The Rollercoaster Functions and Differentiation Assessment

Task 1: Ascent and Descent

Given the following:

• Two straight stretches of track

  • Linear 1 (L1L_1L1): y1=m1x+c1y_1 = m_1x + c_1y1=m1x+c1

  • Linear 2 (L2L_2L2): y2=m2x+c2y_2 = m_2x + c_2y2=m2x+c2

• One curved section of track

  • Quadratic: y=f(x)=ax2+bx+cy = f(x) = ax^2 + bx + cy=f(x)=ax2+bx+c

where xxx and f(x)f(x)f(x) are in metres.

Research suggests that the slope of the ascent should be 0.7 and the descent -1.3 .

The transition from one function to another needs to be smooth. Hence the following conditions apply:

• Tangents to the parabola exist at the transition points P and Q.

• The origin is located at P.

• The horizontal distance from P to Q is 40 metres.

The linear functions and parabola are connected according to the following diagram:

[Diagram showing a parabola connected between two linear segments L1 and L2]

Determine the equations of the three segments of the rollercoaster over the given domains below:

• Linear 1: y1=m1x+c1y_1 = m_1x + c_1y1=m1x+c1 for x≤0x \leq 0x≤0

• Quadratic: y=f(x)=ax2+bx+cy = f(x) = ax^2 + bx + cy=f(x)=ax2+bx+c for 0≤x≤400 \leq x \leq 400≤x≤40

• Linear 2: y2=m2x+c2y_2 = m_2x + c_2y2=m2x+c2 for x≥40x \geq 40x≥40

Handwritten notes from the image:

• Gradient of L1=0.7L_1 = 0.7L1=0.7

• Gradient of L2=−1.3L_2 = -1.3L2=−1.3

• f′(x)=2ax+bf'(x) = 2ax + bf′(x)=2ax+b

• b=0.7b = 0.7b=0.7

• c=0c = 0c=0

• L2:−1.3=?(40)+cL_2: -1.3 = ?(40) + cL2:−1.3=?(40)+c → c=40c = 40c=40

Task 2: Design a rollercoaster

Your task is to design a 2D rollercoaster from the left starting point to the right finish point.

Your design must contain:

• The three functions from Task 1.

• A minimum of three functions from the start to Linear 1.

• A minimum of three functions from Linear 2 to the finish.

Show full working and ensure a smooth transition between each function.

Consider:

• A variety of functions, e.g.
  • polynomials up to degree 3
  • sine/cosine …

• Length and height constraints – what is the maximum height and length of the rollercoaster.

• Reasons for your choice.

Your investigation should make use of the mathematical thinking process:

• Interpreting the task and the key information.

• Choosing the mathematics which could help to complete the task.

• Applying existing mathematical knowledge and strategies to obtain a solution.

• Interpreting the results in relation to the context.

• Communicating the solution to the problem as required.

You will then write a report to showcase your design. As you write your report, take care to clearly identify the underlying mathematics used throughout the process.

Report

Your report should include the following:

• An introduction, that clearly defines the purpose of the task, identifies key information, any assumptions made and an outline of your strategy. (6 marks)

• Evidence of the application of mathematical knowledge and strategies, including calculations and results using appropriate representations (graphs, tables, formulae etc.). (35 marks)

• Your conclusion/summary communicated in a systematic and concise manner, including analysis and interpretation in the context of the problem and consideration of the reasonableness and limitations of the results. (5 marks)

• Use of correct mathematical conventions, symbols and terminology. (5 marks)

Assessment Requirements – Brief Overview

The assessment is divided into two main tasks, both focused on the design and mathematical analysis of a 2D rollercoaster. The key objectives are:

Task 1: Ascent and Descent

• Determine equations of three connected track segments:

  1. Linear segment L1 with slope 0.7 (ascent).

  2. Quadratic (parabolic) section connecting L1 and L2.

  3. Linear segment L2 with slope -1.3 (descent).

• Ensure smooth transitions at points P (origin) and Q (x = 40 m), meaning tangents of connecting functions must match.

