Modelling with Differentiation 📈

Welcome, students! In this lesson, you will learn how calculus can be used to build and test models of real situations. Differentiation is not just about finding slopes on graphs; it is a powerful tool for describing how one quantity changes compared with another. That makes it useful in business, science, medicine, engineering, and everyday decision-making. By the end of this lesson, you should be able to explain the key ideas of modelling with differentiation, use derivatives to study real-world behaviour, and connect this topic to the wider calculus toolkit.

Lesson objectives

Explain the main ideas and terminology behind modelling with differentiation.
Apply IB Mathematics: Analysis and Approaches HL reasoning or procedures related to modelling with differentiation.
Connect modelling with differentiation to the broader topic of calculus.
Summarize how modelling with differentiation fits within calculus.
Use evidence or examples related to modelling with differentiation in IB Mathematics: Analysis and Approaches HL.

What does modelling mean? 🤔

A model is a simplified mathematical description of a real situation. Real life is complicated, so a model focuses on the most important features and ignores details that are less important. For example, if a company wants to estimate profit from selling headphones, it may model profit as a function of the number sold. If a scientist studies the spread of a substance in water, they may use a function to describe how concentration changes over time.

In calculus, modelling often begins by deciding what the variables represent. A variable is a quantity that can change. Common examples include time $t$, distance $s$, height $h$, population $P$, or cost $C$. A function then describes how one variable depends on another. For example, $s(t)$ might represent position as a function of time, and $C(x)$ might represent cost as a function of the number of items $x$.

Differentiation enters the model when we want to know how quickly something is changing. The derivative of a function tells us its rate of change. If $y=f(x)$, then the derivative is written as $f'(x)$ or $\dfrac{dy}{dx}$. In words, the derivative gives the slope of the tangent to the graph and measures the instantaneous rate of change. This is why differentiation is so useful in real-world modelling: it tells us what is happening right now, not just over a long interval.

For example, if $s(t)$ gives the position of a car, then $s'(t)$ is the velocity of the car. If $v(t)=s'(t)$, then $v'(t)=s''(t)$ is the acceleration. These relationships are essential in kinematics, one of the main applications of calculus. 🚗

Building a model from a situation

The first step in modelling with differentiation is understanding the problem. students, ask these questions:

What is changing?
What are the inputs and outputs?
Which quantities matter most?
What assumptions are reasonable?

Assumptions are statements we accept to make the model manageable. For instance, when modelling the motion of a falling object near Earth, we may ignore air resistance at first. This makes the equations simpler and often gives a good approximation. Later, we can improve the model by adding more factors.

A good model should balance simplicity and realism. If it is too simple, it may not fit data well. If it is too complicated, it may be hard to use. In IB Mathematics: Analysis and Approaches HL, you are expected to reason about the structure of the model, not just carry out calculations.

Suppose the height of a plant is modelled by $h(t)=20+3t-0.2t^2$, where $h$ is in centimetres and $t$ is in weeks. Then the growth rate is

$$h'(t)=3-0.4t$$

This tells us how the height changes each week. At $t=0$, the plant is growing at $3$ cm per week. At $t=5$, the growth rate is $3-0.4(5)=1$ cm per week. If $h'(t)=0$, then the plant momentarily stops growing, which may suggest a maximum height in this model.

This is an important idea: the derivative helps us identify key features of the model, such as when growth stops, when a maximum occurs, or when the quantity is increasing or decreasing.

Interpreting derivatives in real contexts 🌍

The meaning of a derivative depends on the situation.

If $f(x)$ represents distance travelled, then $f'(x)$ could represent speed. If $f(x)$ represents revenue, then $f'(x)$ gives marginal revenue. If $f(x)$ represents temperature, then $f'(x)$ measures how quickly temperature changes with respect to time or position.

In business, marginal analysis is a common use of derivatives. If $R(x)$ is revenue and $C(x)$ is cost, then profit is $P(x)=R(x)-C(x)$. The derivative $P'(x)$ tells us how profit is changing as production increases. If $P'(x)>0$, profit is rising. If $P'(x)<0$, profit is falling. If $P'(x)=0$, we may have a maximum or minimum profit.

Example: let $P(x)=-x^2+12x-20$, where $x$ is the number of units produced in hundreds. Then

$$P'(x)=-2x+12$$

Setting the derivative equal to zero gives

$$-2x+12=0$$

so $x=6$. This suggests the profit is stationary when $600$ units are produced. To confirm whether this is a maximum, we can use the second derivative:

$$P''(x)=-2$$

Since $P''(x)<0$, the graph is concave down and the stationary point is a maximum. This is a standard calculus method for optimisation.

