Maths Academic Writing 2011

8/6/2019 Maths Academic Writing 2011

http://slidepdf.com/reader/full/maths-academic-writing-2011 1/26

SMK (P) SULTAN IBRAHIM

JOHOR BAHRU

MATHEMATICS PROJECT

ACADEMIC WRITING TASK

SEMESTER 2 (JANUARY -MAY)

2011

MATHEMATICS INVOLVING

PROBLEM SOLVING

By

NAME : SHALENI A/PANALINGAM

CLASS : UPPER SIX SCIENCE

YEAR : 2011

IC/NO : 920526-01-5824



AKNOWLEDGEMENT

I, Shaleni Analingam would like to regard my thanks to many

parties who helped my in successfully finishing this academic writing.

First of all, I would like to convey my thanks to my mother who

gave moral support to me to finish this project. She helped me to find

some references to aid my academic writing

Secondly, I would like to thanks my project advisor and also

my chemistry teacher, Miss . Norly who assist and guided me in

completing this academic writing.

Next, I would also like to say my thanks to the principal of SMK

(P) SULTAN IBRAHIM, Pn. Hajah Norizah binti Abdul Manap to give

us an opportunity to complete this academic writing.

Lastly, I would like regard my thanks to my friends whom

helped me in finding some references together to finish the chemistry

academic writing task that were given to us.

THANK YOU.



INDEX

ISSUE PAGE

AKNOWELEDGEMENT

TITLE

INTRODUCTION

CONTENT

CONCLUSION

REFERENCES



INTRODUCTION

Problem solving is a mental process and is part of the larger problem

process that includes problem finding and problem shaping.

Considered the most complex of all intellectual functions, problem

solving has been defined as higher-order cognitive process that

requires the modulation and control of more routine or fundamental

skills. Problem solving occurs when an organism or an artificial

intelligence system needs to move from a given state to a desired

goal state.

Everyone must have felt at least once in his or her life how wonderful

it would be if we could solve a problem at hand preferably without

much difficulty or even with some difficulties. Unfortunately the

problem solving is an art at this point and there are no universal

approaches one can take to solving problems. Basically one must

explore possible avenues to a solution one by one until one comes

across a right path to a solution. Thus generally speaking, there is

guessing and hence an element of luck involved in problem solving.

However, in general, as one gains experience in solving problems,

one develops one's own techniques and strategies, though they are

often intangible. Thus the guessing is not an arbitrary guessing but an

educated one.



PROBLEM SOLVING MODEL

Principles of Community Development

• Promote active and representative citizen participation so that

community members can meaningfully influence decisions that

affect their situation.

• Engage community members in problem diagnosis so that

those affected may adequately understand the causes of their

situation.

• Help community members understand the economic, social,

political, environmental, and psychological impact associated

with alternative solutions to the problem.

• Assist community members in designing and implementing a

plan to solve agreed upon problems by emphasizing shared

leadership and active citizen participation.

• Seek alternatives to any effort that is likely to adversely affect

the disadvantaged segments of a community.



• Actively work to increase leadership capacity, skills,

confidence, and aspirations in the community development

process.

POLYA’S FOUR PRINCIPLES

First principle : Understand the problem

"Understand the problem" is often neglected as being obvious and is

not even mentioned in many mathematics classes. Yet students are

often stymied in their efforts to solve it, simply because they don't

understand it fully, or even in part. In order to remedy this oversight,

Pólya taught teachers how to prompt each student with appropriate

questions, depending on the situation, such as:

What are you asked to find or show?

Can you restate the problem in your own words?

Can you think of a picture or a diagram that might help you

understand the problem?

Is there enough information to enable you to find a solution?

Do you understand all the words used in stating the problem?

Do you need to ask a question to get the answer?



The teacher is to select the question with the appropriate level of

difficulty for each student to ascertain if each student understands at

their own level, moving up or down the list to prompt each student,

until each one can respond with something constructive.

Second principle : Devise a plan

Pólya mentions that there are many reasonable ways to solve

problems. The skill at choosing an appropriate strategy is best

learned by solving many problems. You will find choosing a strategy

increasingly easy. A partial list of strategies is included:

Guess and check

Make an orderly list

Eliminate possibilities

Use symmetry

Consider special cases

Use direct reasoning

Solve an equation

Also suggested:

Look for a pattern



Draw a picture

Solve a simpler problem

Use a model

Work backward

Use a formula

Be creative

Use your head

Third principle: Carry out the plan

This step is usually easier than devising the plan. In general, all you

need is care and patience, given that you have the necessary skills.

Persist with the plan that you have chosen. If it continues not to work

discard it and choose another. Don't be misled; this is how

mathematics is done, even by professionals.

Fourth principle : Review/extend

Pólya mentions that much can be gained by taking the time to reflect

and look back at what you have done, what worked and what

didn't. Doing this will enable you to predict what strategy to use to

solve future problems, if these relate to the original problem.



EXAMPLES OF PROBLEM SOLVING USING POLYA’S FOUR PRINCIPLE

DERIVATIVES OF TRIGONOMIC FUNCTIONS

.

Derivatives of the Sine, Cosine and Tangent Functions

It can be shown from first principles that:

In words, we would say:

The derivative of sin x is cos x ,

The derivative of cos x is −sin x (note the negative sign!) and

The derivative of tan x is sec2 x .

Now, if u = f ( x ) is a function of x , then by using the chain rule, we

have:

http://www.intmath.com/differentiation/3-derivative-first-principles.php

http://www.intmath.com/differentiation/3-derivative-first-principles.php



Example 1:

Find the derivative of y = sin( x 2 + 3).

