Rates of change

Rate of change

• Tell me something which …a)Changes

b)Doesn't change

c)Changes at a constant rate

d)Changes slowly / quickly

e)Does not change at a constant rate

f) Whose rate of change is constant

g)Whose rate of change is not constant

Gradient as ‘rate of change’

f(x) = kHow quickly does the height of this line change?

It doesn’t

Hence …

Differentiation as ‘rate of change’

What can you say about:

f’(x)? [the rate of change of

f(x)?]

f’’(x)? [the rate of change of

f’(x)?]

f(x) = kx

g(x) = jx

Differentiation as ‘rate of change’

f(x) = kx2

For each curve, what can you say about:

• f’(x)?• f’’(x)?• f’’’(x)?

Rate of Change

x

f(x) = x2

f ’(x) = 2x

f ’’(x) = 2

f ’’’(x) = 0

1 2 3 4 5 6 7 8 9

1 4 9 16 25 36 49 64 81

3 5 7 9 11 13 15 17

2 2 2 2 2 2 2

0 0 0 0 0 0

Finding the gradient of a line

What is the gradient of this line?

dy/dx

= 20 / 2

=10

Finding the gradient of a curve

• How can we find the gradient at a point? (choose a point)

• We can draw a tangent

• Then find the gradient of the tangent

• However, this is not very accurate

Differentiating to find the gradient

• To find the gradient of a polynomial by differentiation …

• Multiply by the power• Then reduce the

power by 1.

• x^n

• nx^n• nx^(n-1)

Back to our example

Note that f(x) = x^2

Multiply by the power:

2x^2

Reduce the power by 1

2x^1 = 2x

f’(x) = dy/dx = 2x

Gradient at x = 2x

Differentiating to find the gradient

f’(2x) = f’(2x^1)=1(2x^0)= 2

Notice how this also works with straight line graphs

f’(4) = f’(4x^0)= 0(4x^-1)= 0

Integration (the reverse)

If DIFFERENTIATION seeks to find the rate at which something changes

INTEGRATION seeks to find the total change that occurs

For example

A man walks away from his house, and after 2 hours, finds himself 4 km away from his house.

DIFFERENTIATION finds his displacement’s RATE OF CHANGE to be …

In contrast, a man walks away from his house for 2 hours at a rate of 2km / hr.

INTEGRATION finds his displacement’s TOTAL CHANGE to be …

For example

If integration is the opposite of differentiation …

Differentiation involves:

MULTIPLYING by the power, and

REDUCING the power by 1.

Then integration involves:

RAISING the power by 1, and

DIVIDING by the new power.

Integration as ‘the change’

k.dx =

kx

What is changing?

f(x) = k

x

k

Integration as ‘the change’

f(x) = x

x.dx =1/2x2

Does this also apply here?

x

x

Integration as ‘the change’

What is the area under this curve?

x2.dx =1/3x3

f(x) = x2