MEI Conference 2009
Stretching students:
A2 Core
Presenter: Bernard Murphy
bernard.murphy@mei.org.uk
© MEI 2009
Workshop G1
1. The 3-4-5 triangle
How can you prove that these six
right-angled triangles fit together
exactly to make a 3-4-5 triangle?
What does it tell you about the
inscribed circle?
What is the link with the double
angle formulae?
2. Sliding ladders
Ladder A topples away from a wall.
Ladder B slides down a wall.
Compare the paths followed by the mid-point of each ladder.
3. Circle theorem?
The spiral starts at the point (0,-1) and
the perpendicular edges are drawn in
an anticlockwise spiral with a common
ratio r as shown.
Due to similarity, after an even number
of steps the leading point will be on the
diagonal line shown.
If this diagonal makes an angle θ with the first edge as shown, find, in terms
of θ, the coordinates of the point on which the spiral is converging.
© MEI 2009
4. Finding parametric equations of a Cartesian curve
Imagine a point P on the curve as shown.
Then x = r cos θ and y = r sin θ .
If we can write r in terms of θ then we
have our parametric equations.
4 y
2
Find parametric equations for the curve ( 2 − x ) y 2 = x 3 .
x
2
−2
(Substitute x = r cos θ and y = r sin θ and use this to
express r in terms of θ . Finally substitute for r in
x = r cos θ and y = r sin θ )
−4
5. Constructing a regular pentagon
1. Draw a circle, centre O and bisect the radius OP. M is the midpoint of OP.
2. Draw an arc of a circle, centre M, radius MA. This arc crosses OQ at N.
3. Draw an arc of a circle, centre A, radius AN. This gives the first side, AB of
the regular pentagon.
A
A
A
B
P
M
P
M
N
Q
N
Q
Prove this would produce a regular pentagon. What is the link with sin 5θ or
sin18D ?
6. Equilateral triangle on grid points
Prove that an equilateral triangle in the
x-y plane cannot have all three
vertices on grid points (i.e. points
where both coordinates are integers.)
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7. Estimating the harmonic series
1
y
1
y
x
1
2
3
4
x
5
1
The area of the shaded region is
⎛1 1 1 1 ⎞ 5 1
⎜ + + + ⎟ − ∫1 dx
x
⎝1 2 3 4 ⎠
2
3
4
5
The area of the shaded region is
51
⎛1 1 1 1⎞
∫1 x dx − ⎜⎝ 2 + 3 + 4 + 5 ⎟⎠
In each case, imagine sliding the 4 shaded regions left so that each one
touches the y − axis. You can see that these 4 regions fit into the 1 by 1
‘rectangle’ without overlapping and so the shaded areas are both less than 1.
1 1
1
1
Explain how together these lead to 1 + + + ... + ≈ ln N +
2 3
N
2
8. Composite piecewise functions
For the functions f( x) and g( x ) given below, find the composite function fg( x)
x<0
⎧ 0
⎪ 2
f( x) = ⎨ x 0 ≤ x ≤ 4
⎪ 16
x>4
⎩
x<0
⎧ 0
⎪
g( x) = ⎨ 2 x 0 ≤ x ≤ 3
⎪ 6
x>3
⎩
16 y
16 y
12
12
8
8
4
−2 −1
x
1 2 3 4 5 6
4
−2 −1
x
1 2 3 4 5 6
9. Primes of the form 4n+3
Prove by contradiction that there is an infinite number of primes of the form
4n+3
© MEI 2009
10. Which is bigger, ep or pe?
By considering the turning
ex
point on the graph of y = e
x
By considering the turning
ln x
point on the graph of y =
x
y
y
x
x
By considering the gradients of a tangent and chords of the graph y = ln x .
11. A surprising property?
2 y
Look at the graphs of
y = tan x and y = cos x .
It appears that they cross
at right angles to each
other.
1
x
−π/2
π/2
π
3π/2
2π
−1
Is this true?
−2
12. All integers?
4
Find the missing edge length, a .
8
a
6
17
A
22
© MEI 2009
13. A series for ln 2
1
−1
= (1 + x ) = 1 − x + x 2 − x 3 + x 4 − x 5 + x 6 − x 7 + ...
1+ x
Integrating between x = 0 and x = 1 you should be able to find an infinite
1 1
series which converges to ∫
dx = ln 2 .
0 1+ x
Using this idea and starting with other functions generate other infinite series.