Answer

First, let: u = x 2

+ 3 and so y = sin u.

We have:

http://www.intmath.com/differentiation-transcendental/ans-1.php?a=0




IMPORTANT:

cos x 2 + 3

does not equal

cos( x 2 + 3).

The brackets make a big difference. Many students have trouble with

this.

Here are the graphs of y = cos x 2 + 3 (in green) and y = cos( x 2 + 3)

(shown in blue).

The first one, y = cos x 2 + 3, or y = (cos x 2) + 3, means take the curve

y = cos x 2 and move it up by 3 units.



The second one, y = cos( x 2 + 3), means find the value ( x 2 + 3) first,

then find the cosine of the result.

They are quite different!



Example 2:

Find the derivative of y = cos 3 x 4.

Answer

Let u = 3 x 4 and so y = cos u.

Then





EXAMPLE 3:

Find the derivative of y = cos32 x

Answer

This example has a function of a function of a function.

Let u = 2 x and v = cos 2 x

So we can write y = v 3 and v = cos u





EXAMPLE 4 :

Find the derivative of y = 3 sin 4 x + 5 cos 2 x 3

Answer

In the final term, put u = 2 x 3.

We have:





Applications: Derivatives of Trigonometric Functions

We can now use derivatives of trigonometric and inverse

trigonometric functions to solve various types of problems.

Examples

1. Find the equation of the normal to the curve of at x = 3.

Answer

When x = 3, this expression is equal to: 0.153846

So the slope of the tangent at x = 3 is 0.153846.

The slope of the normal at x = 3 is given by:





So the equation of the normal is given by:

(When x = 3, y = 0.9828)

y − 0.9828 = -6.5( x − 3) or

y = -6.5 x + 20.483



2. The apparent power P a of an electric circuit whose power is P and

whose impedance phase angle is θ, is given by

P a= P sec θ.

Given that P is constant at 12 W, find the time rate of change of P a if

θ is changing at the rate of 0.050 rad/min, when θ = 40.0°.

Answer

Using chain rule, we have:

Now P a = P sec θ = 12 sec θ (since P = 12 W)

We are told

So





When θ = 40° , this expression is equal to: 0.657 W/min

3. A machine is programmed to move an etching tool such that the

position of the tool is given by x = 2 cos 3t and y = cos 2t , where the

dimensions are in cm and time is in s. Find the velocity of the tool for

t = 4.1 s.

Answer

At t = 4.1, v x = 1.579 and v y = -1.88

So

For velocity, we need to also indicate direction.





So as we are in the 4th quadrant, the required angle is 310°

4. The television screen at a sports arena is vertical and 2.4 m high.

The lower edge is 8.5 m above an observer's eye level. If the best

view of the screen is obtained when the angle subtended by the

screen at eye level is a maximum, how far from directly below the

screen must the observer be?

Answer

(Diagam not to scale)

We define θ1 and θ2 as shown in the diagram.

So θ = θ2 - θ1. [See diagram]

Let x be the distance from directly under the screen to the observer.

To maximise θ , we will need to find





and then set it to 0.

We note that

This gives:

Now since θ = θ2 - θ1,

We have a function of a function in each term.

Now, in the first term, if we let

then



Similarly for the second term, we will have:

So we have:

Next, we multiply the x 2 in the denominator (bottom) of the first

fraction by the denominators of the 2 fractions in brackets, giving:

To find when this equals 0, we need only determine when the

numerator (the top) is 0.



That is

-2.4 x 2 + 222.36 = 0

This occurs when x = 9.63 (we take positive case only)

So the observer must be 9.63 m from directly below the screen to get

the best view



CONCLUSION

Problem solving is a mental process and is part of the

larger problem process that includes problem finding and problem

shaping. Considered the most complex of all intellectual functions,

problem solving has been defined as higher-order cognitive process

that requires the modulation and control of more routine or

fundamental skills. Problem solving occurs when an organism or

an artificial intelligence system needs to move from a given state to a

desired goal state.

http://en.wikipedia.org/wiki/Problem

http://en.wikipedia.org/wiki/Problem_finding

http://en.wikipedia.org/wiki/Problem_shaping


http://en.wikipedia.org/wiki/Intelligence

http://en.wikipedia.org/wiki/Cognitive

http://en.wikipedia.org/wiki/Organism

http://en.wikipedia.org/wiki/Artificial_intelligence

http://en.wikipedia.org/wiki/System

http://en.wikipedia.org/wiki/Problem

http://en.wikipedia.org/wiki/Problem_finding



http://en.wikipedia.org/wiki/Intelligence

http://en.wikipedia.org/wiki/Cognitive

http://en.wikipedia.org/wiki/Organism

http://en.wikipedia.org/wiki/Artificial_intelligence

http://en.wikipedia.org/wiki/System



REFERENCES

http://en.wikipedia.org/wiki/Problem_solving (21/04/2011)

http://www.education.com/reference/article/problem-solving-

strategies-algorithms/ (19/04/2011)

http://www.ced.msu.edu/probsolvingmodel2.html (11/04/2011)

http://www.pitt.edu/~groups/probsolv.html (11/04/2011)

http://www.math.wichita.edu/history/men/polya.html

(15/04/2011)

http://www.education.com/reference/article/problem-solving-strategies-algorithms/


http://www.ced.msu.edu/probsolvingmodel2.html

http://www.pitt.edu/~groups/probsolv.html



http://www.ced.msu.edu/probsolvingmodel2.html

http://www.pitt.edu/~groups/probsolv.html

Maths Academic Writing 2011

Documents

Transcript of Maths Academic Writing 2011