Transcript HAESE MATHEMATICS th

IB Diploma Higher Level Options
HAESE MATHEMATICS
Specialists in mathematics publishing
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WORKED SOLUTIONS
)
Z
1
3
0
)
Z
0
1
3
1
dx ¼
1 + x3
1
¡
3
1
1
1
+
¡
4 £ 34 7 £ 37 10 £ 310
1
where
¼ 0:000 001 7
10 £ 310
1
dx ¼ 0:3303 (to 4 d.p.)
1 + x3
6 From Example 43, the Maclaurin series for ln(1 + x) is defined
for jxj < 1 and
ln(1 + x) = x ¡
x2
x3
x4
x5
+
¡
+
¡ ::::
2
3
4
5
189
d If f (x) = sin(e¡x )
f 0 (x) = cos(e¡x ) £ ¡e¡x
= ¡e¡x cos(e¡x )
f 00 (x) = e¡x cos(e¡x ) ¡ e¡x (¡ sin(e¡x )) £ ¡e¡x
= e¡x cos(e¡x ) ¡ e¡2x sin(e¡x )
) f (0) = sin 1, f 0 (0) = ¡ cos 1, f 00 (0) = cos 1 ¡ sin 1
Now sin(e¡x ) ¼ f (0) +
f 0 (0)x f 00 (0)x2
+
1!
2!
¼ sin 1 ¡ [cos 1]x + [cos 1 ¡ sin 1]
x2
2!
¼ 0:8415 ¡ 0:5403x ¡ 0:1506x2
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SE
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specialists in mathematics publishing
IGCSE
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th
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Keith Black
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Michael Haese
Mathematics
Robert Haese
Sandra Haese
Mark Humphries
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Core
Keith Black
Alison Ryan
Michael Haese
Robert Haese
Sandra Haese
2014
Mark Humphries
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Statement of inquiry:
Presenting data in tables can help us to perform calculations more efficiently.
Global context:
Identities and relationships
Key concept:
Relationships
Related concepts:
Change, Justification
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Communicating, Applying mathematics in real-life contexts
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SNOWFLAKE
Your favourite mathematics textbooks online
Most human populations consist of people of all different ages, including
babies, children, adults, and the elderly. However, the age composition
of a population differs from country to country. Some countries have
mainly younger people, while others have more elderly citizens. The age
composition of a particular country can also change over time.
It is important that a country is able to measure the age of its population.
For example, if a country is raising many children, it will need to spend
more money on schools and child care. If a country has a mainly elderly
population, the country will need to spend more money on health and
aged care.
The table on page 2 shows the Japanese population of each age, for the years 1920, 1960, and 2010.
1 Find the mode of the data for:
a 1920
b 1960
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c 2010.
2 Is it surprising that the modal age is so low in 1920? Explain your answer.
4 Calculate the mean age of Japan in:
a 1920
b 1960
c 2010.
Assume that the ‘100+’ population are 100 years old.
5 Comment on the change in the mean age of the Japanese population over this time. Can you explain why
this change may have occurred?
6 Do you think the number of births in Japan has increased or decreased over the 90 years from 1920 to 2010?
Use the data to justify your answer.
7 How would these changes to Japan’s age composition affect the structure of a typical family? You should
consider:
² the number of living grandparents a child is likely to have
² the time that children will have with their parents and grandparents
² the number of siblings a child is likely to have.
8 Do you think these changes to the typical family are beneficial? Explain your answer.
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9 Why do you think there is such a large difference between the number of 63 year olds and 64 year olds in
2010?
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CALCULATING THE AGE
OF A POPULATION
Global
context
3 What do you notice about your answers to 1 b and 1 c?
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