# When Will the Population Be a Minimum and the Rate of Change Be a Maximum?

• huan.conchito
In summary, given a function f(t) = (2t-1) / (t^2-t+0.5), the population changes at a rate of f(t) if 75 animals are introduced into the refuge.
huan.conchito
Given a function f(t) = (2t-1) / (t^2-t+0.5)
max supported species = 120, and t is the time.
if 75 animals are introduced into te refuge the population changes at a rate of f(t)
What needs to be done to obtain
a ) Find when the population will be a minimum
and to find
b) When the rate of change of the population will be a maximum

Last edited:
Theoretically

a)f'(x)=0,f''(x)>0
b)f''(x)=0,f'''(x)<0...

Daniel.

i still don't get it

Are you sure you don't mean $f(t)$?

I'll assume that you do. In that case, what conditions are necessary for a differentiable function to have a minimum at a point? Look at your notes if you don't remember.

No need to look at the notes,i wrote them,before he's edited his post & spell out the function...

Daniel.

yup is f(t)

Then solve it...Set those derivatives to zero and verify the nature of the critical points.

Daniel.

i found the minimum at t= 1/2
how do i go about part b?
for part b i got
f''(t)= 2(2t-1)
2(2t-1)=120 -> t =30.5
is that the time of the max population?

Last edited:
huan.conchito said:
Given a function f(x) = (2t-1) / (t^2-t+0.5)
Okay we have a function.

max supported species = 120, 75 are initially introduced.
and t is the time.

What needs to be done to obtain
a ) Find when the population will be a minimum
and to find
b) When the rate of change of the population will be a maximum

?? What happened to f? what does f have to do with the population?
Have you left something out- like f(t) is the population at time t? But I notice that f(0)= -2, not 75 and the maximum of f is 2, not 120. What does f have to do with the population?

if 75 animals are introduced into te refuge the population changes at a rate of f(t)

## 1. What is differentiation and how is it used in scientific applications?

Differentiation is a mathematical concept that involves finding the rate of change of a function. In science, it is used to analyze and model various phenomena that involve continuous change, such as motion, growth, and decay. It can also be used to find maximum and minimum values of a function, which is useful in optimization problems.

## 2. How is differentiation related to calculus?

Differentiation is one of the fundamental concepts in calculus, along with integration. It is the process of finding the derivative of a function, which is the slope of the tangent line at any point on the function's graph. Integration, on the other hand, is the reverse process of finding the original function from its derivative.

## 3. What are some real-world examples of differentiation?

Differentiation is used in a wide range of scientific applications, such as physics, biology, economics, and engineering. Some examples include calculating the velocity of a moving object, determining the growth rate of a population, and optimizing the production process in a manufacturing plant.

## 4. How is differentiation used in data analysis?

In data analysis, differentiation is used to find the rate of change of a dataset, which can provide valuable insights into the underlying patterns and trends. It is also used in curve fitting, where a curve or line is fitted to a set of data points to model the relationship between variables.

## 5. Can differentiation be applied to non-linear functions?

Yes, differentiation can be applied to both linear and non-linear functions. In fact, it is often used to analyze and model non-linear relationships between variables, such as exponential and logarithmic functions. The process of differentiation remains the same, regardless of the type of function.

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