Mathematical Methods

Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

Staying On Track – A Mathematical Investigation

Year

Stage 2

Subject

Mathematical Methods

Type

Investigation

Intention

This subject focuses on the development of mathematical skills that enable students to explore, describe and model aspects of the world around them.

Students will use differential calculus to model a track – such as a car racing track (virtual or real-life), running track or cycling track.

A mathematical report will be written in which students will interpret and justify results, as well as draw conclusions.

Task description

•    You will be using functions to generate a graphical representation of a track.

Examples of tracks:

-     Car racing track

-     A trail for running, hiking or cycling

The track can be in a virtual or real-life context.

•     You can model:

-     A top-down view

-     A side view (which would show height / depth if the track is on mountainous terrain).

•     The track is to be innovative.

However, you can use the following for inspiration: elements of existing tracks or trails, existing landscapes or shapes.

•    You will use differential calculus to ensure each section of the track is connected smoothly whenever possible.

•     Differential calculus could also be demonstrated through consideration of direction, shape (concavity), local minima and maxima, and inflection points.

Tips on… Getting started

•     Look at the examples of functions and relations, including the sub-section entitled “You can also create interesting shapes by:”

•     Hand-draw a sketch of the track.

•     Consider the exemplar.

•     Get working on your model 

Tips on … Functions and relations

As you look at this list, reflect on the features of the graphs presented.

Functions                                     General equation                                           Examples of graphs

Polynomials

f1 (x) = an xn  + an−1xn+1  + ⋯ a2x 2  + a1x + a0

where n is a positive whole number.

 

Trigonometric

functions

f2 (x) = a1 sin(a2x)

 

Exponential and

logarithmic functions *

f3 (x) = a1 ea2x

 

Surge *

f4 (x) = x ex

 

Logistic *

f5 (x) =

 

a1

1 + a2 e −a3x

 


* ‘Playing’ with negatives can change the direction.

•     ‘Playing’ with the powers of x and ory

•    Adding, multiplying and dividing functions, or raising a function to a power

For example, you could explore: y  = √sin x ,  y  = x + sin x ,  y  = xsin x

Tips about… The Results

•     Think about the scale of the graph.

•     For sections to be connected smoothly, the following must be the same at the point at which adjacent functions meet:

(a)  the ，-coordinate

(b)  the tangent

Tips about… The Communication of mathematical ideas

•     Let the mathematics speak for itself. Text should explain what conclusions can be made from the results.

•     In other words, the process used does not need to be written in words.

To help with maintaining a logical structure with your calculations, use sub-headings and still explain where values have come from.

Tips about… Indicators of higher level of achievement

•     The range of functions used

•    The complexity of functions used (and the complexity of the calculus used)

•     The range of techniques used to connect sections smoothly

•     The breadth of calculus considered

Tips on… The report

Leave this until after the results have been completed.

•     State the aim of the investigation.

•     Describe the background information / context for your track.

•     Summarise the mathematical method used to model the track.

You do not necessarily need to use this template but you are encouraged to include the following elements.

Function 1:

f1 (x) =

Graph

Development of the model

This is where the mathematics could go.

 

Interpretation of findings

This is where you could describe the features of the graph.

Assumptions and limitations specific to this section of the track could also be   considered here.

Assumptions are beliefs adopted in order to proceed with an investigation. They are values which have not been measured or researched but accepted.

Limitations explain the differences, or discrepancies, between the mathematics and reality.

A limitation is different to …

•    An error that could be rectified by the person conducting the investigation.

•    A constraint evident in the context of the investigation.

•     Outline the main findings of the investigation.

•     Discuss the overall reasonableness of the results.

•     Describe further investigations that could be undertaken.

This is not assessed.

The following can go in the Appendix:

•     The hand-drawn sketch

•     Supporting evidence or repetitive calculations, provided reference to this is made in the report.

Example

Two methods of ensuring smooth connections will be demonstrated.

Part 1 / Method 1

f1 (x) = −2x {−3 ≤ x ≤ 3}, which has already been drawn.

I only have the type of function I want to join to f1 (x) at x  =  1 in mind. It is to be a quadratic

A quadratic has the form f2 (x) = ax2  + bx + C  .

Since there are three unknowns, three pieces of information were needed.

We already have two pieces of information from the desired point of connection.

Piece of information #1:     They-coordinate of the adjoining functions must be the same at the desired

point of connection.

f1 (3) = f2 (3)

f1 (3) = −2(3) = −6

f2 (1) = a(3)2  + b(1) + C

f2 (1) = 9a + 3b + C

Therefore:

9a + 3b + C = −6

Piece of information #2:     The slope of the tangent must be the same.

 f1′ (3) = f2 ′ (3)

f1′ (x) = −2

f1′ (1) = −2

f2′ (x) = 2ax + b

f2′ (1) = 2a(3) + b

f2′ (1) = 6a + b

Therefore:

6a + b = −2

As another piece of information is needed, let us say the quadratic is to finish at (8, −1).

f2 (8) = a(8)2  + b(8) + C

An online calculator for solving simultaneous equations can be used.

a =  , b = −  , C = 

Part 2 / Method 1

I have a specific function in mind to be joined tof2 (x) atx  = 8. It is f3 (x) = e 2x 

The slope of the tangent must be the same.


They-coordinate of the adjoining functions must be the same at the desired point of connection.

 f2 (8) = f3 (x)

=  (8)2  −  (8) + 

= −1

f3 (x) = e 2(8−7.653) = 2


Practicing Method 2

Work through the prompts provided for the example given.

I have a specific function in mind to be joined tof2 (x) = x 2  − x +  atx  = 8. It is f3 (x) = e 2x  .

Find f2′ (8).

Find f3′ (x).

Find the value(s) of xfor which f2′ (8) = f3 ′ (8).

Find the horizontal translation needed.

Find f2 (8).

Find f3 (8) (with the horizontal translation included).

Find the vertical translation needed.

Practicing Method 2

Here is another example.

I have a specific function in mind to be joined tof1 (x)  =  x 2  atx  = 1. It is f2 (x) =  sin x .

Find f1′ (1).

Find f2′ (x).

Find the value(s) of xfor which f2′ (x) = f1 ′ (1).

Find the horizontal translation needed.

Find f1 (1).

Find f2 (1) (with the horizontal translation included).



 

2024-03-21