14. Series of binomial coefficients
⎛n⎞ ⎛n⎞
⎛ n⎞
⎛ n⎞
n
The binomial expansion: (1 + x ) = ⎜ ⎟ + ⎜ ⎟ x + ⎜ ⎟ x 2 + ... + ⎜ ⎟ x n
⎝0⎠ ⎝1⎠
⎝ 2⎠
⎝ n⎠
By using calculus and/or substitution, prove the following:
⎛n⎞ ⎛n⎞ ⎛n⎞
n −1
⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ + ... = 2
0
2
4
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎛n⎞ ⎛n⎞ ⎛n⎞
⎛ n⎞
n
⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ + ... + ⎜ ⎟ = 2
0
1
2
n
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎝ ⎠
⎛n⎞ ⎛ n⎞ ⎛ n⎞ ⎛ n⎞
⎜ ⎟ − ⎜ ⎟ + ⎜ ⎟ − ⎜ ⎟ + ... = 0
⎝ 0⎠ ⎝ 1⎠ ⎝ 2⎠ ⎝ 2⎠
⎛n⎞ ⎛n⎞ ⎛n⎞
⎛ n⎞
n −1
⎜ ⎟ + 2 ⎜ ⎟ + 3 ⎜ ⎟ + ... + n ⎜ ⎟ = n × 2
⎝1⎠ ⎝ 2⎠ ⎝ 3⎠
⎝ n⎠
⎛n⎞ ⎛n⎞ ⎛n⎞
⎛ n ⎞ ⎛ 2n ⎞
⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ + ... + ⎜ ⎟ = ⎜ ⎟
⎝ 0⎠ ⎝1⎠ ⎝ 2⎠
⎝n⎠ ⎝ n ⎠
2
2
2
2
Can you find any more?
15. Binomial theorem and differentiation
y = (1 − x ) = 1 + x + x 2 + x 3 + x 4 + ... ⇒
−1
⇒
So the expansion of (1 − x )
−3
dy
1
=
= 1 + 2 x + 3 x 2 + 4 x 3 + ...
dx (1 − x )2
1 d2 y
1
=
= 1 + 3 x + 6 x 2 + 10 x3 + ...
3
2
2 dx
(1 − x )
has triangular numbers as coefficients.
•
Find the rational function whose expansion has square numbers as
coefficients.
•
Evaluate
1 2 3 4
n
+ 2 + 3 + 4 + ... + n + ... .
2 2 2 2
2
© MEI 2009
Lesson idea 1: Newton’s approximation to π
y
0.5
x
0.25
0.5
The diagram shows a semi-circle with centre
0.75
1
( 12 , 0 ) and radius
1. Show that the area of the shaded region is
π
24
−
1
2
.
3
32
2. Show that the semicircle has equation y = x 2 (1 − x ) 2 and use the
binomial theorem to find the first five terms in the expansion.
1
1
3. Using these terms, and integration, find an approximate value for the
shaded area.
4. Compare this with the exact answer found in 1 above*. To what level of
accuracy does this give the value of π ?
*How would Newton have evaluated 3 ?
He might have used the binomial theorem on
1
16 49
49 48 7 ⎛
1 ⎞2
=
×
= ⎜ 1 − ⎟ since this would converge quickly.
3 = 3× ×
16 49
16 49 4 ⎝ 49 ⎠
© MEI 2009
Lesson idea 2: Bounds on n!
To calculate, say, 100! you need to perform 99 multiplications. Is there a
quicker way to find the approximate value of n! where n is a large number?
Here is one method.
Think about the area under the graph y = ln x between x = 1 and x = n .
This is
∫
n
1
1
n
dx = [ x ln x − x ]1 = n ln n − n + 1
x
ln x dx = ∫ 1× ln x dx = [ x ln x ]1 − ∫ x ×
n
n
n
1
1
We can get lower and upper bounds for this by approximating the area of the
region under the graph y = ln x as shown below.
3
y
y
3
2
2
1
1
x
x
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
ln 2 + ln 3 + ... + ln ( n − 1) <
⇒
ln ( ( n − 1) !) <
⇒
ln ( ( n − 1) !) <
n
∫ ln x dx
∫ ln x dx
< ln ( n !)
n ln n − n + 1
< ln ( n !)
< ln 2 + ln 3 + ... + ln n
1
n
1
Taking the two inequalities separately:
n ln n − n + 1 < ln ( n !) ⇒ n ! > e
n ln n − n +1
nn
⎛n⎞
= n −1 = e ⎜ ⎟
e
⎝e⎠
⎛n⎞
ln ( ( n − 1) !) < n ln n − n + 1 ⇒ ( n − 1) ! < e ⎜ ⎟
⎝e⎠
n
n
⎛n⎞
⎛n⎞
e ⎜ ⎟ < n ! < en ⎜ ⎟
⎝e⎠
⎝e⎠
Combining these gives
100
⎛ 100 ⎞
For example, e ⎜
⎟
⎝ e ⎠
n
⎛n⎞
⇒ n ! < en ⎜ ⎟
⎝e⎠
n
n
100
⎛ 100 ⎞
< 100! < 100e ⎜
⎟
⎝ e ⎠
⇒ 1.01×10157 < 100! < 1.01× 10159
In fact, 100! ≈ 9.33 × 10157 .
