The Concept of Differentiation
C2 differentiation
Maximum points, minimum points and points of inflection
All 3 types of point are easy enough to spot on a graph:
• Maximum points are the tops of ‘peaks’
• Minimum points are bottoms of ‘troughs’
• Points of inflection are where a curve stops turning ‘left’ and starts turning ‘right’ (or vice versa). An example is the point (0,1) on the curve [pic]+1
Notes (i) Any point on a curve where the gradient is zero can be called a ‘stationary point’ (which means that stationary points include maximum and minimum turning points and also any points of inflection at which the curve is horizontal, as it is in the example given above)
(ii) If a maximum or minimum are not actually the highest or lowest values that a curve ever reaches, but are just the highest or lowest value on the curve near to that point (as is the case for the maximum and minimum points on a cubic curve) then they may be called a ‘local maximum’ or ‘local minimum’. If on the other hand a curve just has a single maximum or just a single minimum (eg a quadratic function) then the value of y at that point will be the greatest or least value of that function.
The task in C2 is to use differentiation to find where these points are on different curves. In the case of maximum and minimum points, we also have to be able to use differentiation to determine which of the two it is.
Finding Maximum and Minimum points
It is fairly obvious that at these points on a curve the gradient is zero.
At a maximum or a minimum point [pic]
Therefore to find a maximum or minimum point (or points), the procedure will be:
i) get the gradient function by differentiating the equation of the curve, then ii) find the value (or values) of x which make the gradient function equal zero
Example: Find
Maximum points, minimum points and points of inflection
All 3 types of point are easy enough to spot on a graph:
• Maximum points are the tops of ‘peaks’
• Minimum points are bottoms of ‘troughs’
• Points of inflection are where a curve stops turning ‘left’ and starts turning ‘right’ (or vice versa). An example is the point (0,1) on the curve [pic]+1
Notes (i) Any point on a curve where the gradient is zero can be called a ‘stationary point’ (which means that stationary points include maximum and minimum turning points and also any points of inflection at which the curve is horizontal, as it is in the example given above)
(ii) If a maximum or minimum are not actually the highest or lowest values that a curve ever reaches, but are just the highest or lowest value on the curve near to that point (as is the case for the maximum and minimum points on a cubic curve) then they may be called a ‘local maximum’ or ‘local minimum’. If on the other hand a curve just has a single maximum or just a single minimum (eg a quadratic function) then the value of y at that point will be the greatest or least value of that function.
The task in C2 is to use differentiation to find where these points are on different curves. In the case of maximum and minimum points, we also have to be able to use differentiation to determine which of the two it is.
Finding Maximum and Minimum points
It is fairly obvious that at these points on a curve the gradient is zero.
At a maximum or a minimum point [pic]
Therefore to find a maximum or minimum point (or points), the procedure will be:
i) get the gradient function by differentiating the equation of the curve, then ii) find the value (or values) of x which make the gradient function equal zero
Example: Find