Transcript Year 11 Maths enrichment

Year 11 Maths enrichment
Gradients
Contents
This enrichment work is meant to develop your understanding of some
GCSE topics and extend them into more advanced ideas. All the methods
used are ones that you have learnt or will learn in the Higher GCSE course,
but the outcome is part of the A level. Each section should take about 30
mins. Click on the links below to go straight to the section.
1. The gradient at a point
2. The gradient function
3. Pattern hunting
4. Proving the results
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1. The gradient at a point
How can you calculate the gradient at a point on a graph?
We’ll start with the curve y=x2, and try to find the gradient at (1, 1).
You could draw a tangent, and measure the gradient (Now click)
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4
y=x2
2
-2
2
4
1
0
-4
1.8
gradient 
0
1.8
 1.8
1
6
-2
but this would not be accurate.
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1. The gradient at a point
We can calculate exactly the gradient of a chord (a line between two
points on the curve)
Look at the graph and check the calculations below – click to move the
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position of B.
B
44
B
3
3
B 1.25
0.44 gradient = 1 =3
A C C C
1.25
0.44
A
gradient
=
=2.5
1
0.5
0.5 =2.2
0.2
0.2
22
-2
-2
00
The graph shows the chord from
y=x
(1,2 1) to (1.2,
(2, 4)1.2
(1.5,
1.5
on2)the
on the
graph
graph
of
y=x
Coordinates y=x
2 which
2 which
of y=x
hashas
gradient:
gradient:
a=1.5
a=1.2
2
00
22
44
Coordinates
12..44
25411
13.44
0
25
  3 2.52
1.25 2 1 1 01.25
66
To find the-2 gradient we bring B in closer to A and hope to spot a pattern…
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Does it make sense?
Try for yourself.
Complete the table below – use your calculator to work out the
gradients from the point in the table to the point (1,1).
x
y=x2
2
1.5
1.4
1.3
1.2
1.1
22=4
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Gradient
of chord
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y 1 4 1

x 1 2 1
Compare the values of
the gradient with the
values of x.
What do you think the
gradient will be at x=1?
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2. The gradient function
Here, you are going to try to find a formula for the gradient from the formula
of the curve.
Find the gradient of the curve for a number of values of x, put these in a
table. Can you spot the formula that connects
gradient of
x
y
chord to (2,4)
the values of x to the values of the gradient?
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2.9
2.8
2.7
2.6
2.5
2.4
2.3
2.2
2.1
2
9
8.41
7.84
7.29
6.76
6.25
5.76
5.29
4.84
4.41
4
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4.9
4.8
4.7
4.6
4.5
4.4
4.3
4.2
4.1
????
The spreadsheet on the left will do all the
calculations for you. Click on it to open it – you
should get the “enable macros” option. If you
don’t, click here!
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3. Pattern hunting
Here you are going to try to find a pattern that connects the formula to
its gradient.
Use the spreadsheet on the last page to help you find the formulas for the
gradients to the curves in the table below
Equation
y=x
y=x2
y=x3
y=x4
y=x5
y=x6
Gradient
function
1
2x
3x2
4x3
5x4
6x5
Now generalise this to write down a rule:
The gradient function of xn is =nxn-1
Test your rule on the spreadsheet for other powers, like x-1, or x0.5
Click i for answers
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4. Proving the results
Now you are actually going to prove some of the earlier results.
To do this you need to check some algebra first.
Expand these brackets and then check the answers by double clicking on the i
reference.
(a + b)2= a2 + 2ab + b2
(a + b)3= a3 + 3a2b + 3ab2 + b3
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4. Proving the results
Ok, here goes…
The x-coordinate of point A is x.
What is the y-coordinate? (hint
B
y=x2
Coordinates
– the equation tells you).
x2
A
What is the x-coordinate of
point C? x+h
C
x
What is the y-coordinate of point B?
What is the length BC?
h
(x+h)2
(x+h)2-x2=2xh+h2
What is the gradient of AB?
2 xh  h 2
 2x  h
h
Now you can see what the gradient is when h=0. This should agree with the
work you have done in the previous sections.
Repeat the work for y=x3
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Possible extension questions
Now that you have been through the theory
you could apply it to these questions
1. What is the gradient function of a
multiple of the examples you have done
e.g. y=4x3
2. What is the gradient function when you
add these examples? E.g. y=2x3+x2
3. Can you prove the results for x3, x4 etc.
using the ideas in section 4 (Look up
binomial theorem to help you.
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