# DE equation modeling growth off a tan function

• Outlaw747
In summary, the conversation discusses the rate of growth of bacteria, which is proportional to the tangent function evaluated on the amount of bacteria present at time t, up to 1.5 mg. After 1.5 mg, the value is maintained. The growth in 1 hour ranges from pi/6 to pi/3 mg. The question is how to write a DE equation based on this information, and the solution involves using a piecewise definition for the rate of growth.
Outlaw747

## Homework Statement

Data suggests rate of growth of bacteria is proportional to the tangent function evaluated on the amount of bacteria present at time t, up to 1.5 mg. After 1.5 mg, value maintained. pi/6 to pi/3 mg growth in 1 hour. How would I write a DE equation based on this? I'm not asking how to solve it but to set the equation up.

## Homework Equations

dP/dt = kP
P(t) = P-initial * e^(kt)

## The Attempt at a Solution

I am not really sure how to go about this. If we use the second equation isn't that based off another growth model? If I have dP/dt = tan (t) , that doesn't really make sense. Just need a push in the right direction if possible. Tried a few different attempts but none really worked.

The question stated "is proportional to", not "equals", so that should be dP/dt = k tan(t). That's only until 1.5mg though. If I understand correctly, after 1.5 mg, P increases by at least pi/6 mg, and at most pi/3 mg, each hour? For that part I would suggest something like a piecewise definition- Before 1.5mg, dP/dt = k tan(t), after 1.5mg, $$\pi/6 < dP/dt < \pi/3$$.

K I'll try that, thanks.

## 1. What is a differential equation (DE)?

A differential equation is a mathematical equation that relates the rate of change of a quantity to its current value. It is commonly used to model dynamic systems in various fields, such as physics, chemistry, and biology.

## 2. What is a tan function?

The tan function, short for tangent function, is a trigonometric function that calculates the ratio of the length of the opposite side to the length of the adjacent side in a right triangle.

## 3. How does DE equation model growth off a tan function?

The DE equation models growth off a tan function by relating the rate of change of a quantity, such as population growth, to its current value using the tangent function. This allows for a more accurate prediction of the growth pattern over time.

## 4. What are the key parameters in DE equation modeling growth off a tan function?

The key parameters in DE equation modeling growth off a tan function include the initial value, the growth rate, and the carrying capacity. The initial value represents the starting point of the growth, the growth rate determines the speed of growth, and the carrying capacity represents the maximum limit of growth.

## 5. How is DE equation modeling growth off a tan function useful in real-life applications?

DE equation modeling growth off a tan function has various real-life applications, such as predicting population growth, analyzing stock market trends, and forecasting the spread of diseases. It can also be used to optimize processes in fields such as engineering and economics.

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