Task Sheet
STAGE 2 Mathematical Studies
Name of assessment task: Designing a Rollercoaster
Assessment type:
Date Issued: 9th April 2014
Due Date: 11:00 pm 8th May 2014 Electronic
9:00am 11th May 2014 Hardcopy
Draft Due Date: 5pm 27th April 2014
Student’s Full Name:
Folio Task
Task Weighting: 12.5%
Task Description:
This assessment allows you to show your understanding of the mathematical concepts and relationships in the
following topics, and the algorithmic and problem solving skills needed to solve a range of calculus questions.
2.4: Rate of Change
2.6: The Derivative
2.7: Differentiation
2.8: Using Derivatives
Assessment design criteria and specific features:
Mathematical Knowledge and Skills and Their Application
The specific features are as follows:

MKSA1 Knowledge of content and understanding of mathematical concepts and relationships

MKSA2 Use of mathematical algorithms and techniques (implemented electronically where appropriate) to find
solutions to routine and complex questions

MKSA3 Application of knowledge and skills to answer questions set in applied and theoretical contexts.
Mathematical Modelling and Problem-solving
The specific features are as follows:

MMP1 Application of mathematical models

MMP2 Development of solutions to mathematical problems set in applied and theoretical contexts.

MMP3 Interpretation of the mathematical results in the context of the problem

MMP4 Understanding of the reasonableness and possible limitations of the interpreted results, and recognition of
assumptions made

MMP5 Development and testing of conjectures, with some attempt at proof
Communication of Mathematical Information
The specific features are as follows:

CMI1 Communication of mathematical ideas and reasoning to develop logical arguments
 CMI2 Use of appropriate mathematical notation, representations, and terminology
Assessment conditions:
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Time: 2 weeks plus homework time.
This is a partially supervised assessment.
Provide complete working for all.
Use electronic technology where appropriate.
This assignment contains no material which has been submitted for another SACE assessment and to the best of
my knowledge and belief it contains no material previously published or written by another person except when
due reference is made in the text.
Signature…………………………………………………………Date……………………
Performance standards for Stage 2 Mathematical Studies
Mathematical Knowledge and
Skills and Their Application
A
Comprehensive knowledge of content and
understanding of concepts and relationships
Appropriate selection and use of mathematical
algorithms and techniques (implemented electronically
where appropriate) to find efficient solutions to
complex problems
Highly effective and accurate application of knowledge
and skills to answer questions set in applied and
theoretical contexts
B
Some depth of knowledge of content and understanding
of concepts and relationships.
Use of mathematical algorithms and techniques
(implemented electronically where appropriate) to find
some correct solutions to complex problems.
Accurate application of knowledge and skills to answer
questions set in applied and theoretical contexts.
Mathematical Modelling and Problem Solving
Development and effective application of mathematical models
Complete, concise, and accurate solutions to mathematical problems set in applied and
theoretical contexts.
Concise interpretation of the mathematical results in the context of the problem
In-depth understanding of the reasonableness and possible limitations of the interpreted
results and recognition of assumptions made.
Communication of
Mathematical
Information
Highly effective communication
of mathematical ideas and
reasoning to develop logical
arguments.
Proficient and accurate use of
appropriate notation,
representations, and terminology
Development and testing of valid conjectures, with proof
Attempted development and appropriate application of mathematical models.
Mostly accurate and complete solutions to mathematical problems set in applied and
theoretical contexts.
Complete interpretation of the mathematical results in the context of the problem.
Some depth of understanding of the reasonableness and possible limitations of the
interpreted results, and recognition of assumptions made.
Effective communication of
mathematical ideas and
reasoning to develop mostly
logical arguments.
Mostly accurate use of
appropriate notation,
representations, and
terminology.
Development and testing of reasonable conjectures, with substantial attempt at proof
C
Generally competent knowledge of content and
understanding of concepts and relationships
Use of mathematical algorithms and techniques
(implemented electronically where appropriate) to find
mostly correct solutions to routine questions
Generally accurate application of knowledge and skills
to answer questions set in applied and theoretical
contexts
Appropriate application of mathematical models.
Some accurate and generally complete solutions to mathematical problems set in applied
and theoretical contexts.
Generally appropriate interpretation of the mathematical results in the context of the
problem.
Some understanding of the reasonableness and possible limitations of the interpreted
results and some recognition of assumptions made.
Appropriate communication of
mathematical ideas and
reasoning to develop some
logical arguments.
Use of generally appropriate
notation, representations, and
terminology, with some
inaccuracies.
Development and testing of reasonable conjectures, with some attempt at proof.
D
Basic knowledge of content and some understanding of
concepts and relationships
Some use of mathematical algorithms and techniques
(implemented electronically where appropriate) to find
some correct solutions to routine questions.
Sometimes accurate application of knowledge and
skills to answer questions set in applied or theoretical
contexts
E
Limited knowledge of content
Attempted use of mathematical algorithms and
techniques (implemented electronically where
appropriate) to find limited correct solutions to routine
questions.
Attempted application of knowledge and skills to
answer questions set in applied or theoretical contexts,
with limited effectiveness.
Application of a mathematical model, with partial effectiveness.
Partly accurate and generally incomplete solutions to mathematical problems set in applied
or theoretical contexts.
Attempted interpretation of the mathematical results in the context of the problem.
Some awareness of the reasonableness and possible limitations of the interpreted results.
Some appropriate
communication of mathematical
ideas and reasoning
Some attempt to use appropriate
notation, representations, and
terminology, with occasional
accuracy.
Attempted development or testing of a reasonable conjecture.
Attempted application of a basic mathematical model.
Limited accuracy in solutions to one or more mathematical problems set in applied or
theoretical contexts.
Limited attempt at interpretation of the mathematical results in the context of the problem.
Limited awareness of the reasonableness and possible limitations of the results
Limited attempt to develop or test a conjecture
Attempted communication of
emerging mathematical ideas and
reasoning.
Limited attempt to use
appropriate notation,
representations, or terminology,
and with limited accuracy.
Designing a Rollercoaster
Suppose you are asked to design the first ascent
and drop for a new rollercoaster. By studying
photographs of your favourite coasters, you decide
to make the slope of the ascent 0.7 and the slope of
the drop -1.3.
You decide to connect these two straight stretches
𝑦 = 𝐿1 (𝑥) = 𝑚1 𝑥 + 𝑘1 and 𝑦 = 𝐿2 (𝑥) = 𝑚2 𝑥 +
𝑘2 with part of a parabola 𝑦 = 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 +
𝑐, where 𝑥 and 𝑓(𝑥) are measured in metres.
For the track to be smooth there can’t be
abrupt changes in direction, so you want
the linear segments 𝐿1 and 𝐿2 to be
tangents to the parabola at the transition
points P and Q.
To simplify the equations, you decide to
place the origin at P. Furthermore the
horizontal distance between P and Q is 40
metres.
Your task is to:

