The document discusses how the first derivative can be used to analyze curves geometrically. It indicates that the first derivative measures the slope of the tangent line to a curve. If the derivative is positive, the curve is increasing; if negative, decreasing; and if zero, the curve is stationary. As an example, it finds the stationary points of the curve y = 3x^2 - x^3 and determines that they are a minimum at (0,0) and an inflection point at (2,4).
2. Geometrical Applications
of Differentiation d dy
The First Derivative y, f x , f x ,
dx dx
3. Geometrical Applications
of Differentiation d dy
The First Derivative y, f x , f x ,
dx dx
dy
measures the slope of the tangent to a curve
dx
4. Geometrical Applications
of Differentiation d dy
The First Derivative y, f x , f x ,
dx dx
dy
measures the slope of the tangent to a curve
dx
If f x 0, the curve is increasing
5. Geometrical Applications
of Differentiation d dy
The First Derivative y, f x , f x ,
dx dx
dy
measures the slope of the tangent to a curve
dx
If f x 0, the curve is increasing
If f x 0, the curve is decreasing
6. Geometrical Applications
of Differentiation d dy
The First Derivative y, f x , f x ,
dx dx
dy
measures the slope of the tangent to a curve
dx
If f x 0, the curve is increasing
If f x 0, the curve is decreasing
If f x 0, the curve is stationary
7. Geometrical Applications
of Differentiation d dy
The First Derivative y, f x , f x ,
dx dx
dy
measures the slope of the tangent to a curve
dx
If f x 0, the curve is increasing
If f x 0, the curve is decreasing
If f x 0, the curve is stationary
e.g. For the curve y 3x 2 x 3 , find all of the stationary points
and determine their nature.
Hence sketch the curve
8. Geometrical Applications
of Differentiation d dy
The First Derivative y, f x , f x ,
dx dx
dy
measures the slope of the tangent to a curve
dx
If f x 0, the curve is increasing
If f x 0, the curve is decreasing
If f x 0, the curve is stationary
e.g. For the curve y 3x 2 x 3 , find all of the stationary points
and determine their nature.
Hence sketch the curve
dy
6 x 3x 2
dx
11. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
12. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
stationary points occur at 0,0 and 2,4
13. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
stationary points occur at 0,0 and 2,4
0,0
14. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
stationary points occur at 0,0 and 2,4
0,0
x
dy
dx
15. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
stationary points occur at 0,0 and 2,4
0,0
x 0
dy
dx
16. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
stationary points occur at 0,0 and 2,4
0,0
x 0
dy 0
dx
17. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
stationary points occur at 0,0 and 2,4
0,0
x 0 0
dy 0
dx
18. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
stationary points occur at 0,0 and 2,4
0,0
x 0 0 0
dy 0
dx
19. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
stationary points occur at 0,0 and 2,4
0,0
x 0(-1) 0 0
dy 0
dx
20. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
stationary points occur at 0,0 and 2,4
0,0
x 0(-1) 0 0
dy (-9)
0
dx
21. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
stationary points occur at 0,0 and 2,4
0,0
x 0(-1) 0 0 (1)
dy (-9)
0
dx
22. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
stationary points occur at 0,0 and 2,4
0,0
x 0(-1) 0 0 (1)
dy (-9)
0 (3)
dx
23. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
stationary points occur at 0,0 and 2,4
0,0
x 0(-1) 0 0 (1)
dy (-9)
0 (3)
dx 0,0 is a minimum turning point
24. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
stationary points occur at 0,0 and 2,4
0,0
x 0(-1) 0 0 (1)
dy (-9)
0 (3)
dx 0,0 is a minimum turning point
2,4
25. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
stationary points occur at 0,0 and 2,4
0,0
x 0(-1) 0 0 (1)
dy (-9)
0 (3)
dx 0,0 is a minimum turning point
2,4
x
dy
dx
26. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
stationary points occur at 0,0 and 2,4
0,0
x 0(-1) 0 0 (1)
dy (-9)
0 (3)
dx 0,0 is a minimum turning point
2,4
x 2
dy
dx
27. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
stationary points occur at 0,0 and 2,4
0,0
x 0(-1) 0 0 (1)
dy (-9)
0 (3)
dx 0,0 is a minimum turning point
2,4
x 2
dy 0
dx
28. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
stationary points occur at 0,0 and 2,4
0,0
x 0(-1) 0 0 (1)
dy (-9)
0 (3)
dx 0,0 is a minimum turning point
2,4
x 2 2
dy 0
dx
29. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
stationary points occur at 0,0 and 2,4
0,0
x 0(-1) 0 0 (1)
dy (-9)
0 (3)
dx 0,0 is a minimum turning point
2,4
x 2 2 2
dy 0
dx
30. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
stationary points occur at 0,0 and 2,4
0,0
x 0(-1) 0 0 (1)
dy (-9)
0 (3)
dx 0,0 is a minimum turning point
2,4
x 2(1) 2 2
dy 0
dx
31. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
stationary points occur at 0,0 and 2,4
0,0
x 0(-1) 0 0 (1)
dy (-9)
0 (3)
dx 0,0 is a minimum turning point
2,4
x 2(1) 2 2
dy (3)
0
dx
32. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
stationary points occur at 0,0 and 2,4
0,0
x 0(-1) 0 0 (1)
dy (-9)
0 (3)
dx 0,0 is a minimum turning point
2,4
x 2(1) 2 2 (3)
dy (3)
0
dx
33. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
stationary points occur at 0,0 and 2,4
0,0
x 0(-1) 0 0 (1)
dy (-9)
0 (3)
dx 0,0 is a minimum turning point
2,4
x 2(1) 2 2 (3)
dy (3)
0 (-9)
dx
34. dy
Stationary points occur when 0
dx
i.e. 6 x 3x 2 0
3x2 x 0
x 0 or x 2
stationary points occur at 0,0 and 2,4
0,0
x 0(-1) 0 0 (1)
dy (-9)
0 (3)
dx 0,0 is a minimum turning point
2,4
x 2(1) 2 2 (3)
dy (3)
0 (-9)
dx 2,4 is a maximum turning point