HSC Mathematics
Workshop 1
Presented by
Richard D. Kenderdine
BSc, GradDipAppSc(IndMaths), SurvCert, MAppStat, GStat
School of Mathematics and Applied Statistics
University of Wollongong
Moss Vale Centre
July 2009
HSC Mathematics - Workshop 1
Richard D Kenderdine
University of Wollongong
July 2009 1
Series
Problem:
The sum of the first n terms of a series is given by Sn = 4n2 + 5. Find an
expression for the n-th term, Tn .
One method is to use the relationship:
Sn−1 + Tn = Sn ⇒ Tn = Sn − Sn−1
(1)
Another method is to obtain the first few terms of the series directly.
Arithmetic Series
An Arithmetic Series is a series of numbers that differ by a common difference. It is the discrete counterpart to the continuous linear function given by
y = mx + b.
The formula for the nth term is given by:
Tn = a + (n − 1)d
= dn + (a − d)
(2)
where a and d are fixed and n is the variable (integer ≥ 1).
This corresponds to y = mx + b where m and b are fixed and x variable.
Suppose we have a question that asks to find T21 given T5 and T9 . One way
would be to set up a pair of equations for T5 and T9 , solve for a and d, then
find T21 .
Is there a better way?
1. Exercise:
Find the 25th term of an AP given that the seventh and twelth terms are
35 and 20 respectively.
1 email:
[email protected]
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The sum of an AP:
Sn = n(
=
a+l
)
2
n
(a + l)
2
(3)
2. Exercise:
How many pieces can be cut from a rope of length 100m if the first piece
is 8cm and each subsequent piece 3cm longer than the preceding piece?
What is the length of the remainder?
Geometric Series
An Geometric Series is a series of numbers that differ by a common ratio,
r. It is the discrete counterpart to the continuous exponential function
given by y = arx .
The formula for the nth term is given by:
Tn = arn−1
(4)
where a and r are fixed and n is the variable (integer ≥ 1).
3. Exercise :
Find the 15th term of a geometric series given that the 6th term is 729
and the 9th term is 27.
Sum of a GP
The sum of a geometric series is given by:
a(rn − 1)
r−1
a(1 − rn )
=
1−r
Sn =
f or |r| > 1
(5)
f or |r| < 1
As n → ∞ we have a Limiting Sum if |r| < 1
a(1 − 0)
1−r
a
=
1−r
lim Sn =
n→∞
2
(6)
4. Exercise: (1) Determine whether a limiting sum exists for the following
series, and if so find its value:
1
1
+ ( √2−1
)2 + ...
(a)1 + √2−1
1
1
(b)1 + √2+1
+ ( √2+1
)2 + ...
(2) Determine the infinite series if the first two terms add to 2 and each
term is twice the sum of all the subsequent terms.
(3) An aeroplane climbs 500 metres in the first minute after take-off. How
high can it fly if in each subsequent minute the rate of climb reduces by
5%?
Sigma notation
Sigma notation is used to express the sum of a series in compact form.
The expression inside the summation is Tn .
5. Exercise:
Evaluate the following:
(a)
25
X
4k − 5
(b)
P∞
−k
k=3 4
k=6
(c)
10
X
3k + 3k
(7)
k=5
Financial Applications of Series
The compound interest formula A = P (1 + i)n is the same form as the
general term of a GP, Tn = arn−1 . Therefore we can use the sum of a GP
to evaluate the accumulation of regular payments or calculate the instalment to repay a loan.
Note
Some questions specifically ask to develop the formula for the accumulated
amount/amount outstanding at the end of year n. Other questions aren’t so
restrictive and hence we can use alternative approaches.
Consider:
(a) Superannuation, where $1000 is paid into a fund at the start of each year
for 6 years at an interest rate of 8% pa compound. We need to find the
accumulated amount at the end of the 6th year.
(b) Loan repayment, where a loan of $20000 is to be repaid by annual repayments over 5 years at an interest rate of 6% pa.
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6. Exercise:
A woman opens a trust account with an initial deposit of $10000 upon the birth
of her first grand-daughter. At each subsequent birthday up to age 17 (ie ages 1
to 17 inclusive) she makes additional deposits of $1000. The account pays interest at 5% pa compound. On the girl’s 18th birthday the accumulated amount
is transferred to another account which pays interest at 6% pa compounded
monthly. The girl wants to withdraw a fixed amount each month commencing
on her 18th birthday. How much can she withdraw each month so that the account is exhausted after exactly 5 years?
7. Exercise:
This comes from the 2003 HSC and was described as an unfair question at the
time:
Barbara borrows $120000 to be repaid over a period of 25 years at 6% pa reducible interest. Each year there are k regular payments of $F. Interest is
calculated and charged just before each repayment.
(i) Write down an expression for the amount owing after two repayments.
(ii) Show that the amount owing after n repayments is
An = 120000αn −
kF (αn − 1)
0.06
where α = 1 + 0.06
. (iii) Calculate the amount of each repayment if repayments
k
are made quarterly (ie k = 4)
Geometrical applications of calculus
Given a function y = f (x), the first derivative, f 0 (x), calculates the gradient of
the tangent to the curve. That is, the rate of change in the function.
The second derivative, f 00 (x), calculates the rate of change in the first derivative
or the concavity of the function.
We find stationary points by solving f 0 (x) = 0.
An inflection point is where the concavity changes. So the full test for an inflection point involves calculating the sign of f 00 (x) either side of the x-value
obtained from solving f 00 (x) = 0.
At an ordinary inflection point the gradient is steepest so the maximum positive
value, or minimum negative value, of f 0 (x) is achieved.
At a horizontal inflection point the gradient is zero, so both f 0 (x) and f 00 (x) are
0.
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8. Exercise:
Draw graphs for functions that: (1) Increase at an increasing rate (2) Increase at
a decreasing rate (3) Decrease at an increasing rate (4) Decrease at a decreasing
rate.
9. Exercise:
Now determine the signs of both f 0 (x) and f 00 (x) for each of the above 4 functions.
10. Exercise:
The population of a town was 20000 at the end of 1998. During the next 10
years the population increased at a decreasing rate to 24000. Draw a possible
graph of the population in the town over the period.
11. Exercise:
30
The plot below shows y = f 0 (x). Draw a possible graph of y = f (x) given that
f (0) = 1.
y = f '(x)
x
−1
1
2
3
4
−30
−2
12. Exercise:
The plot below shows y = f (x). Draw a possible graph of y = f 0 (x).
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y = f(x)
x
13. Exercise:
For the function f (x) = x3 (x − 2)2 : (i) calculate the stationary points (ii)
determine their nature (iii) plot the function
14. Exercise:
Plot the function whose derivative function is f ´ (x) = x(x − 3) and passes
through the point (0, 1).
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