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8/6/2019 Maths Academic Writing 2011
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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
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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.
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INDEX
ISSUE PAGE
AKNOWELEDGEMENT
TITLE
INTRODUCTION
CONTENT
CONCLUSION
REFERENCES
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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.
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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.
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• 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?
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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
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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.
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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:
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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:
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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.
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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!
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Example 2:
Find the derivative of y = cos 3 x 4.
Answer
Let u = 3 x 4 and so y = cos u.
Then
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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
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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:
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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:
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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
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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
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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.
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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
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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
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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.
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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
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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.
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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)