Nuffield
Free-Standing
Mathematics
Activity
Gradients
© Nuffield Foundation 2011
Gradients
5
6
7
4
8
3
1
2
©2011 Google – Map data
Walking the dog
Kerry goes on a walk. Where is the gradient of Kerry’s walk positive?
Where is it negative? Is there any part of the walk with a zero gradient?
Where is the gradient steepest?
© Nuffield Foundation 2011
Gradients
Height of a child on a swing
Height
Time
When is the gradient positive? negative?
zero?
What is happening then?
This activity shows how to find accurate values for the
gradients of curves.
Measuring gradients
Straight lines
y
Curves
y = mx + c
y
tangent
P
y step
y step
c
x step
x step
0
x
m = gradient =
0
x
y step
x step
Gradient of y =
y step
x step
gives an
approximate
value for the
gradient
Graph of y = x2
y
25
20
15
P (3, 9)
10
y step
5
x step
0
-5
-4
-3
-2
-1
0
1
2
2
x
3
4
5
x
It can be
calculated
more
accurately
Incremental changes
Gradient of PQ1

difference in
difference in
y
x
y = x2
values
values
Q1 (4, 16)
 16  9 = 7
4 3
12.25  9
Gradient of PQ2  3.5  3
Q2 (3.5, 12.25)
Q3 (3.25, 10.5625)
P(3, 9)
= 6.5
10.5625 9
Gradient of PQ3  3.25  3
= 6.25
As Q  P
gradient 6
Gradients of functions of the
form y = xn
Equation of curve
Gradient function
y = x2
2x
y = x3
3x2
y = x4
4x3
y = x5
Think about
• What do you think is the gradient function for y = x5?
How can you prove it?
• What about y = x6?
• Can you suggest an expression for the gradient of the
general function y = xn ?
Gradients
Reflect on your work
•
Describe the way in which the gradient of a
curve can be found using a spreadsheet.
•
What advantages does this have on drawing a
tangent to a hand-drawn graph?
•
What is the gradient function of y = xn ?
Extension: Differentiation
Gradient of PQ


difference in y values
difference in x values
y = x2
Q(x + dx, (x + dx)2)
x  dx2  x2
x  dx  x
2
2
2
 x  2xdx  dx  x
dx
2


2
x
d
x

d
x

dx
P(x, x2)
As Q  P
dx  0
 2x  dx
gradient  2x
Rules of differentiation
Function
Derivative
x2
dy
dx
y=
x3
dy
dx
=
y=
x4
dy
dx
= 4x3
y=
x5
dy
dx
=
5x4
y = mx
dy
dx
=m
y=c
dy
dx
=0
y=
= 2x
3x2
General rules
y = xn
dy
dx
= nx n – 1
y = ax n
dy
dx
= nax n – 1
General Rule for y = ax n
dy
= nax n – 1
dx
Example y = 2x3 – 9x2 + 12x + 1
dy
dx
= 6x2 – 18x + 12
x
0
0.5
y
1
5
1
1.5
2
2.5
6
5.5
5
6
gradient
12
4.5
0
– 1.5
0
4.5
y
y = 2x3 – 9x2 + 12x + 1
maximum
minimum
0
x
Example
y
maximum
y = 2x3 – 9x2 + 12x + 1
minimum
0
Gradient function
dy
dx
1
2
x
dy
dx
= 6x2 – 18x + 12
0
1
2
x