In many IB questions, you may be asked to explain the meaning of your answer in context. For example, do not stop at saying $x=6$. You should interpret it as the production level that gives the maximum profit in the model. That interpretation is essential. ✅

Differentiation and optimisation

Optimisation means finding the largest or smallest value of a quantity. This is one of the most important uses of differentiation in modelling. Common optimisation problems include:

maximising profit
minimising cost
finding the shortest path
maximising volume or area
designing containers or structures efficiently

The general process is:

Define variables.
Write an expression for the quantity to optimise.
Use constraints to reduce the expression to one variable.
Differentiate.
Set the derivative equal to zero and solve.
Test the result and interpret it.

For instance, imagine a rectangular pen is made with $40$ metres of fencing and one side is against a wall. If the width is $x$ and the length is $y$, then the fencing constraint is

$$2x+y=40$$

$$y=40-2x$$

The area is

$$A(x)=x(40-2x)=40x-2x^2$$

Differentiate:

$$A'(x)=40-4x$$

Set $A'(x)=0$:

$$40-4x=0$$

So $x=10$. Then

$$y=40-2(10)=20$$

The maximum area occurs when the pen is $10$ metres by $20$ metres. This result is not just algebra; it tells us how to use limited resources efficiently.

In HL-level work, you may also need to justify why the solution is a maximum. You can use the second derivative or explain the shape of the function. For this example,

$$A''(x)=-4<0$$

which confirms a maximum.

Modelling growth and change over time

Differentiation is especially useful when modelling situations that evolve continuously over time. Population growth, spread of disease, cooling, and chemical reactions all involve changing rates. A derivative can describe the rate at which a quantity is increasing or decreasing.

Suppose the population of a town is modelled by $P(t)=5000e^{0.03t}$, where $t$ is years. Then

$$P'(t)=150e^{0.03t}$$

This means the population is increasing faster as time passes, because the growth rate itself depends on the current population size. Such models are common in science and economics.

Sometimes the rate of change depends on the amount present. For example, in simple cooling models, the rate of change of temperature is proportional to the difference between the object’s temperature and the surrounding temperature. In a derivative-based model, this idea is written as a differential equation. While full differential equations are a larger topic, this lesson shows how derivatives naturally lead to them.

For instance, if a quantity $y$ changes at a rate proportional to itself, then

$$\frac{dy}{dt}=ky$$

where $k$ is a constant. This is a basic model for exponential growth or decay. The derivative tells us the rule of change, which is the heart of the model.

Limitations of models and checking reasonableness

No model is perfect. students, it is important to check whether a model is reasonable.

A derivative-based model may fail if:

the situation changes suddenly
assumptions are unrealistic
the data are noisy or incomplete
the model is used outside its valid domain

For example, a quadratic model may fit a short-term trend but fail over a long time because real-world growth may slow down or reverse. Similarly, a linear approximation can be useful near one point but not far away from it.

This is why calculus connects with limits and continuity. The derivative is defined using a limit:

$$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$

This definition shows that differentiation is built on the idea of making the change in input extremely small. It also explains why smoothness matters. If a function has a sharp corner or jump, the derivative may not exist there.

When modelling, always ask whether the function is differentiable in the region you are studying. If it is not, then derivative-based reasoning may break down at that point.

Conclusion 🎯

Modelling with differentiation is about using derivatives to describe and solve real-world problems. The derivative gives the rate of change of a function, helping us interpret motion, growth, profit, and many other situations. In IB Mathematics: Analysis and Approaches HL, you should be able to build a function from a context, differentiate it, analyse what the derivative means, and explain your answer clearly in context. This topic connects directly to other parts of calculus, especially limits, continuity, optimisation, kinematics, and differential equations. When used carefully, differentiation turns a real situation into a powerful mathematical model.

Study Notes

A model is a simplified mathematical description of a real situation.
A derivative such as $f'(x)$ gives the instantaneous rate of change of $f(x)$.
In context, derivatives can represent velocity, marginal cost, marginal revenue, growth rate, or acceleration.
Optimisation uses derivatives to find maximum or minimum values.
A common method is: define variables, write the function, differentiate, solve $f'(x)=0$, and interpret the result.
The second derivative can help confirm maxima or minima.
Many models rely on assumptions, so checking realism is important.
The derivative is defined by the limit $$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$
Differentiation connects strongly to kinematics, business, engineering, and population modelling.
Always state the meaning of your mathematical answer in the real-world context.