Find the equation of each segment, showing all appropriate steps of logic, namely:
𝐿1 (𝑥) = 𝑚1 𝑥 + 𝑘1
𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
𝐿2 (𝑥) = 𝑚2 𝑥 + 𝑘2
for 𝑥 < 0
for 0 ≤ 𝑥 ≤ 40
for 𝑥 > 40

Design the section immediately before this one joining the track at 𝐿1 . Determine
polynomial equations (minimum of 3, and at least one cubic) for this section ensuring
a smooth transition between points. Show detailed working, equations, constraints,
etc.

Using FX Graph or Geogebra draw a detailed, fully labeled, smooth graph showing
what this section looks like.

Calculate the second derivative function for each section of your rollercoaster and
discuss the implications of these functions on the shape of that section. Generalise
your results for all functions.
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A discussion of any assumptions, limitations, improvements and suggestions of
further work.
Your report should be word processed and
structured to include:
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Introduction
Mathematical Investigations
Analysis/Discussion
Conclusion
Note: your report should be in past tense, third
person and have a passive voice e.g. “ The
analysis…” rather than “When I analysed” or
“When you analyse”
A completed folio task should include:
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An introduction that outlines the problem to be explored, including its significance, its
features and the context.
The method of solution in terms or the mathematical model or strategy to be used
The appropriate application of the mathematical model or strategy including:
 The generation or collection of relevant data
 The analysis and interpretation of results
 Reference to the limitations of the original problem as well as appropriate
refinements and/or extensions
A statement of the solution and outcome in the context of the original problem
Appendices and bibliography as appropriate.