© MEI 2009
Lesson idea 3: Wallis’ formula for π
π
Define I m = ∫ 2 sin m x dx
0
1. Evaluate I 0 and I1 .
2. Writing sin m x = sin m −1 x sin x and using integration by parts, show that
mI m = ( m − 1) I m− 2 .
3. Using your two answers above, evaluate I 2 , I 4 , I 6 ,... and I 3 , I 5 , I 7 ,...
4. Using the fact that 0 < sin x < 1 for 0 < x <
∫
π
2
0
π
π
0
0
π
2
, explain why
sin 2 m +1 x dx < ∫ 2 sin 2 m x dx < ∫ 2 sin 2 m −1 x dx
5. Hence show that
π
2 2 4 4 6 6
= . . . . . ...
2 1 3 3 5 5 7
Lesson idea 4: Binet’s formula in three steps
The Fibonacci sequence: f1 = 1, f 2 = 1,
Consider
∞
∑f x
n =1
n
n
f n +1 = f n + f n −1 for n ≥ 2
=x + x 2 + 2 x 3 + 3 x 4 + 5 x5 + 8 x 6 + ...
∞
1. Verify that (1 − x − x 2 ) ∑ f n x n = x then divide throughout by x :
n =1
∞
∑f x
n −1
n
1
2. Derive, using partial fractions:
α=
=
1
(1 − x − x 2 )
1
1 ⎛ −1
1 ⎞
≡
+
where
⎜
2
(1 − x − x ) 5 ⎝ x − α x − β ⎟⎠
−1 + 5
−1 − 5
and β =
.
2
2
3. Using the binomial expansion of the terms on the RHS and considering
n
n
1 ⎛⎜ ⎛ 1 + 5 ⎞ ⎛ 1 − 5 ⎞ ⎞⎟
n −1
coefficients of x show that f n =
⎜
⎟ −⎜
⎟
5 ⎜ ⎜⎝ 2 ⎟⎠ ⎜⎝ 2 ⎟⎠ ⎟
⎝
⎠
© MEI 2009
1.
3 = 42 + 62 − 7 2
4 = 62 + 7 2 − 92
5 = 4 2 + 52 − 6 2
Prove that every positive integer can be written in the form a 2 + b 2 − c 2
2. Take every integer power (greater than the first power) of every positive
integer greater than 1 and add the reciprocals together. What do you get?
1
1
1
1
+ ... + 17 + ... +
+ ... +
+ ...
2
3
2
5
29583
4588935290109207
3. Which of the following numbers is bigger?
∫
1
0
4
1 − x 7 dx or
∫
1
0
7
1 − x 4 dx
4. Prove that every positive rational number can be written as the sum of
distinct unit fractions (i.e. fractions with numerator 1).
3 1 1 1 1
1
1
1
= + + + + +
+
7 7 8 9 56 57 72 3192
5. Let f ( x ) = (1 − x + x 2 − ... + x100 )(1 + x + x 2 + ... + x100 ) .
Show that, after multiplying out, only even powers of x remain.
6. If you use a graph plotter to plot x3 + 3xy + y 3 = 1 you will find it seems be a
straight line. Is it?
7. Prove that tan 50D + tan 60D + tan 70D = tan 80D .
8. Some positive numbers add up to 19. What is the maximum product?
9. Using the sine rule and the compound angle formulae, prove that, in any
triangle,
a + b tan ( A+2 B )
=
a − b tan ( A−2 B )
10. Varignon's theorem: Prove that joining the midpoints of the sides of any
quadrilateral in order produces a parallelogram.
11. u1 , u2 , u3 ,..., u2 n +1 is a sequence of 2n + 1 positive integers.
v1 , v2 , v3 ,..., v2 n +1 is a rearrangement of u1 , u2 , u3 ,..., u2 n +1 .
Prove that the sequence {tk } where tk = uk − vk
k = 1, 2,3,..., 2n + 1 contains at
least one even number.
12. The particular function f : ` → ` It has the following two properties:
• The function is increasing; i.e. f ( n + 1) > f ( n ) for all n ∈ `
•
The composite function f ( f ( n ) ) = 3n for all n ∈ ` .
Find f (100 )
© MEI 2009