Download Differentiation and Integration Differentiation and Integration

1
Differentiation and Integration
Differentiation and
Integration together
are called Calculus.
Maths revision course by Miriam Hanks
2
Differentiation
How do you know when to differentiate?
Differentiate if the question says:
 Derive
 Find the derivative
 Find the rate of change
dy
 Find
or f ’(x)
dx
 Maximise or minimise
Maths revision course by Miriam Hanks
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Differentiation
How do you prepare for differentiation?
1


Change x to x 2
Get rid of fractions with x on the bottom:
3
2
eg
 3x
x

2
Multiply out brackets
Maths revision course by Miriam Hanks
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Differentiation
How do you differentiate?
Bring the power
to the front (to
multiply)
Subtract 1
from the
power
Maths revision course by Miriam Hanks
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Differentiation in real life
Why was differentiation developed originally?
In Isaac Newton's day,
shipwrecks occurred regularly
because the ship was not where
the captain thought it was using
only the stars for guidance.
Calculus was developed to
improve navigation techniques.
Maths revision course by Miriam Hanks
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Differentiation in real life
Differentiation is used in
economics and business to
find optimum solutions.
eg to use the minimum
amount of material to
make the packaging, or to
maximise efficiency of
products or to minimise
their costs.
Maths revision course by Miriam Hanks
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Differentiation and Integration
Click for calculus video
Maths revision course by Miriam Hanks
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Graphs of the derivative
If you are asked to draw the graph of the derivative,
keep in mind that this is a graph of the gradient.
 Start by marking points on the axes where there
are turning points on the original graph.
 Now look at the gradient before and after each of
these points: Is it up (in which case, draw above
the x- axis) or down (draw below x-axis)?
 Make sure you curve is smooth
Maths revision course by Miriam Hanks
9
Integration
How do you know when to integrate?
Integrate if the question says:
 Find the integral
 Find the area under a curve
 Find the area between 2 curves
 Find
dx
dy
 Find y when you are given

dx
Maths revision course by Miriam Hanks
10
Integration
How do you prepare for integration?
1


Change x to x 2
Get rid of fractions with x on the bottom:
3
2
eg
 3x
x

2
Multiply out brackets
Maths revision course by Miriam Hanks
11
Integration
How do you integrate?
Add 1 to the
power
Divide by the
new power
+c
Maths revision course by Miriam Hanks
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Integration in real life
Integration is used to
analyse crash test data, to
improve safety of cars.
It is also used by
engineers to find the force
of water from the surface
area of a reservoir, to
design dams.
Maths revision course by Miriam Hanks
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Further Calculus
To differentiate or integrate a trig function,
use the formula sheet:
sin ax
cos ax
sin ax
cos ax
differentiates to
integrates to
a cos ax
-a sin ax
1
a
1
a
- cos ax
sin ax
Maths revision course by Miriam Hanks
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Further Calculus
To differentiate or integrate a bracket with a power:
1. Think of the whole bracket as an “x” and
differentiate or integrate it as usual
2. Multiply by the derivative of the bracket
(if differentiating)
or Divide by the derivative of the bracket
(if integrating)
Maths revision course by Miriam Hanks
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Integration
Click for an integration joke
Maths revision course by Miriam Hanks