As the interval gets smaller and approaches $x = 2$, the slope of the secant line seems to approach a value near $-16$. Therefore, we can conjecture that the slope of the tangent line at $x = 2$ is $-16$.

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Would you like more details on the calculations or have any questions?

Here are some related questions for you to explore:

1. What is the general formula for the derivative of $f(x) = -4x^2$?
2. How does the secant line differ from the tangent line in terms of geometric interpretation?
3. What happens to the slope of the secant line as the interval becomes infinitesimally small?
4. How would you use limits to formally prove the slope of the tangent line at a point?
5. Can the method used above be applied to any quadratic function?

Tip: When calculating slopes of secant lines, smaller intervals between the points give a more accurate approximation of the slope of the tangent line.