Significance of the second derivative

• SithsNGiggles
In summary, the conversation discusses the interpretation of the conditions s' = 0 and s" > 0 in the context of a problem involving the number of students with measles over time. These conditions indicate that no students are contracting measles anymore, but the value of s' is changing from either positive to negative or negative to positive. This may seem counterintuitive, but it simply means that the rate at which the number of students contracting measles is changing is also changing.

Homework Statement

Let y = s(t) represent the number of students who have contracted measles at time t (days). Give an interpretation for each condition:

e) s' = 0, s" > 0

The Attempt at a Solution

This seems counterintuitive to me, to think that the second derivative is also not zero. In the context of the problem, what does this mean? So far I've said that no students are contracting measles anymore, but I don't know where s" fits in here.

well for a start what does s'=0 tells you about the problem?

its not counter-intuitive to have s" non-zero whilst s'=0. If s" is non-zero it simply means the value of s' is changing

HINT: if you know s'=0, and s" is non zero, then s' must be changing from either (+ve to -ve) or (-ve to +ve)

What is the significance of the second derivative?

The second derivative is a measure of the rate of change of the first derivative. It can give insight into the curvature of a function and how it is changing over time or space.

How is the second derivative calculated?

The second derivative is calculated by taking the derivative of the first derivative. This can be done using the power rule, product rule, quotient rule, or chain rule depending on the form of the function.

What does a positive second derivative indicate?

A positive second derivative indicates that the function is concave up, meaning it is curving upwards. This can also mean that the rate of change of the function is increasing.

What does a negative second derivative indicate?

A negative second derivative indicates that the function is concave down, meaning it is curving downwards. This can also mean that the rate of change of the function is decreasing.

Why is the second derivative important in calculus?

The second derivative is important because it can help identify critical points, inflection points, and the behavior of a function. It can also be used to optimize functions and determine the concavity of curves.