Differentiation
differentiation
1. The similarity of the turning points and the points of inflexion is both of them can be stationary point, but not all the stationary points are turning points or points of inflexion.
A turning point is a point which is the point of the sign of the derivative changes. And the turning points are the local maximum and minimum where the derivative of the function changes from positive to negative or from negative to positive. When the shape of the function is smooth, the turning point will be a stationary point.
(As shown in the graph, the points are the turning points of this graph.)
The points of inflexion are the points on the curve which is the sign of the curvature changes. The points of inflexion can be the stationary points but not the local maximum or minimum. After the first derivative, points of inflexion can be categorised to two different kinds: if f’(x) =0, this point is a stationary point of inflexion; if f’(x) ≠0, this point is a non-stationary point of inflexion. There is a condition for the points of inflexion in second derivative which is f”(x) need to equal to zero or not exist. On the both sides of the inflexion points, the graph will be increasing or decreasing on both sides.
As showing in the graph, the point (0, 0) is the inflexion point. 2. People use differential equations to predict the spread of diseases through a population.
Populations usually grow in an exponential fashion at first:
However, populations do not continue to grow forever, because food, water and other resources get used up over time. Differential equations are used to predict populations of people, animals, bacteria and viruses that are being affected by external events.
The logistic equation (developed in the mid-19th century) allows for a growth term AND an inhibition term. It is predicted that the AIDS epidemic will follow the pattern of the logistic equation.
If
A = number of people affected by the virus at time t,
P
1. The similarity of the turning points and the points of inflexion is both of them can be stationary point, but not all the stationary points are turning points or points of inflexion.
A turning point is a point which is the point of the sign of the derivative changes. And the turning points are the local maximum and minimum where the derivative of the function changes from positive to negative or from negative to positive. When the shape of the function is smooth, the turning point will be a stationary point.
(As shown in the graph, the points are the turning points of this graph.)
The points of inflexion are the points on the curve which is the sign of the curvature changes. The points of inflexion can be the stationary points but not the local maximum or minimum. After the first derivative, points of inflexion can be categorised to two different kinds: if f’(x) =0, this point is a stationary point of inflexion; if f’(x) ≠0, this point is a non-stationary point of inflexion. There is a condition for the points of inflexion in second derivative which is f”(x) need to equal to zero or not exist. On the both sides of the inflexion points, the graph will be increasing or decreasing on both sides.
As showing in the graph, the point (0, 0) is the inflexion point. 2. People use differential equations to predict the spread of diseases through a population.
Populations usually grow in an exponential fashion at first:
However, populations do not continue to grow forever, because food, water and other resources get used up over time. Differential equations are used to predict populations of people, animals, bacteria and viruses that are being affected by external events.
The logistic equation (developed in the mid-19th century) allows for a growth term AND an inhibition term. It is predicted that the AIDS epidemic will follow the pattern of the logistic equation.
If
A = number of people affected by the virus at time t,
P