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The document is a mathematical investigation for designing a roller coaster track, focusing on further differentiation and applications. The problem involves three main parts:

1. **Ascent Segment (L1)**: A linear function with a slope of 0.7, passing through the origin \((0, 0)\).
   \[
   L_1(x) = 0.7x
   \]

2. **Quadratic Segment (f(x))**: A parabolic segment connecting the ascent and drop. The general form of the quadratic is:
   \[
   f(x) = ax^2 + bx + c
   \]
   Using given conditions and slope information at specific points, the quadratic becomes:
   \[
   f(x) = -0.078125x^2 + 0.7x
   \]

3. **Drop Segment (L2)**: A linear function representing the drop with a slope of -1.8, passing through the point \(Q(16, -8.8)\).
   \[
   L_2(x) = -1.8x + 20
   \]

In **Part B**, students are required to expand the design by determining additional equations for sections before \(L_1\) and after \(L_2\), ensuring smooth transitions between the segments. This involves using a variety of functions and tools such as GeoGebra or FX Graph to create a full roller coaster track model. The task also requires discussing assumptions, constraints, and potential improvements to the design.

Would you like a detailed breakdown of any of these steps or assistance with calculations?

Here are 5 related questions you might explore:
1. How are the slopes of linear and quadratic functions connected at the transition points?
2. What are the limitations of using only parabolas for roller coaster designs?
3. How could trigonometric functions be used in roller coaster modeling?
4. What software tools are best for graphing and analyzing such designs?
5. How does changing the steepness of the drop affect the overall roller coaster experience?

**Tip**: When designing smooth transitions between curves, ensure that both the function and its derivative (slope) match at the points where they meet.

Designing a roller coaster track, focusing on further differentiation and applications. The user is tasked with finding the equations for the ascent, parabolic, and drop sections of a track, ensuring smooth transitions between them.

Learn how to design a roller coaster track using further differentiation. The task involves connecting two linear functions and a parabolic segment, ensuring smooth transitions by solving for tangents and derivatives. This is ideal for students studying advanced mathematics at the Grade 12 level.

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