• Work within defined domains:

  • L1: x≤0x \le 0x≤0

  • Quadratic: 0≤x≤400 \le x \le 400≤x≤40

  • L2: x≥40x \ge 40x≥40

Task 2: Full Rollercoaster Design

• Extend the track to a complete 2D rollercoaster:

  • Include at least three functions before L1 and after L2.

  • Ensure smooth transitions throughout.

• Consider a variety of functions (e.g., polynomials up to degree 3, sine/cosine functions).

• Respect height and length constraints.

• Document the mathematical thinking process: interpreting information, applying mathematical strategies, solving, interpreting results, and communicating findings.

Report Requirements

• Introduction : Purpose, assumptions, key information, and strategy outline (6 marks).

• Mathematical Application : Calculations, results, and representations (35 marks).

• Conclusion/Summary : Systematic interpretation, reasonableness, and limitations (5 marks).

• Mathematical Conventions : Correct symbols, terminology, and formatting (5 marks).

Assessment Approach by Academic Mentor

The academic mentor guided the student step-by-step as follows:

Step 1: Understanding the Problem

• Reviewed the task brief to identify key slopes, domains, and continuity requirements.

• Discussed the importance of smooth transitions between segments and how this affects derivative matching.

Step 2: Task 1 – Segment Equations

• Linear Segment L1 : Slope = 0.7; intercept at origin = 0 → equation: y1=0.7xy_1 = 0.7xy1=0.7x.

• Quadratic Section : Form y=ax2+bx+cy = ax^2 + bx + cy=ax2+bx+c.

  • Condition 1: Tangent at P matches L1 → derivative at x=0: f′(0)=b=0.7f'(0) = b = 0.7f′(0)=b=0.7.

  • Condition 2: Tangent at Q matches L2 → derivative at x=40: f′(40)=2a(40)+0.7=−1.3f'(40) = 2a(40) + 0.7 = -1.3f′(40)=2a(40)+0.7=−1.3 → solve for a.

  • Condition 3: Pass through origin → c = 0.

• Linear Segment L2 : Slope = -1.3, passes through Q → solve for c → equation: y2=−1.3x+52.3y_2 = -1.3x + 52.3y2=−1.3x+52.3.

Step 3: Task 2 – Complete Rollercoaster Design

• Chose additional functions before L1 and after L2 to create a visually and mathematically smooth ride.

• Ensured derivatives matched at all junctions for continuity.

• Evaluated height and length constraints to prevent unrealistic designs.

• Incorporated varied functions (polynomials, sine/cosine) to demonstrate mathematical versatility.

Step 4: Mathematical Documentation

• Graphs, tables, and formulae were prepared to visually and numerically validate the design.

• Stepwise calculations showed how parameters were derived and how smooth transitions were ensured.

Step 5: Report Compilation

• Introduction : Defined purpose, identified key information, assumptions, and outlined the approach.

• Application Section : Detailed equations, derivations, and graphs.

• Conclusion : Summarized design, analyzed feasibility, and addressed limitations (e.g., maximum height, curvature).

• Mathematical Conventions : Maintained correct symbols, notation, and clear explanations throughout.

Outcome Achieved

• Successfully derived equations for all track segments with smooth transitions.

• Designed a complete 2D rollercoaster respecting all mathematical and physical constraints.

• Demonstrated mathematical thinking: interpreting data, applying functions, ensuring continuity, and validating results.

• Report fully aligned with assessment marking criteria, covering all learning objectives:

  • Application of derivatives and tangents

  • Continuity and smooth transitions between functions

  • Mathematical modeling of real-world scenarios

  • Clear communication of results

Boost Your Understanding with Our Sample Solution – Safely!

Unlock insights into your assignment with our sample solution, crafted to help you understand the approach, structure, and methodology. Use it as a reference guide only to learn how to tackle the problem effectively. Submitting this sample as your own work is considered plagiarism and can have serious academic consequences.

If you want a completely original, plagiarism-free solution tailored to your requirements, our team of professional academic writers is ready to create it for you. Ordering a fresh solution ensures:

• 100% unique content designed just for your assignment.

• Step-by-step explanations to improve your learning.

• Adherence to your institution’s guidelines and marking criteria.

• Stress-free submission with confidence in originality.

Take action now:

Download Sample Solution                              Order Fresh Assignment