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MATHEMATICS GRADE 12 DIFFERENTIAL CALCULUS PART 2
1
D I F F E R E N T I A L C A L C U L U S
THE GRAPH OF THE CUBIC FUNCTION
Turning Points (also called ‘Stationary Points’ or ‘Critical Points’)
When we determine ( ) we are dealing with the gradient of which can be
increasing, decreasing or equal to zero.
Minimum turning points
Maximum turning points
𝑓(𝑥) = 𝑎𝑥3 + 𝑏𝑥2 + 𝑐𝑥 + 𝑑
+ +
+ +
+ +
+ + +
+ +
+ +
- - -
- - - -
m = positive
∴ 𝒇 is increasing
m = 0 ∴Turning Point
m = positive
∴ 𝒇 is increasing
m =negative
∴ 𝒇 decreasing
m = 0 ∴Turning Point
𝒇(𝒙) = 𝒂𝒙𝟑 + 𝒃𝒙𝟐 + 𝒄𝒙 + 𝒅
𝑓 (𝑥) = 0
𝑓 (𝑥) < 0 𝑓 (𝑥) > 0 At a minimum turning point the sign
of the gradient changes from negative
to positive.
𝑓 (𝑥) < 0 𝑓 (𝑥) > 0
𝑓 (𝑥) = 0
At a maximum turning point the sign
of the gradient changes from positive
to negative.
MATHEMATICS GRADE 12 DIFFERENTIAL CALCULUS PART 2
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TO SKETCH THE CUBIC FUNCTION
To sketch the graph of ( ) = 3 + 2 + + , first determine the following:
1. SHAPE
If > 0 (positive), then
If < 0 (negative), then
2. TURNING POINTS
The x – coordinates: AND The y – coordinates:
Let ( ) = 0 Calculate ( ) and ( )
2 + + = 0
∴ = =
The turning points are ( ( )) and ( ( )).
3. THE x – INTERCEPT
*Let = 0, then factorize if possible (e.g. take out a common factor).
Otherwise….
* Let = 0
* “Guess” the first factor, e.g.
If f (2) = 0 then (x – 2) is a factor of f
If f (–3) = 0 then (x + 3) is a factor of f
* long division / synthetic division
* factorize to get ( )( )( ) = 0
* solve for to get the – intercepts
MATHEMATICS GRADE 12 DIFFERENTIAL CALCULUS PART 2
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4. THE y – INTERCEPT (d – value)
Let = 0, then calculate y.
5. Point of INFLECTION
The x – coordinate: The y – coordinate:
*Let ( ) = 0 *Substitute the x – value into ( )
*Solve for x *Simplify
OR
* Determine x = ( ) ( )
2
CONCAVITY & THE POINT OF INFLECTION
MATHEMATICS GRADE 12 DIFFERENTIAL CALCULUS PART 2
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EXAMPLES:
MATHEMATICS GRADE 12 DIFFERENTIAL CALCULUS PART 2
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ACTIVITY 1
MATHEMATICS GRADE 12 DIFFERENTIAL CALCULUS PART 2
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MATHEMATICS GRADE 12 DIFFERENTIAL CALCULUS PART 2
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MATHEMATICS GRADE 12 DIFFERENTIAL CALCULUS PART 2
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FINDING THE EQUATION OF A CUBIC CURVE
The graphs of ( ) vs the graph of ( )
MATHEMATICS GRADE 12 DIFFERENTIAL CALCULUS PART 2
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MAKE SURE YOU UNDERSTAND…..
Zeros at –5
‘Zeros’ = x – intercepts. In this case only one, at =
Can also be given as ( ) = 0 or ( 0)
Stationary points at x = – 4 and x = 0
‘Stationary points’ = Turning points
Can also be given as ( ) = 0 (0) = 0
f (–4) = 8 and f (0) = 1
tell us that for the function ……
= at = and
= at = 0
Can also be given as ( ) (0 )
Increasing on the interval (– ; – 4) and (0 ; )
’increasing’ means that the gradient will be positive for these intervals
Can also be written as ( ) > 0 for these intervals
Decreasing on the interval (– 4 ; 0) ’decreasing’ means that the gradient will be negative for the interval Can
also be written as ( ) < 0 for the interval
Now, a rough sketch of can be drawn:
-5 0
(- 4 ; 8)
(0 ; 1)
𝑓
MATHEMATICS GRADE 12 DIFFERENTIAL CALCULUS PART 2
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ACTIVITY 2
MATHEMATICS GRADE 12 DIFFERENTIAL CALCULUS PART 2
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MATHEMATICS GRADE 12 DIFFERENTIAL CALCULUS PART 2
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MATHEMATICS GRADE 12 DIFFERENTIAL CALCULUS PART 2
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