Introduction to Differentiation

1
Introduction to Differentiation

• Motion Graphs

2
Travel Graph
4
3
5
2
1
Describe what is happening at each stage of this
travel graph.
3
Travel Graph
4
3
5
2
1
4
Travel Graph
4
3
What is the link between the speed of the car and
the graph?
5
2
1
What is the average speed of the car in each
section?
5
Projectile Motion
6
Projectile Motion
What shape is the flight path of the basketball?
7
Projectile Motion
h(t) (m)
3
2
4
1
5
t (s)
Describe what is happening at each stage of this
projectile motion graph.
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Projectile Motion
h(t) (m)
3
2
4
5
1
t (s)
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Projectile Motion
h(t) (m)
3
2
4
How can we calculate this?
1
5
t (s)
What is the instantaneous speed of the projectile
at each point?
10
Calculating the Gradient of a Curve
tangent
h(t) (m)
3
normal
2
4
1
5
t (s)
Draw a normal to the curve then a tangent at that
point.
11
Calculating the Gradient of a Curve
tangent
h(t) (m)
3
normal
4
1
5
t (s)
Calculate the gradient of the tangent.
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Calculating the Gradient of a Curve
tangent
h(t) (m)
(1.5,20)
The basketball is travelling at 10m/s at this
point.
(0.5,10)
The gradient of the tangent equals the gradient
of the curve at this point.
t (s)
The Rate of Change of the graph is equal to the
gradient at that point.
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Gradient Function
Use your calculated values for the gradients to
complete the following table.
x 0 1 2 3 4
gradient
20
10
0
10
20
Now plot these points on squared paper.
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Gradient Function
You have plotted the value of the gradient at
each point x on the curve for 0 x 4.
This is called the Gradient Function
Find the equation of this function.
15
Using (0,20) and (2,0) we get
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Projectile Motion
Use your knowledge of quadratic functions to
obtain the equation of this function.
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Roots at x 0 an x 4
When x 2, y 20
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Equation of the projectile curve
Equation of the gradient function
Can you spot a link?
Here is another pair that might help you
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Gradient Function, Secant Tangent
Click image to open Flash animation Differentiati
on Animation
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Differentiation the gradient of f(x)
y
y f(x)
Q (x h, f(x h))
P (x, f(x))
x
x
x h
h
The gradient of PQ is given by
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Differentiation the derivative of f(x)
y
y f(x)
Q (x h, f(x h))
T
P (x, f(x))
x
x
x h
The gradient of tangent at P is given by