1. Nuffield Free-Standing Mathematics Activity
Gradients

2. Gradients Walking the dog6 5 7 4 8 3 1 2
Walking the dog
©2011 Google – Map data
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?

3. Gradients When is the gradient positive? negative? zero?
Height of a child on a swing
When is the gradient
positive?
negative?
zero?
What is happening then?
This activity shows how to find accurate values for the gradients of curves.

4. Measuring gradients Straight lines Curves m = gradient = y y = mx + cx
Curves y x
tangent
y step
x step P
y step
x step
m = gradient =

5. Gradient of y = x2 gives an approximate value for the gradient step x
y step
x step
P (3, 9)
It can be calculated more accurately

6. Incremental changes y = x2 Gradient of PQ1 = 7 Gradient of PQ2 = 6.5
As Q ® P
gradient ® 6
= 6.25

7. 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 ?

8. 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 ?

9. Extension: Differentiation
P(x, x2)
y = x2
Q(x + dx, (x + dx)2)
Gradient of PQ
As Q ® P
dx ® 0
gradient ® 2x

10. Rules of differentiation
Function
Derivative
General rules
y = x2
= 2x
y = x n
= nx n – 1
= 3x2
y = x3
= 4x3
y = x4
y = ax n
= 5x4
= nax n – 1
y = x5
= m
y = mx
= 0
y = c

11. General Rule for y = ax n Example y = 2x3 – 9x2 + 12x + 1 = 6x2 – 18x
= nax n – 1
Example
y = 2x3 – 9x2 + 12x + 1
= 6x2
– 18x
+ 12
0.5 1
1.5 2
2.5 x
gradient y y x
y = 2x3 – 9x2 + 12x + 1 12 1
4.5 5
maximum 6
– 1.5
5.5
minimum 5
4.5 6

12. Example y = 2x3 – 9x2 + 12x + 1 Gradient function = 6x2 – 18x + 12 y 11
maximum 1 2
minimum
= 6x2 – 18x + 12
Gradient function x